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              </function>
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          </quantifiedformula>
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        </sequent>
        <rule type="perml" symbol="\pi:l" param="(1 2)">
         <sequent>
          <formulalist>
           <constantatomformula symbol="=">
            <function symbol="f">
             <function symbol="+">
              <variable symbol="n_0"/>
              <variable symbol="k_0"/>
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            <quantifiedformula type="exists">
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              <function symbol="f">
               <function symbol="+">
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                <variable symbol="k"/>
               </function>
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          <formulalist>
           <quantifiedformula type="exists">
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            <quantifiedformula type="exists">
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              <constantatomformula symbol="=">
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                <variable symbol="q"/>
               </function>
              </constantatomformula>
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           </quantifiedformula>
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         </sequent>
         <rule type="foralll" symbol="\all:l">
          <sequent>
           <formulalist>
            <quantifiedformula type="all">
             <variable symbol="n"/>
             <quantifiedformula type="exists">
              <variable symbol="k"/>
              <constantatomformula symbol="=">
               <function symbol="f">
                <function symbol="+">
                 <variable symbol="n"/>
                 <variable symbol="k"/>
                </function>
               </function>
               <constant symbol="1"/>
              </constantatomformula>
             </quantifiedformula>
            </quantifiedformula>
            <constantatomformula symbol="=">
             <function symbol="f">
              <function symbol="+">
               <variable symbol="n_0"/>
               <variable symbol="k_0"/>
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             <constant symbol="1"/>
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            <quantifiedformula type="exists">
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             <quantifiedformula type="exists">
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               <constantatomformula symbol="=">
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          <rule type="existsl" symbol="\ex:l">
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            <formulalist>
             <quantifiedformula type="exists">
              <variable symbol="k"/>
              <constantatomformula symbol="=">
               <function symbol="f">
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                 <function symbol="+">
                  <function symbol="+">
                   <variable symbol="n_0"/>
                   <variable symbol="k_0"/>
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             <constantatomformula symbol="=">
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                <variable symbol="k_0"/>
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            <formulalist>
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              <quantifiedformula type="exists">
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                <constantatomformula symbol="=">
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             </quantifiedformula>
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           <rule type="perml" symbol="\pi:l" param="(1 2)">
            <sequent>
             <formulalist>
              <constantatomformula symbol="=">
               <function symbol="f">
                <function symbol="+">
                 <function symbol="+">
                  <function symbol="+">
                   <variable symbol="n_0"/>
                   <variable symbol="k_0"/>
                  </function>
                  <constant symbol="1"/>
                 </function>
                 <variable symbol="k_1"/>
                </function>
               </function>
               <constant symbol="1"/>
              </constantatomformula>
              <constantatomformula symbol="=">
               <function symbol="f">
                <function symbol="+">
                 <variable symbol="n_0"/>
                 <variable symbol="k_0"/>
                </function>
               </function>
               <constant symbol="1"/>
              </constantatomformula>
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             <formulalist>
              <quantifiedformula type="exists">
               <variable symbol="p"/>
               <quantifiedformula type="exists">
                <variable symbol="q"/>
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                 <conjunctiveformula type="neg">
                  <constantatomformula symbol="=">
                   <variable symbol="p"/>
                   <variable symbol="q"/>
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                 </conjunctiveformula>
                 <constantatomformula symbol="=">
                  <function symbol="f">
                   <variable symbol="p"/>
                  </function>
                  <function symbol="f">
                   <variable symbol="q"/>
                  </function>
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                </conjunctiveformula>
               </quantifiedformula>
              </quantifiedformula>
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            </sequent>
            <rule type="existsr" symbol="\ex:r">
             <sequent>
              <formulalist>
               <constantatomformula symbol="=">
                <function symbol="f">
                 <function symbol="+">
                  <variable symbol="n_0"/>
                  <variable symbol="k_0"/>
                 </function>
                </function>
                <constant symbol="1"/>
               </constantatomformula>
               <constantatomformula symbol="=">
                <function symbol="f">
                 <function symbol="+">
                  <function symbol="+">
                   <function symbol="+">
                    <variable symbol="n_0"/>
                    <variable symbol="k_0"/>
                   </function>
                   <constant symbol="1"/>
                  </function>
                  <variable symbol="k_1"/>
                 </function>
                </function>
                <constant symbol="1"/>
               </constantatomformula>
              </formulalist>
              <formulalist>
               <quantifiedformula type="exists">
                <variable symbol="p"/>
                <quantifiedformula type="exists">
                 <variable symbol="q"/>
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                   <constantatomformula symbol="=">
                    <variable symbol="p"/>
                    <variable symbol="q"/>
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                  </conjunctiveformula>
                  <constantatomformula symbol="=">
                   <function symbol="f">
                    <variable symbol="p"/>
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                   <function symbol="f">
                    <variable symbol="q"/>
                   </function>
                  </constantatomformula>
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               </quantifiedformula>
              </formulalist>
             </sequent>
             <rule type="existsr" symbol="\ex:r">
              <sequent>
               <formulalist>
                <constantatomformula symbol="=">
                 <function symbol="f">
                  <function symbol="+">
                   <variable symbol="n_0"/>
                   <variable symbol="k_0"/>
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                 <constant symbol="1"/>
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                <constantatomformula symbol="=">
                 <function symbol="f">
                  <function symbol="+">
                   <function symbol="+">
                    <function symbol="+">
                     <variable symbol="n_0"/>
                     <variable symbol="k_0"/>
                    </function>
                    <constant symbol="1"/>
                   </function>
                   <variable symbol="k_1"/>
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                 <constant symbol="1"/>
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               </formulalist>
               <formulalist>
                <quantifiedformula type="exists">
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                   <constantatomformula symbol="=">
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                     <variable symbol="n_0"/>
                     <variable symbol="k_0"/>
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                  <constantatomformula symbol="=">
                   <function symbol="f">
                    <function symbol="+">
                     <variable symbol="n_0"/>
                     <variable symbol="k_0"/>
                    </function>
                   </function>
                   <function symbol="f">
                    <variable symbol="q"/>
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                  </constantatomformula>
                 </conjunctiveformula>
                </quantifiedformula>
               </formulalist>
              </sequent>
              <rule type="andr" symbol="\and:r">
               <sequent>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="f">
                   <function symbol="+">
                    <variable symbol="n_0"/>
                    <variable symbol="k_0"/>
                   </function>
                  </function>
                  <constant symbol="1"/>
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                 <constantatomformula symbol="=">
                  <function symbol="f">
                   <function symbol="+">
                    <function symbol="+">
                     <function symbol="+">
                      <variable symbol="n_0"/>
                      <variable symbol="k_0"/>
                     </function>
                     <constant symbol="1"/>
                    </function>
                    <variable symbol="k_1"/>
                   </function>
                  </function>
                  <constant symbol="1"/>
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                </formulalist>
                <formulalist>
                 <conjunctiveformula type="and">
                  <conjunctiveformula type="neg">
                   <constantatomformula symbol="=">
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                    <function symbol="+">
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
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                      <constant symbol="1"/>
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                     <variable symbol="k_1"/>
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                  <constantatomformula symbol="=">
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                    <function symbol="+">
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                     <variable symbol="k_0"/>
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                   </function>
                   <function symbol="f">
                    <function symbol="+">
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <constant symbol="1"/>
                     </function>
                     <variable symbol="k_1"/>
                    </function>
                   </function>
                  </constantatomformula>
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                </formulalist>
               </sequent>
               <rule type="negr" symbol="\neg:r">
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                 <formulalist/>
                 <formulalist>
                  <conjunctiveformula type="neg">
                   <constantatomformula symbol="=">
                    <function symbol="+">
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                     <variable symbol="k_0"/>
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                    <function symbol="+">
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                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <constant symbol="1"/>
                     </function>
                     <variable symbol="k_1"/>
                    </function>
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                </sequent>
                <rule type="eql1" symbol="=:l1" param="(2)">
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                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="+">
                     <variable symbol="n_0"/>
                     <variable symbol="k_0"/>
                    </function>
                    <function symbol="+">
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <constant symbol="1"/>
                     </function>
                     <variable symbol="k_1"/>
                    </function>
                   </constantatomformula>
                  </formulalist>
                  <formulalist/>
                 </sequent>
                 <rule type="axiom" symbol="axiom">
                  <sequent>
                   <formulalist/>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <function symbol="+">
                       <constant symbol="1"/>
                       <variable symbol="k_1"/>
                      </function>
                     </function>
                     <function symbol="+">
                      <function symbol="+">
                       <function symbol="+">
                        <variable symbol="n_0"/>
                        <variable symbol="k_0"/>
                       </function>
                       <constant symbol="1"/>
                      </function>
                      <variable symbol="k_1"/>
                     </function>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                 </rule>
                 <rule type="axiom" symbol="axiom">
                  <sequent>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <function symbol="+">
                      <variable symbol="n_0"/>
                      <variable symbol="k_0"/>
                     </function>
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <function symbol="+">
                       <constant symbol="1"/>
                       <variable symbol="k_1"/>
                      </function>
                     </function>
                    </constantatomformula>
                   </formulalist>
                   <formulalist/>
                  </sequent>
                 </rule>
                </rule>
               </rule>
               <rule type="eqr2" symbol="=:r2" param="(1)">
                <sequent>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="f">
                    <function symbol="+">
                     <variable symbol="n_0"/>
                     <variable symbol="k_0"/>
                    </function>
                   </function>
                   <constant symbol="1"/>
                  </constantatomformula>
                  <constantatomformula symbol="=">
                   <function symbol="f">
                    <function symbol="+">
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <constant symbol="1"/>
                     </function>
                     <variable symbol="k_1"/>
                    </function>
                   </function>
                   <constant symbol="1"/>
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                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="f">
                    <function symbol="+">
                     <variable symbol="n_0"/>
                     <variable symbol="k_0"/>
                    </function>
                   </function>
                   <function symbol="f">
                    <function symbol="+">
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <constant symbol="1"/>
                     </function>
                     <variable symbol="k_1"/>
                    </function>
                   </function>
                  </constantatomformula>
                 </formulalist>
                </sequent>
                <rule type="axiom" symbol="axiom">
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                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="f">
                     <function symbol="+">
                      <variable symbol="n_0"/>
                      <variable symbol="k_0"/>
                     </function>
                    </function>
                    <constant symbol="1"/>
                   </constantatomformula>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="f">
                     <function symbol="+">
                      <variable symbol="n_0"/>
                      <variable symbol="k_0"/>
                     </function>
                    </function>
                    <constant symbol="1"/>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                </rule>
                <rule type="eqr2" symbol="=:r2" param="(2)">
                 <sequent>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="f">
                     <function symbol="+">
                      <function symbol="+">
                       <function symbol="+">
                        <variable symbol="n_0"/>
                        <variable symbol="k_0"/>
                       </function>
                       <constant symbol="1"/>
                      </function>
                      <variable symbol="k_1"/>
                     </function>
                    </function>
                    <constant symbol="1"/>
                   </constantatomformula>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <constant symbol="1"/>
                    <function symbol="f">
                     <function symbol="+">
                      <function symbol="+">
                       <function symbol="+">
                        <variable symbol="n_0"/>
                        <variable symbol="k_0"/>
                       </function>
                       <constant symbol="1"/>
                      </function>
                      <variable symbol="k_1"/>
                     </function>
                    </function>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                 <rule type="axiom" symbol="axiom">
                  <sequent>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <function symbol="f">
                      <function symbol="+">
                       <function symbol="+">
                        <function symbol="+">
                         <variable symbol="n_0"/>
                         <variable symbol="k_0"/>
                        </function>
                        <constant symbol="1"/>
                       </function>
                       <variable symbol="k_1"/>
                      </function>
                     </function>
                     <constant symbol="1"/>
                    </constantatomformula>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <function symbol="f">
                      <function symbol="+">
                       <function symbol="+">
                        <function symbol="+">
                         <variable symbol="n_0"/>
                         <variable symbol="k_0"/>
                        </function>
                        <constant symbol="1"/>
                       </function>
                       <variable symbol="k_1"/>
                      </function>
                     </function>
                     <constant symbol="1"/>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                 </rule>
                 <rule type="axiom" symbol="axiom">
                  <sequent>
                   <formulalist/>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <constant symbol="1"/>
                     <constant symbol="1"/>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="the-proof_{sk}" calculus="LK">
  <rule type="contrr" symbol="c:r" param="2">
   <sequent>
    <formulalist>
     <constantatomformula symbol="A" definition="A"/>
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    <formulalist>
     <quantifiedformula type="exists">
      <variable symbol="p"/>
      <quantifiedformula type="exists">
       <variable symbol="q"/>
       <conjunctiveformula type="and">
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         <constantatomformula symbol="=">
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          <variable symbol="q"/>
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        </conjunctiveformula>
        <constantatomformula symbol="=">
         <function symbol="f">
          <variable symbol="p"/>
         </function>
         <function symbol="f">
          <variable symbol="q"/>
         </function>
        </constantatomformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
    </formulalist>
   </sequent>
   <rule type="cut" symbol="cut">
    <sequent>
     <formulalist>
      <constantatomformula symbol="A" definition="A"/>
     </formulalist>
     <formulalist>
      <quantifiedformula type="exists">
       <variable symbol="p"/>
       <quantifiedformula type="exists">
        <variable symbol="q"/>
        <conjunctiveformula type="and">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="=">
           <variable symbol="p"/>
           <variable symbol="q"/>
          </constantatomformula>
         </conjunctiveformula>
         <constantatomformula symbol="=">
          <function symbol="f">
           <variable symbol="p"/>
          </function>
          <function symbol="f">
           <variable symbol="q"/>
          </function>
         </constantatomformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
      <quantifiedformula type="exists">
       <variable symbol="p"/>
       <quantifiedformula type="exists">
        <variable symbol="q"/>
        <conjunctiveformula type="and">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="=">
           <variable symbol="p"/>
           <variable symbol="q"/>
          </constantatomformula>
         </conjunctiveformula>
         <constantatomformula symbol="=">
          <function symbol="f">
           <variable symbol="p"/>
          </function>
          <function symbol="f">
           <variable symbol="q"/>
          </function>
         </constantatomformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
     </formulalist>
    </sequent>
    <rule type="permr" symbol="\pi:r" param="(1 2)">
     <sequent>
      <formulalist>
       <constantatomformula symbol="A" definition="A"/>
      </formulalist>
      <formulalist>
       <quantifiedformula type="exists">
        <variable symbol="p"/>
        <quantifiedformula type="exists">
         <variable symbol="q"/>
         <conjunctiveformula type="and">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="=">
            <variable symbol="p"/>
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           </constantatomformula>
          </conjunctiveformula>
          <constantatomformula symbol="=">
           <function symbol="f">
            <variable symbol="p"/>
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           <function symbol="f">
            <variable symbol="q"/>
           </function>
          </constantatomformula>
         </conjunctiveformula>
        </quantifiedformula>
       </quantifiedformula>
       <constantatomformula symbol="I" definition="I">
        <constant symbol="1"/>
       </constantatomformula>
      </formulalist>
     </sequent>
     <rule type="cut" symbol="cut">
      <sequent>
       <formulalist>
        <constantatomformula symbol="A" definition="A"/>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="I" definition="I">
         <constant symbol="1"/>
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        <quantifiedformula type="exists">
         <variable symbol="p"/>
         <quantifiedformula type="exists">
          <variable symbol="q"/>
          <conjunctiveformula type="and">
           <conjunctiveformula type="neg">
            <constantatomformula symbol="=">
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             <variable symbol="q"/>
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           </conjunctiveformula>
           <constantatomformula symbol="=">
            <function symbol="f">
             <variable symbol="p"/>
            </function>
            <function symbol="f">
             <variable symbol="q"/>
            </function>
           </constantatomformula>
          </conjunctiveformula>
         </quantifiedformula>
        </quantifiedformula>
       </formulalist>
      </sequent>
      <rule type="defr" symbol="d:r" param="();[(1)(2 2 2)];()">
       <sequent>
        <formulalist>
         <constantatomformula symbol="A" definition="A"/>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="I" definition="I">
          <constant symbol="1"/>
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         <constantatomformula symbol="I" definition="I">
          <constant symbol="0"/>
         </constantatomformula>
        </formulalist>
       </sequent>
       <rule type="permr" symbol="\pi:r" param="(1 2)">
        <sequent>
         <formulalist>
          <constantatomformula symbol="A" definition="A"/>
         </formulalist>
         <formulalist>
          <constantatomformula symbol="I" definition="I">
           <constant symbol="1"/>
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          <quantifiedformula type="all">
           <variable symbol="n"/>
           <quantifiedformula type="exists">
            <variable symbol="k"/>
            <constantatomformula symbol="=">
             <function symbol="f">
              <function symbol="+">
               <variable symbol="n"/>
               <variable symbol="k"/>
              </function>
             </function>
             <constant symbol="0"/>
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           </quantifiedformula>
          </quantifiedformula>
         </formulalist>
        </sequent>
        <rule type="defr" symbol="d:r" param="();[(1)(2 2 2)];()">
         <sequent>
          <formulalist>
           <constantatomformula symbol="A" definition="A"/>
          </formulalist>
          <formulalist>
           <quantifiedformula type="all">
            <variable symbol="n"/>
            <quantifiedformula type="exists">
             <variable symbol="k"/>
             <constantatomformula symbol="=">
              <function symbol="f">
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                  <constantatomformula symbol="=">
                   <function symbol="f">
                    <variable symbol="p"/>
                   </function>
                   <function symbol="f">
                    <variable symbol="q"/>
                   </function>
                  </constantatomformula>
                 </conjunctiveformula>
                </quantifiedformula>
               </quantifiedformula>
              </formulalist>
             </sequent>
             <rule type="existsr" symbol="\ex:r">
              <sequent>
               <formulalist>
                <constantatomformula symbol="=">
                 <function symbol="f">
                  <function symbol="+">
                   <variable symbol="n_0"/>
                   <variable symbol="k_0"/>
                  </function>
                 </function>
                 <constant symbol="1"/>
                </constantatomformula>
                <constantatomformula symbol="=">
                 <function symbol="f">
                  <function symbol="+">
                   <function symbol="+">
                    <function symbol="+">
                     <variable symbol="n_0"/>
                     <variable symbol="k_0"/>
                    </function>
                    <constant symbol="1"/>
                   </function>
                   <variable symbol="k_1"/>
                  </function>
                 </function>
                 <constant symbol="1"/>
                </constantatomformula>
               </formulalist>
               <formulalist>
                <quantifiedformula type="exists">
                 <variable symbol="q"/>
                 <conjunctiveformula type="and">
                  <conjunctiveformula type="neg">
                   <constantatomformula symbol="=">
                    <function symbol="+">
                     <variable symbol="n_0"/>
                     <variable symbol="k_0"/>
                    </function>
                    <variable symbol="q"/>
                   </constantatomformula>
                  </conjunctiveformula>
                  <constantatomformula symbol="=">
                   <function symbol="f">
                    <function symbol="+">
                     <variable symbol="n_0"/>
                     <variable symbol="k_0"/>
                    </function>
                   </function>
                   <function symbol="f">
                    <variable symbol="q"/>
                   </function>
                  </constantatomformula>
                 </conjunctiveformula>
                </quantifiedformula>
               </formulalist>
              </sequent>
              <rule type="andr" symbol="\and:r">
               <sequent>
                <formulalist>
                 <constantatomformula symbol="=">
                  <function symbol="f">
                   <function symbol="+">
                    <variable symbol="n_0"/>
                    <variable symbol="k_0"/>
                   </function>
                  </function>
                  <constant symbol="1"/>
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                 <constantatomformula symbol="=">
                  <function symbol="f">
                   <function symbol="+">
                    <function symbol="+">
                     <function symbol="+">
                      <variable symbol="n_0"/>
                      <variable symbol="k_0"/>
                     </function>
                     <constant symbol="1"/>
                    </function>
                    <variable symbol="k_1"/>
                   </function>
                  </function>
                  <constant symbol="1"/>
                 </constantatomformula>
                </formulalist>
                <formulalist>
                 <conjunctiveformula type="and">
                  <conjunctiveformula type="neg">
                   <constantatomformula symbol="=">
                    <function symbol="+">
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                     <variable symbol="k_0"/>
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                    <function symbol="+">
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <constant symbol="1"/>
                     </function>
                     <variable symbol="k_1"/>
                    </function>
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                  </conjunctiveformula>
                  <constantatomformula symbol="=">
                   <function symbol="f">
                    <function symbol="+">
                     <variable symbol="n_0"/>
                     <variable symbol="k_0"/>
                    </function>
                   </function>
                   <function symbol="f">
                    <function symbol="+">
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <constant symbol="1"/>
                     </function>
                     <variable symbol="k_1"/>
                    </function>
                   </function>
                  </constantatomformula>
                 </conjunctiveformula>
                </formulalist>
               </sequent>
               <rule type="negr" symbol="\neg:r">
                <sequent>
                 <formulalist/>
                 <formulalist>
                  <conjunctiveformula type="neg">
                   <constantatomformula symbol="=">
                    <function symbol="+">
                     <variable symbol="n_0"/>
                     <variable symbol="k_0"/>
                    </function>
                    <function symbol="+">
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <constant symbol="1"/>
                     </function>
                     <variable symbol="k_1"/>
                    </function>
                   </constantatomformula>
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                </sequent>
                <rule type="eql1" symbol="=:l1" param="(2)">
                 <sequent>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="+">
                     <variable symbol="n_0"/>
                     <variable symbol="k_0"/>
                    </function>
                    <function symbol="+">
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <constant symbol="1"/>
                     </function>
                     <variable symbol="k_1"/>
                    </function>
                   </constantatomformula>
                  </formulalist>
                  <formulalist/>
                 </sequent>
                 <rule type="axiom" symbol="axiom">
                  <sequent>
                   <formulalist/>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <function symbol="+">
                       <constant symbol="1"/>
                       <variable symbol="k_1"/>
                      </function>
                     </function>
                     <function symbol="+">
                      <function symbol="+">
                       <function symbol="+">
                        <variable symbol="n_0"/>
                        <variable symbol="k_0"/>
                       </function>
                       <constant symbol="1"/>
                      </function>
                      <variable symbol="k_1"/>
                     </function>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                 </rule>
                 <rule type="axiom" symbol="axiom">
                  <sequent>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <function symbol="+">
                      <variable symbol="n_0"/>
                      <variable symbol="k_0"/>
                     </function>
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <function symbol="+">
                       <constant symbol="1"/>
                       <variable symbol="k_1"/>
                      </function>
                     </function>
                    </constantatomformula>
                   </formulalist>
                   <formulalist/>
                  </sequent>
                 </rule>
                </rule>
               </rule>
               <rule type="eqr2" symbol="=:r2" param="(1)">
                <sequent>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="f">
                    <function symbol="+">
                     <variable symbol="n_0"/>
                     <variable symbol="k_0"/>
                    </function>
                   </function>
                   <constant symbol="1"/>
                  </constantatomformula>
                  <constantatomformula symbol="=">
                   <function symbol="f">
                    <function symbol="+">
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <constant symbol="1"/>
                     </function>
                     <variable symbol="k_1"/>
                    </function>
                   </function>
                   <constant symbol="1"/>
                  </constantatomformula>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="=">
                   <function symbol="f">
                    <function symbol="+">
                     <variable symbol="n_0"/>
                     <variable symbol="k_0"/>
                    </function>
                   </function>
                   <function symbol="f">
                    <function symbol="+">
                     <function symbol="+">
                      <function symbol="+">
                       <variable symbol="n_0"/>
                       <variable symbol="k_0"/>
                      </function>
                      <constant symbol="1"/>
                     </function>
                     <variable symbol="k_1"/>
                    </function>
                   </function>
                  </constantatomformula>
                 </formulalist>
                </sequent>
                <rule type="axiom" symbol="axiom">
                 <sequent>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="f">
                     <function symbol="+">
                      <variable symbol="n_0"/>
                      <variable symbol="k_0"/>
                     </function>
                    </function>
                    <constant symbol="1"/>
                   </constantatomformula>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="f">
                     <function symbol="+">
                      <variable symbol="n_0"/>
                      <variable symbol="k_0"/>
                     </function>
                    </function>
                    <constant symbol="1"/>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                </rule>
                <rule type="eqr2" symbol="=:r2" param="(2)">
                 <sequent>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <function symbol="f">
                     <function symbol="+">
                      <function symbol="+">
                       <function symbol="+">
                        <variable symbol="n_0"/>
                        <variable symbol="k_0"/>
                       </function>
                       <constant symbol="1"/>
                      </function>
                      <variable symbol="k_1"/>
                     </function>
                    </function>
                    <constant symbol="1"/>
                   </constantatomformula>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="=">
                    <constant symbol="1"/>
                    <function symbol="f">
                     <function symbol="+">
                      <function symbol="+">
                       <function symbol="+">
                        <variable symbol="n_0"/>
                        <variable symbol="k_0"/>
                       </function>
                       <constant symbol="1"/>
                      </function>
                      <variable symbol="k_1"/>
                     </function>
                    </function>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                 <rule type="axiom" symbol="axiom">
                  <sequent>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <function symbol="f">
                      <function symbol="+">
                       <function symbol="+">
                        <function symbol="+">
                         <variable symbol="n_0"/>
                         <variable symbol="k_0"/>
                        </function>
                        <constant symbol="1"/>
                       </function>
                       <variable symbol="k_1"/>
                      </function>
                     </function>
                     <constant symbol="1"/>
                    </constantatomformula>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <function symbol="f">
                      <function symbol="+">
                       <function symbol="+">
                        <function symbol="+">
                         <variable symbol="n_0"/>
                         <variable symbol="k_0"/>
                        </function>
                        <constant symbol="1"/>
                       </function>
                       <variable symbol="k_1"/>
                      </function>
                     </function>
                     <constant symbol="1"/>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                 </rule>
                 <rule type="axiom" symbol="axiom">
                  <sequent>
                   <formulalist/>
                   <formulalist>
                    <constantatomformula symbol="=">
                     <constant symbol="1"/>
                     <constant symbol="1"/>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{1}" calculus="LK">
  <rule type="contrr" symbol="c:r" param="2">
   <sequent>
    <formulalist>
     <constantatomformula symbol="A" definition="A"/>
     <constantatomformula symbol="=">
      <function symbol="f">
       <function symbol="+">
        <variable symbol="n_0"/>
        <variable symbol="k_0"/>
       </function>
      </function>
      <constant symbol="1"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="f">
       <function symbol="+">
        <function symbol="+">
         <function symbol="+">
          <variable symbol="n_0"/>
          <variable symbol="k_0"/>
         </function>
         <constant symbol="1"/>
        </function>
        <variable symbol="k_1"/>
       </function>
      </function>
      <constant symbol="1"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <quantifiedformula type="exists">
      <variable symbol="p"/>
      <quantifiedformula type="exists">
       <variable symbol="q"/>
       <conjunctiveformula type="and">
        <conjunctiveformula type="neg">
         <constantatomformula symbol="=">
          <variable symbol="p"/>
          <variable symbol="q"/>
         </constantatomformula>
        </conjunctiveformula>
        <constantatomformula symbol="=">
         <function symbol="f">
          <variable symbol="p"/>
         </function>
         <function symbol="f">
          <variable symbol="q"/>
         </function>
        </constantatomformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
    </formulalist>
   </sequent>
   <rule type="permr" symbol="\pi:r" param="(1 2)">
    <sequent>
     <formulalist>
      <constantatomformula symbol="A" definition="A"/>
      <constantatomformula symbol="=">
       <function symbol="f">
        <function symbol="+">
         <variable symbol="n_0"/>
         <variable symbol="k_0"/>
        </function>
       </function>
       <constant symbol="1"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="f">
        <function symbol="+">
         <function symbol="+">
          <function symbol="+">
           <variable symbol="n_0"/>
           <variable symbol="k_0"/>
          </function>
          <constant symbol="1"/>
         </function>
         <variable symbol="k_1"/>
        </function>
       </function>
       <constant symbol="1"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <quantifiedformula type="exists">
       <variable symbol="p"/>
       <quantifiedformula type="exists">
        <variable symbol="q"/>
        <conjunctiveformula type="and">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="=">
           <variable symbol="p"/>
           <variable symbol="q"/>
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         </conjunctiveformula>
         <constantatomformula symbol="=">
          <function symbol="f">
           <variable symbol="p"/>
          </function>
          <function symbol="f">
           <variable symbol="q"/>
          </function>
         </constantatomformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
      <quantifiedformula type="exists">
       <variable symbol="p"/>
       <quantifiedformula type="exists">
        <variable symbol="q"/>
        <conjunctiveformula type="and">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="=">
           <variable symbol="p"/>
           <variable symbol="q"/>
          </constantatomformula>
         </conjunctiveformula>
         <constantatomformula symbol="=">
          <function symbol="f">
           <variable symbol="p"/>
          </function>
          <function symbol="f">
           <variable symbol="q"/>
          </function>
         </constantatomformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
     </formulalist>
    </sequent>
    <rule type="weakr" symbol="w:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="A" definition="A"/>
       <constantatomformula symbol="=">
        <function symbol="f">
         <function symbol="+">
          <variable symbol="n_0"/>
          <variable symbol="k_0"/>
         </function>
        </function>
        <constant symbol="1"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="f">
         <function symbol="+">
          <function symbol="+">
           <function symbol="+">
            <variable symbol="n_0"/>
            <variable symbol="k_0"/>
           </function>
           <constant symbol="1"/>
          </function>
          <variable symbol="k_1"/>
         </function>
        </function>
        <constant symbol="1"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <quantifiedformula type="exists">
        <variable symbol="p"/>
        <quantifiedformula type="exists">
         <variable symbol="q"/>
         <conjunctiveformula type="and">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="=">
            <variable symbol="p"/>
            <variable symbol="q"/>
           </constantatomformula>
          </conjunctiveformula>
          <constantatomformula symbol="=">
           <function symbol="f">
            <variable symbol="p"/>
           </function>
           <function symbol="f">
            <variable symbol="q"/>
           </function>
          </constantatomformula>
         </conjunctiveformula>
        </quantifiedformula>
       </quantifiedformula>
       <quantifiedformula type="exists">
        <variable symbol="p"/>
        <quantifiedformula type="exists">
         <variable symbol="q"/>
         <conjunctiveformula type="and">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="=">
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            <variable symbol="q"/>
           </constantatomformula>
          </conjunctiveformula>
          <constantatomformula symbol="=">
           <function symbol="f">
            <variable symbol="p"/>
           </function>
           <function symbol="f">
            <variable symbol="q"/>
           </function>
          </constantatomformula>
         </conjunctiveformula>
        </quantifiedformula>
       </quantifiedformula>
      </formulalist>
     </sequent>
     <rule type="weakl" symbol="w:l">
      <sequent>
       <formulalist>
        <constantatomformula symbol="A" definition="A"/>
        <constantatomformula symbol="=">
         <function symbol="f">
          <function symbol="+">
           <variable symbol="n_0"/>
           <variable symbol="k_0"/>
          </function>
         </function>
         <constant symbol="1"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="f">
          <function symbol="+">
           <function symbol="+">
            <function symbol="+">
             <variable symbol="n_0"/>
             <variable symbol="k_0"/>
            </function>
            <constant symbol="1"/>
           </function>
           <variable symbol="k_1"/>
          </function>
         </function>
         <constant symbol="1"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <quantifiedformula type="exists">
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         <quantifiedformula type="exists">
          <variable symbol="q"/>
          <conjunctiveformula type="and">
           <conjunctiveformula type="neg">
            <constantatomformula symbol="=">
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             <variable symbol="q"/>
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           </conjunctiveformula>
           <constantatomformula symbol="=">
            <function symbol="f">
             <variable symbol="p"/>
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            <function symbol="f">
             <variable symbol="q"/>
            </function>
           </constantatomformula>
          </conjunctiveformula>
         </quantifiedformula>
        </quantifiedformula>
       </formulalist>
      </sequent>
      <rule type="existsr" symbol="\ex:r">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="f">
           <function symbol="+">
            <variable symbol="n_0"/>
            <variable symbol="k_0"/>
           </function>
          </function>
          <constant symbol="1"/>
         </constantatomformula>
         <constantatomformula symbol="=">
          <function symbol="f">
           <function symbol="+">
            <function symbol="+">
             <function symbol="+">
              <variable symbol="n_0"/>
              <variable symbol="k_0"/>
             </function>
             <constant symbol="1"/>
            </function>
            <variable symbol="k_1"/>
           </function>
          </function>
          <constant symbol="1"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <quantifiedformula type="exists">
          <variable symbol="p"/>
          <quantifiedformula type="exists">
           <variable symbol="q"/>
           <conjunctiveformula type="and">
            <conjunctiveformula type="neg">
             <constantatomformula symbol="=">
              <variable symbol="p"/>
              <variable symbol="q"/>
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            </conjunctiveformula>
            <constantatomformula symbol="=">
             <function symbol="f">
              <variable symbol="p"/>
             </function>
             <function symbol="f">
              <variable symbol="q"/>
             </function>
            </constantatomformula>
           </conjunctiveformula>
          </quantifiedformula>
         </quantifiedformula>
        </formulalist>
       </sequent>
       <rule type="existsr" symbol="\ex:r">
        <sequent>
         <formulalist>
          <constantatomformula symbol="=">
           <function symbol="f">
            <function symbol="+">
             <variable symbol="n_0"/>
             <variable symbol="k_0"/>
            </function>
           </function>
           <constant symbol="1"/>
          </constantatomformula>
          <constantatomformula symbol="=">
           <function symbol="f">
            <function symbol="+">
             <function symbol="+">
              <function symbol="+">
               <variable symbol="n_0"/>
               <variable symbol="k_0"/>
              </function>
              <constant symbol="1"/>
             </function>
             <variable symbol="k_1"/>
            </function>
           </function>
           <constant symbol="1"/>
          </constantatomformula>
         </formulalist>
         <formulalist>
          <quantifiedformula type="exists">
           <variable symbol="q"/>
           <conjunctiveformula type="and">
            <conjunctiveformula type="neg">
             <constantatomformula symbol="=">
              <function symbol="+">
               <variable symbol="n_0"/>
               <variable symbol="k_0"/>
              </function>
              <variable symbol="q"/>
             </constantatomformula>
            </conjunctiveformula>
            <constantatomformula symbol="=">
             <function symbol="f">
              <function symbol="+">
               <variable symbol="n_0"/>
               <variable symbol="k_0"/>
              </function>
             </function>
             <function symbol="f">
              <variable symbol="q"/>
             </function>
            </constantatomformula>
           </conjunctiveformula>
          </quantifiedformula>
         </formulalist>
        </sequent>
        <rule type="andr" symbol="\and:r">
         <sequent>
          <formulalist>
           <constantatomformula symbol="=">
            <function symbol="f">
             <function symbol="+">
              <variable symbol="n_0"/>
              <variable symbol="k_0"/>
             </function>
            </function>
            <constant symbol="1"/>
           </constantatomformula>
           <constantatomformula symbol="=">
            <function symbol="f">
             <function symbol="+">
              <function symbol="+">
               <function symbol="+">
                <variable symbol="n_0"/>
                <variable symbol="k_0"/>
               </function>
               <constant symbol="1"/>
              </function>
              <variable symbol="k_1"/>
             </function>
            </function>
            <constant symbol="1"/>
           </constantatomformula>
          </formulalist>
          <formulalist>
           <conjunctiveformula type="and">
            <conjunctiveformula type="neg">
             <constantatomformula symbol="=">
              <function symbol="+">
               <variable symbol="n_0"/>
               <variable symbol="k_0"/>
              </function>
              <function symbol="+">
               <function symbol="+">
                <function symbol="+">
                 <variable symbol="n_0"/>
                 <variable symbol="k_0"/>
                </function>
                <constant symbol="1"/>
               </function>
               <variable symbol="k_1"/>
              </function>
             </constantatomformula>
            </conjunctiveformula>
            <constantatomformula symbol="=">
             <function symbol="f">
              <function symbol="+">
               <variable symbol="n_0"/>
               <variable symbol="k_0"/>
              </function>
             </function>
             <function symbol="f">
              <function symbol="+">
               <function symbol="+">
                <function symbol="+">
                 <variable symbol="n_0"/>
                 <variable symbol="k_0"/>
                </function>
                <constant symbol="1"/>
               </function>
               <variable symbol="k_1"/>
              </function>
             </function>
            </constantatomformula>
           </conjunctiveformula>
          </formulalist>
         </sequent>
         <rule type="negr" symbol="\neg:r">
          <sequent>
           <formulalist/>
           <formulalist>
            <conjunctiveformula type="neg">
             <constantatomformula symbol="=">
              <function symbol="+">
               <variable symbol="n_0"/>
               <variable symbol="k_0"/>
              </function>
              <function symbol="+">
               <function symbol="+">
                <function symbol="+">
                 <variable symbol="n_0"/>
                 <variable symbol="k_0"/>
                </function>
                <constant symbol="1"/>
               </function>
               <variable symbol="k_1"/>
              </function>
             </constantatomformula>
            </conjunctiveformula>
           </formulalist>
          </sequent>
          <rule type="eql1" symbol="=:l1" param="(2)">
           <sequent>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="+">
               <variable symbol="n_0"/>
               <variable symbol="k_0"/>
              </function>
              <function symbol="+">
               <function symbol="+">
                <function symbol="+">
                 <variable symbol="n_0"/>
                 <variable symbol="k_0"/>
                </function>
                <constant symbol="1"/>
               </function>
               <variable symbol="k_1"/>
              </function>
             </constantatomformula>
            </formulalist>
            <formulalist/>
           </sequent>
           <rule type="axiom" symbol="axiom">
            <sequent>
             <formulalist/>
             <formulalist>
              <constantatomformula symbol="=">
               <function symbol="+">
                <function symbol="+">
                 <variable symbol="n_0"/>
                 <variable symbol="k_0"/>
                </function>
                <function symbol="+">
                 <constant symbol="1"/>
                 <variable symbol="k_1"/>
                </function>
               </function>
               <function symbol="+">
                <function symbol="+">
                 <function symbol="+">
                  <variable symbol="n_0"/>
                  <variable symbol="k_0"/>
                 </function>
                 <constant symbol="1"/>
                </function>
                <variable symbol="k_1"/>
               </function>
              </constantatomformula>
             </formulalist>
            </sequent>
           </rule>
           <rule type="axiom" symbol="axiom">
            <sequent>
             <formulalist>
              <constantatomformula symbol="=">
               <function symbol="+">
                <variable symbol="n_0"/>
                <variable symbol="k_0"/>
               </function>
               <function symbol="+">
                <function symbol="+">
                 <variable symbol="n_0"/>
                 <variable symbol="k_0"/>
                </function>
                <function symbol="+">
                 <constant symbol="1"/>
                 <variable symbol="k_1"/>
                </function>
               </function>
              </constantatomformula>
             </formulalist>
             <formulalist/>
            </sequent>
           </rule>
          </rule>
         </rule>
         <rule type="eqr2" symbol="=:r2" param="(1)">
          <sequent>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="f">
              <function symbol="+">
               <variable symbol="n_0"/>
               <variable symbol="k_0"/>
              </function>
             </function>
             <constant symbol="1"/>
            </constantatomformula>
            <constantatomformula symbol="=">
             <function symbol="f">
              <function symbol="+">
               <function symbol="+">
                <function symbol="+">
                 <variable symbol="n_0"/>
                 <variable symbol="k_0"/>
                </function>
                <constant symbol="1"/>
               </function>
               <variable symbol="k_1"/>
              </function>
             </function>
             <constant symbol="1"/>
            </constantatomformula>
           </formulalist>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="f">
              <function symbol="+">
               <variable symbol="n_0"/>
               <variable symbol="k_0"/>
              </function>
             </function>
             <function symbol="f">
              <function symbol="+">
               <function symbol="+">
                <function symbol="+">
                 <variable symbol="n_0"/>
                 <variable symbol="k_0"/>
                </function>
                <constant symbol="1"/>
               </function>
               <variable symbol="k_1"/>
              </function>
             </function>
            </constantatomformula>
           </formulalist>
          </sequent>
          <rule type="axiom" symbol="axiom">
           <sequent>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="f">
               <function symbol="+">
                <variable symbol="n_0"/>
                <variable symbol="k_0"/>
               </function>
              </function>
              <constant symbol="1"/>
             </constantatomformula>
            </formulalist>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="f">
               <function symbol="+">
                <variable symbol="n_0"/>
                <variable symbol="k_0"/>
               </function>
              </function>
              <constant symbol="1"/>
             </constantatomformula>
            </formulalist>
           </sequent>
          </rule>
          <rule type="eqr2" symbol="=:r2" param="(2)">
           <sequent>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="f">
               <function symbol="+">
                <function symbol="+">
                 <function symbol="+">
                  <variable symbol="n_0"/>
                  <variable symbol="k_0"/>
                 </function>
                 <constant symbol="1"/>
                </function>
                <variable symbol="k_1"/>
               </function>
              </function>
              <constant symbol="1"/>
             </constantatomformula>
            </formulalist>
            <formulalist>
             <constantatomformula symbol="=">
              <constant symbol="1"/>
              <function symbol="f">
               <function symbol="+">
                <function symbol="+">
                 <function symbol="+">
                  <variable symbol="n_0"/>
                  <variable symbol="k_0"/>
                 </function>
                 <constant symbol="1"/>
                </function>
                <variable symbol="k_1"/>
               </function>
              </function>
             </constantatomformula>
            </formulalist>
           </sequent>
           <rule type="axiom" symbol="axiom">
            <sequent>
             <formulalist>
              <constantatomformula symbol="=">
               <function symbol="f">
                <function symbol="+">
                 <function symbol="+">
                  <function symbol="+">
                   <variable symbol="n_0"/>
                   <variable symbol="k_0"/>
                  </function>
                  <constant symbol="1"/>
                 </function>
                 <variable symbol="k_1"/>
                </function>
               </function>
               <constant symbol="1"/>
              </constantatomformula>
             </formulalist>
             <formulalist>
              <constantatomformula symbol="=">
               <function symbol="f">
                <function symbol="+">
                 <function symbol="+">
                  <function symbol="+">
                   <variable symbol="n_0"/>
                   <variable symbol="k_0"/>
                  </function>
                  <constant symbol="1"/>
                 </function>
                 <variable symbol="k_1"/>
                </function>
               </function>
               <constant symbol="1"/>
              </constantatomformula>
             </formulalist>
            </sequent>
           </rule>
           <rule type="axiom" symbol="axiom">
            <sequent>
             <formulalist/>
             <formulalist>
              <constantatomformula symbol="=">
               <constant symbol="1"/>
               <constant symbol="1"/>
              </constantatomformula>
             </formulalist>
            </sequent>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{2}" calculus="LK">
  <rule type="contrr" symbol="c:r" param="2">
   <sequent>
    <formulalist>
     <constantatomformula symbol="A" definition="A"/>
     <constantatomformula symbol="=">
      <function symbol="f">
       <function symbol="+">
        <variable symbol="n_0"/>
        <variable symbol="k_0"/>
       </function>
      </function>
      <constant symbol="0"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="f">
       <function symbol="+">
        <function symbol="+">
         <function symbol="+">
          <variable symbol="n_0"/>
          <variable symbol="k_0"/>
         </function>
         <constant symbol="1"/>
        </function>
        <variable symbol="k_1"/>
       </function>
      </function>
      <constant symbol="0"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <quantifiedformula type="exists">
      <variable symbol="p"/>
      <quantifiedformula type="exists">
       <variable symbol="q"/>
       <conjunctiveformula type="and">
        <conjunctiveformula type="neg">
         <constantatomformula symbol="=">
          <variable symbol="p"/>
          <variable symbol="q"/>
         </constantatomformula>
        </conjunctiveformula>
        <constantatomformula symbol="=">
         <function symbol="f">
          <variable symbol="p"/>
         </function>
         <function symbol="f">
          <variable symbol="q"/>
         </function>
        </constantatomformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
    </formulalist>
   </sequent>
   <rule type="weakr" symbol="w:r">
    <sequent>
     <formulalist>
      <constantatomformula symbol="A" definition="A"/>
      <constantatomformula symbol="=">
       <function symbol="f">
        <function symbol="+">
         <variable symbol="n_0"/>
         <variable symbol="k_0"/>
        </function>
       </function>
       <constant symbol="0"/>
      </constantatomformula>
      <constantatomformula symbol="=">
       <function symbol="f">
        <function symbol="+">
         <function symbol="+">
          <function symbol="+">
           <variable symbol="n_0"/>
           <variable symbol="k_0"/>
          </function>
          <constant symbol="1"/>
         </function>
         <variable symbol="k_1"/>
        </function>
       </function>
       <constant symbol="0"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <quantifiedformula type="exists">
       <variable symbol="p"/>
       <quantifiedformula type="exists">
        <variable symbol="q"/>
        <conjunctiveformula type="and">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="=">
           <variable symbol="p"/>
           <variable symbol="q"/>
          </constantatomformula>
         </conjunctiveformula>
         <constantatomformula symbol="=">
          <function symbol="f">
           <variable symbol="p"/>
          </function>
          <function symbol="f">
           <variable symbol="q"/>
          </function>
         </constantatomformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
      <quantifiedformula type="exists">
       <variable symbol="p"/>
       <quantifiedformula type="exists">
        <variable symbol="q"/>
        <conjunctiveformula type="and">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="=">
           <variable symbol="p"/>
           <variable symbol="q"/>
          </constantatomformula>
         </conjunctiveformula>
         <constantatomformula symbol="=">
          <function symbol="f">
           <variable symbol="p"/>
          </function>
          <function symbol="f">
           <variable symbol="q"/>
          </function>
         </constantatomformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
     </formulalist>
    </sequent>
    <rule type="weakl" symbol="w:l">
     <sequent>
      <formulalist>
       <constantatomformula symbol="A" definition="A"/>
       <constantatomformula symbol="=">
        <function symbol="f">
         <function symbol="+">
          <variable symbol="n_0"/>
          <variable symbol="k_0"/>
         </function>
        </function>
        <constant symbol="0"/>
       </constantatomformula>
       <constantatomformula symbol="=">
        <function symbol="f">
         <function symbol="+">
          <function symbol="+">
           <function symbol="+">
            <variable symbol="n_0"/>
            <variable symbol="k_0"/>
           </function>
           <constant symbol="1"/>
          </function>
          <variable symbol="k_1"/>
         </function>
        </function>
        <constant symbol="0"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <quantifiedformula type="exists">
        <variable symbol="p"/>
        <quantifiedformula type="exists">
         <variable symbol="q"/>
         <conjunctiveformula type="and">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="=">
            <variable symbol="p"/>
            <variable symbol="q"/>
           </constantatomformula>
          </conjunctiveformula>
          <constantatomformula symbol="=">
           <function symbol="f">
            <variable symbol="p"/>
           </function>
           <function symbol="f">
            <variable symbol="q"/>
           </function>
          </constantatomformula>
         </conjunctiveformula>
        </quantifiedformula>
       </quantifiedformula>
      </formulalist>
     </sequent>
     <rule type="existsr" symbol="\ex:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="=">
         <function symbol="f">
          <function symbol="+">
           <variable symbol="n_0"/>
           <variable symbol="k_0"/>
          </function>
         </function>
         <constant symbol="0"/>
        </constantatomformula>
        <constantatomformula symbol="=">
         <function symbol="f">
          <function symbol="+">
           <function symbol="+">
            <function symbol="+">
             <variable symbol="n_0"/>
             <variable symbol="k_0"/>
            </function>
            <constant symbol="1"/>
           </function>
           <variable symbol="k_1"/>
          </function>
         </function>
         <constant symbol="0"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <quantifiedformula type="exists">
         <variable symbol="p"/>
         <quantifiedformula type="exists">
          <variable symbol="q"/>
          <conjunctiveformula type="and">
           <conjunctiveformula type="neg">
            <constantatomformula symbol="=">
             <variable symbol="p"/>
             <variable symbol="q"/>
            </constantatomformula>
           </conjunctiveformula>
           <constantatomformula symbol="=">
            <function symbol="f">
             <variable symbol="p"/>
            </function>
            <function symbol="f">
             <variable symbol="q"/>
            </function>
           </constantatomformula>
          </conjunctiveformula>
         </quantifiedformula>
        </quantifiedformula>
       </formulalist>
      </sequent>
      <rule type="existsr" symbol="\ex:r">
       <sequent>
        <formulalist>
         <constantatomformula symbol="=">
          <function symbol="f">
           <function symbol="+">
            <variable symbol="n_0"/>
            <variable symbol="k_0"/>
           </function>
          </function>
          <constant symbol="0"/>
         </constantatomformula>
         <constantatomformula symbol="=">
          <function symbol="f">
           <function symbol="+">
            <function symbol="+">
             <function symbol="+">
              <variable symbol="n_0"/>
              <variable symbol="k_0"/>
             </function>
             <constant symbol="1"/>
            </function>
            <variable symbol="k_1"/>
           </function>
          </function>
          <constant symbol="0"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <quantifiedformula type="exists">
          <variable symbol="q"/>
          <conjunctiveformula type="and">
           <conjunctiveformula type="neg">
            <constantatomformula symbol="=">
             <function symbol="+">
              <variable symbol="n_0"/>
              <variable symbol="k_0"/>
             </function>
             <variable symbol="q"/>
            </constantatomformula>
           </conjunctiveformula>
           <constantatomformula symbol="=">
            <function symbol="f">
             <function symbol="+">
              <variable symbol="n_0"/>
              <variable symbol="k_0"/>
             </function>
            </function>
            <function symbol="f">
             <variable symbol="q"/>
            </function>
           </constantatomformula>
          </conjunctiveformula>
         </quantifiedformula>
        </formulalist>
       </sequent>
       <rule type="andr" symbol="\and:r">
        <sequent>
         <formulalist>
          <constantatomformula symbol="=">
           <function symbol="f">
            <function symbol="+">
             <variable symbol="n_0"/>
             <variable symbol="k_0"/>
            </function>
           </function>
           <constant symbol="0"/>
          </constantatomformula>
          <constantatomformula symbol="=">
           <function symbol="f">
            <function symbol="+">
             <function symbol="+">
              <function symbol="+">
               <variable symbol="n_0"/>
               <variable symbol="k_0"/>
              </function>
              <constant symbol="1"/>
             </function>
             <variable symbol="k_1"/>
            </function>
           </function>
           <constant symbol="0"/>
          </constantatomformula>
         </formulalist>
         <formulalist>
          <conjunctiveformula type="and">
           <conjunctiveformula type="neg">
            <constantatomformula symbol="=">
             <function symbol="+">
              <variable symbol="n_0"/>
              <variable symbol="k_0"/>
             </function>
             <function symbol="+">
              <function symbol="+">
               <function symbol="+">
                <variable symbol="n_0"/>
                <variable symbol="k_0"/>
               </function>
               <constant symbol="1"/>
              </function>
              <variable symbol="k_1"/>
             </function>
            </constantatomformula>
           </conjunctiveformula>
           <constantatomformula symbol="=">
            <function symbol="f">
             <function symbol="+">
              <variable symbol="n_0"/>
              <variable symbol="k_0"/>
             </function>
            </function>
            <function symbol="f">
             <function symbol="+">
              <function symbol="+">
               <function symbol="+">
                <variable symbol="n_0"/>
                <variable symbol="k_0"/>
               </function>
               <constant symbol="1"/>
              </function>
              <variable symbol="k_1"/>
             </function>
            </function>
           </constantatomformula>
          </conjunctiveformula>
         </formulalist>
        </sequent>
        <rule type="negr" symbol="\neg:r">
         <sequent>
          <formulalist/>
          <formulalist>
           <conjunctiveformula type="neg">
            <constantatomformula symbol="=">
             <function symbol="+">
              <variable symbol="n_0"/>
              <variable symbol="k_0"/>
             </function>
             <function symbol="+">
              <function symbol="+">
               <function symbol="+">
                <variable symbol="n_0"/>
                <variable symbol="k_0"/>
               </function>
               <constant symbol="1"/>
              </function>
              <variable symbol="k_1"/>
             </function>
            </constantatomformula>
           </conjunctiveformula>
          </formulalist>
         </sequent>
         <rule type="eql1" symbol="=:l1" param="(2)">
          <sequent>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="+">
              <variable symbol="n_0"/>
              <variable symbol="k_0"/>
             </function>
             <function symbol="+">
              <function symbol="+">
               <function symbol="+">
                <variable symbol="n_0"/>
                <variable symbol="k_0"/>
               </function>
               <constant symbol="1"/>
              </function>
              <variable symbol="k_1"/>
             </function>
            </constantatomformula>
           </formulalist>
           <formulalist/>
          </sequent>
          <rule type="axiom" symbol="axiom">
           <sequent>
            <formulalist/>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="+">
               <function symbol="+">
                <variable symbol="n_0"/>
                <variable symbol="k_0"/>
               </function>
               <function symbol="+">
                <constant symbol="1"/>
                <variable symbol="k_1"/>
               </function>
              </function>
              <function symbol="+">
               <function symbol="+">
                <function symbol="+">
                 <variable symbol="n_0"/>
                 <variable symbol="k_0"/>
                </function>
                <constant symbol="1"/>
               </function>
               <variable symbol="k_1"/>
              </function>
             </constantatomformula>
            </formulalist>
           </sequent>
          </rule>
          <rule type="axiom" symbol="axiom">
           <sequent>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="+">
               <variable symbol="n_0"/>
               <variable symbol="k_0"/>
              </function>
              <function symbol="+">
               <function symbol="+">
                <variable symbol="n_0"/>
                <variable symbol="k_0"/>
               </function>
               <function symbol="+">
                <constant symbol="1"/>
                <variable symbol="k_1"/>
               </function>
              </function>
             </constantatomformula>
            </formulalist>
            <formulalist/>
           </sequent>
          </rule>
         </rule>
        </rule>
        <rule type="eqr2" symbol="=:r2" param="(1)">
         <sequent>
          <formulalist>
           <constantatomformula symbol="=">
            <function symbol="f">
             <function symbol="+">
              <variable symbol="n_0"/>
              <variable symbol="k_0"/>
             </function>
            </function>
            <constant symbol="0"/>
           </constantatomformula>
           <constantatomformula symbol="=">
            <function symbol="f">
             <function symbol="+">
              <function symbol="+">
               <function symbol="+">
                <variable symbol="n_0"/>
                <variable symbol="k_0"/>
               </function>
               <constant symbol="1"/>
              </function>
              <variable symbol="k_1"/>
             </function>
            </function>
            <constant symbol="0"/>
           </constantatomformula>
          </formulalist>
          <formulalist>
           <constantatomformula symbol="=">
            <function symbol="f">
             <function symbol="+">
              <variable symbol="n_0"/>
              <variable symbol="k_0"/>
             </function>
            </function>
            <function symbol="f">
             <function symbol="+">
              <function symbol="+">
               <function symbol="+">
                <variable symbol="n_0"/>
                <variable symbol="k_0"/>
               </function>
               <constant symbol="1"/>
              </function>
              <variable symbol="k_1"/>
             </function>
            </function>
           </constantatomformula>
          </formulalist>
         </sequent>
         <rule type="axiom" symbol="axiom">
          <sequent>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="f">
              <function symbol="+">
               <variable symbol="n_0"/>
               <variable symbol="k_0"/>
              </function>
             </function>
             <constant symbol="0"/>
            </constantatomformula>
           </formulalist>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="f">
              <function symbol="+">
               <variable symbol="n_0"/>
               <variable symbol="k_0"/>
              </function>
             </function>
             <constant symbol="0"/>
            </constantatomformula>
           </formulalist>
          </sequent>
         </rule>
         <rule type="eqr2" symbol="=:r2" param="(2)">
          <sequent>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="f">
              <function symbol="+">
               <function symbol="+">
                <function symbol="+">
                 <variable symbol="n_0"/>
                 <variable symbol="k_0"/>
                </function>
                <constant symbol="1"/>
               </function>
               <variable symbol="k_1"/>
              </function>
             </function>
             <constant symbol="0"/>
            </constantatomformula>
           </formulalist>
           <formulalist>
            <constantatomformula symbol="=">
             <constant symbol="0"/>
             <function symbol="f">
              <function symbol="+">
               <function symbol="+">
                <function symbol="+">
                 <variable symbol="n_0"/>
                 <variable symbol="k_0"/>
                </function>
                <constant symbol="1"/>
               </function>
               <variable symbol="k_1"/>
              </function>
             </function>
            </constantatomformula>
           </formulalist>
          </sequent>
          <rule type="axiom" symbol="axiom">
           <sequent>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="f">
               <function symbol="+">
                <function symbol="+">
                 <function symbol="+">
                  <variable symbol="n_0"/>
                  <variable symbol="k_0"/>
                 </function>
                 <constant symbol="1"/>
                </function>
                <variable symbol="k_1"/>
               </function>
              </function>
              <constant symbol="0"/>
             </constantatomformula>
            </formulalist>
            <formulalist>
             <constantatomformula symbol="=">
              <function symbol="f">
               <function symbol="+">
                <function symbol="+">
                 <function symbol="+">
                  <variable symbol="n_0"/>
                  <variable symbol="k_0"/>
                 </function>
                 <constant symbol="1"/>
                </function>
                <variable symbol="k_1"/>
               </function>
              </function>
              <constant symbol="0"/>
             </constantatomformula>
            </formulalist>
           </sequent>
          </rule>
          <rule type="axiom" symbol="axiom">
           <sequent>
            <formulalist/>
            <formulalist>
             <constantatomformula symbol="=">
              <constant symbol="0"/>
              <constant symbol="0"/>
             </constantatomformula>
            </formulalist>
           </sequent>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{3}" calculus="LK">
  <rule type="contrr" symbol="c:r" param="1,1,2">
   <sequent>
    <formulalist>
     <constantatomformula symbol="A" definition="A"/>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="=">
      <function symbol="f">
       <function symbol="+">
        <variable symbol="n_0"/>
        <variable symbol="n_1"/>
       </function>
      </function>
      <constant symbol="0"/>
     </constantatomformula>
     <constantatomformula symbol="=">
      <function symbol="f">
       <function symbol="+">
        <variable symbol="n_1"/>
        <variable symbol="n_0"/>
       </function>
      </function>
      <constant symbol="1"/>
     </constantatomformula>
     <quantifiedformula type="exists">
      <variable symbol="p"/>
      <quantifiedformula type="exists">
       <variable symbol="q"/>
       <conjunctiveformula type="and">
        <conjunctiveformula type="neg">
         <constantatomformula symbol="=">
          <variable symbol="p"/>
          <variable symbol="q"/>
         </constantatomformula>
        </conjunctiveformula>
        <constantatomformula symbol="=">
         <function symbol="f">
          <variable symbol="p"/>
         </function>
         <function symbol="f">
          <variable symbol="q"/>
         </function>
        </constantatomformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
    </formulalist>
   </sequent>
   <rule type="weakr" symbol="w:r">
    <sequent>
     <formulalist>
      <constantatomformula symbol="A" definition="A"/>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="=">
       <function symbol="f">
        <function symbol="+">
         <variable symbol="n_0"/>
         <variable symbol="n_1"/>
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      <formulalist>
       <constantatomformula symbol="=">
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       <formulalist>
        <constantatomformula symbol="=">
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           <formulalist/>
           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="+">
              <variable symbol="n_0"/>
              <variable symbol="n_1"/>
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             <function symbol="+">
              <variable symbol="n_1"/>
              <variable symbol="n_0"/>
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           <formulalist>
            <constantatomformula symbol="=">
             <function symbol="f">
              <function symbol="+">
               <variable symbol="n_1"/>
               <variable symbol="n_0"/>
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           <formulalist>
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       </rule>
      </rule>
     </rule>
    </rule>
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  </rule>
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  <rule type="axset">
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       <variable symbol="l"/>
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     <rule type="axset">
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       <formulalist/>
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     </rule>
    </rule>
   </rule>
  </rule>
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  <rule type="res" symbol="r">
   <sequent>
    <formulalist/>
    <formulalist/>
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     <formulalist/>
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      <constantatomformula symbol="=">
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        <function symbol="+">
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      <formulalist/>
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       <formulalist/>
       <formulalist>
        <constantatomformula symbol="=">
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           <function symbol="+">
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          <function symbol="+">
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         <function symbol="+">
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         <constantatomformula symbol="=">
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      <sequent>
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                          <function symbol="+">
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                        <function symbol="+">
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                           <function symbol="+">
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                           <function symbol="+">
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                           <constant symbol="1"/>
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                           <function symbol="+">
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                            <function symbol="+">
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                          <function symbol="+">
                           <function symbol="+">
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                            <constant symbol="1"/>
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                           <constant symbol="1"/>
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                         <function symbol="+">
                          <function symbol="+">
                           <constant symbol="1"/>
                           <function symbol="+">
                            <variable symbol="x_{110}"/>
                            <constant symbol="1"/>
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                          <function symbol="+">
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                           <function symbol="+">
                            <constant symbol="1"/>
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                           </function>
                           <function symbol="+">
                            <variable symbol="x_{109}"/>
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                       <constantatomformula symbol="=">
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                            <constant symbol="1"/>
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                            <function symbol="+">
                             <variable symbol="x_{109}"/>
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                            <function symbol="+">
                             <function symbol="+">
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                               <variable symbol="x_{110}"/>
                               <constant symbol="1"/>
                              </function>
                             </function>
                             <function symbol="+">
                              <variable symbol="x_{109}"/>
                              <constant symbol="1"/>
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                            <constant symbol="1"/>
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                            <function symbol="+">
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                             <function symbol="+">
                              <variable symbol="x_{110}"/>
                              <constant symbol="1"/>
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                            </function>
                            <variable symbol="x_{109}"/>
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                         <formulalist/>
                         <formulalist>
                          <constantatomformula symbol="=">
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                            <function symbol="+">
                             <function symbol="+">
                              <constant symbol="1"/>
                              <function symbol="+">
                               <variable symbol="x_{110}"/>
                               <constant symbol="1"/>
                              </function>
                             </function>
                             <variable symbol="x_{109}"/>
                            </function>
                            <constant symbol="1"/>
                           </function>
                           <function symbol="+">
                            <function symbol="+">
                             <constant symbol="1"/>
                             <function symbol="+">
                              <variable symbol="x_{110}"/>
                              <constant symbol="1"/>
                             </function>
                            </function>
                            <function symbol="+">
                             <variable symbol="x_{109}"/>
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                          <formulalist>
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                             <function symbol="+">
                              <constant symbol="1"/>
                              <function symbol="+">
                               <variable symbol="x_{110}"/>
                               <constant symbol="1"/>
                              </function>
                             </function>
                             <function symbol="+">
                              <variable symbol="x_{109}"/>
                              <constant symbol="1"/>
                             </function>
                            </function>
                            <function symbol="+">
                             <function symbol="+">
                              <function symbol="+">
                               <constant symbol="1"/>
                               <function symbol="+">
                                <variable symbol="x_{110}"/>
                                <constant symbol="1"/>
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                              </function>
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                             <function symbol="+">
                              <variable symbol="x_{109}"/>
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                            </function>
                            <function symbol="+">
                             <function symbol="+">
                              <constant symbol="1"/>
                              <function symbol="+">
                               <variable symbol="x_{110}"/>
                               <constant symbol="1"/>
                              </function>
                             </function>
                             <function symbol="+">
                              <variable symbol="x_{109}"/>
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                             <function symbol="+">
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                               </function>
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                               <constant symbol="1"/>
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                             <variable symbol="x_{109}"/>
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                            <function symbol="+">
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                            <function symbol="+">
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                           <function symbol="+">
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                            <function symbol="+">
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                            <variable symbol="x_{109}"/>
                            <function symbol="+">
                             <constant symbol="1"/>
                             <function symbol="+">
                              <variable symbol="x_{110}"/>
                              <function symbol="+">
                               <constant symbol="1"/>
                               <function symbol="+">
                                <constant symbol="1"/>
                                <variable symbol="x_{111}"/>
                               </function>
                              </function>
                             </function>
                            </function>
                           </function>
                          </function>
                          <constant symbol="1"/>
                         </constantatomformula>
                        </formulalist>
                        <formulalist/>
                       </sequent>
                       <rule type="init" symbol="">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="f">
                            <function symbol="+">
                             <variable symbol="x_{109}"/>
                             <function symbol="+">
                              <constant symbol="1"/>
                              <function symbol="+">
                               <variable symbol="x_{110}"/>
                               <function symbol="+">
                                <constant symbol="1"/>
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <variable symbol="x_{111}"/>
                                </function>
                               </function>
                              </function>
                             </function>
                            </function>
                           </function>
                           <constant symbol="1"/>
                          </constantatomformula>
                          <constantatomformula symbol="=">
                           <function symbol="f">
                            <function symbol="+">
                             <function symbol="+">
                              <function symbol="+">
                               <variable symbol="x_{109}"/>
                               <function symbol="+">
                                <constant symbol="1"/>
                                <function symbol="+">
                                 <variable symbol="x_{110}"/>
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <variable symbol="x_{111}"/>
                                  </function>
                                 </function>
                                </function>
                               </function>
                              </function>
                              <constant symbol="1"/>
                             </function>
                             <variable symbol="x_{112}"/>
                            </function>
                           </function>
                           <constant symbol="1"/>
                          </constantatomformula>
                         </formulalist>
                         <formulalist/>
                        </sequent>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="acnf" calculus="LK">
  <rule type="contrr" symbol="c:r" param="2">
   <sequent>
    <formulalist>
     <constantatomformula symbol="A" definition="A"/>
    </formulalist>
    <formulalist>
     <quantifiedformula type="exists">
      <variable symbol="p"/>
      <quantifiedformula type="exists">
       <variable symbol="q"/>
       <conjunctiveformula type="and">
        <conjunctiveformula type="neg">
         <constantatomformula symbol="=">
          <variable symbol="p"/>
          <variable symbol="q"/>
         </constantatomformula>
        </conjunctiveformula>
        <constantatomformula symbol="=">
         <function symbol="f">
          <variable symbol="p"/>
         </function>
         <function symbol="f">
          <variable symbol="q"/>
         </function>
        </constantatomformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
    </formulalist>
   </sequent>
   <rule type="permr" symbol="\pi:r" param="">
    <sequent>
     <formulalist>
      <constantatomformula symbol="A" definition="A"/>
     </formulalist>
     <formulalist>
      <quantifiedformula type="exists">
       <variable symbol="p"/>
       <quantifiedformula type="exists">
        <variable symbol="q"/>
        <conjunctiveformula type="and">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="=">
           <variable symbol="p"/>
           <variable symbol="q"/>
          </constantatomformula>
         </conjunctiveformula>
         <constantatomformula symbol="=">
          <function symbol="f">
           <variable symbol="p"/>
          </function>
          <function symbol="f">
           <variable symbol="q"/>
          </function>
         </constantatomformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
      <quantifiedformula type="exists">
       <variable symbol="p"/>
       <quantifiedformula type="exists">
        <variable symbol="q"/>
        <conjunctiveformula type="and">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="=">
           <variable symbol="p"/>
           <variable symbol="q"/>
          </constantatomformula>
         </conjunctiveformula>
         <constantatomformula symbol="=">
          <function symbol="f">
           <variable symbol="p"/>
          </function>
          <function symbol="f">
           <variable symbol="q"/>
          </function>
         </constantatomformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
     </formulalist>
    </sequent>
    <rule type="contrl" symbol="c:l" param="2">
     <sequent>
      <formulalist>
       <constantatomformula symbol="A" definition="A"/>
      </formulalist>
      <formulalist>
       <quantifiedformula type="exists">
        <variable symbol="p"/>
        <quantifiedformula type="exists">
         <variable symbol="q"/>
         <conjunctiveformula type="and">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="=">
            <variable symbol="p"/>
            <variable symbol="q"/>
           </constantatomformula>
          </conjunctiveformula>
          <constantatomformula symbol="=">
           <function symbol="f">
            <variable symbol="p"/>
           </function>
           <function symbol="f">
            <variable symbol="q"/>
           </function>
          </constantatomformula>
         </conjunctiveformula>
        </quantifiedformula>
       </quantifiedformula>
       <quantifiedformula type="exists">
        <variable symbol="p"/>
        <quantifiedformula type="exists">
         <variable symbol="q"/>
         <conjunctiveformula type="and">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="=">
            <variable symbol="p"/>
            <variable symbol="q"/>
           </constantatomformula>
          </conjunctiveformula>
          <constantatomformula symbol="=">
           <function symbol="f">
            <variable symbol="p"/>
           </function>
           <function symbol="f">
            <variable symbol="q"/>
           </function>
          </constantatomformula>
         </conjunctiveformula>
        </quantifiedformula>
       </quantifiedformula>
      </formulalist>
     </sequent>
     <rule type="perml" symbol="\pi:l" param="">
      <sequent>
       <formulalist>
        <constantatomformula symbol="A" definition="A"/>
        <constantatomformula symbol="A" definition="A"/>
       </formulalist>
       <formulalist>
        <quantifiedformula type="exists">
         <variable symbol="p"/>
         <quantifiedformula type="exists">
          <variable symbol="q"/>
          <conjunctiveformula type="and">
           <conjunctiveformula type="neg">
            <constantatomformula symbol="=">
             <variable symbol="p"/>
             <variable symbol="q"/>
            </constantatomformula>
           </conjunctiveformula>
           <constantatomformula symbol="=">
            <function symbol="f">
             <variable symbol="p"/>
            </function>
            <function symbol="f">
             <variable symbol="q"/>
            </function>
           </constantatomformula>
          </conjunctiveformula>
         </quantifiedformula>
        </quantifiedformula>
        <quantifiedformula type="exists">
         <variable symbol="p"/>
         <quantifiedformula type="exists">
          <variable symbol="q"/>
          <conjunctiveformula type="and">
           <conjunctiveformula type="neg">
            <constantatomformula symbol="=">
             <variable symbol="p"/>
             <variable symbol="q"/>
            </constantatomformula>
           </conjunctiveformula>
           <constantatomformula symbol="=">
            <function symbol="f">
             <variable symbol="p"/>
            </function>
            <function symbol="f">
             <variable symbol="q"/>
            </function>
           </constantatomformula>
          </conjunctiveformula>
         </quantifiedformula>
        </quantifiedformula>
       </formulalist>
      </sequent>
      <rule type="cut" symbol="cut">
       <sequent>
        <formulalist>
         <constantatomformula symbol="A" definition="A"/>
         <constantatomformula symbol="A" definition="A"/>
        </formulalist>
        <formulalist>
         <quantifiedformula type="exists">
          <variable symbol="p"/>
          <quantifiedformula type="exists">
           <variable symbol="q"/>
           <conjunctiveformula type="and">
            <conjunctiveformula type="neg">
             <constantatomformula symbol="=">
              <variable symbol="p"/>
              <variable symbol="q"/>
             </constantatomformula>
            </conjunctiveformula>
            <constantatomformula symbol="=">
             <function symbol="f">
              <variable symbol="p"/>
             </function>
             <function symbol="f">
              <variable symbol="q"/>
             </function>
            </constantatomformula>
           </conjunctiveformula>
          </quantifiedformula>
         </quantifiedformula>
         <quantifiedformula type="exists">
          <variable symbol="p"/>
          <quantifiedformula type="exists">
           <variable symbol="q"/>
           <conjunctiveformula type="and">
            <conjunctiveformula type="neg">
             <constantatomformula symbol="=">
              <variable symbol="p"/>
              <variable symbol="q"/>
             </constantatomformula>
            </conjunctiveformula>
            <constantatomformula symbol="=">
             <function symbol="f">
              <variable symbol="p"/>
             </function>
             <function symbol="f">
              <variable symbol="q"/>
             </function>
            </constantatomformula>
           </conjunctiveformula>
          </quantifiedformula>
         </quantifiedformula>
        </formulalist>
       </sequent>
       <rule type="contrr" symbol="c:r" param="2,1">
        <sequent>
         <formulalist>
          <constantatomformula symbol="A" definition="A"/>
         </formulalist>
         <formulalist>
          <quantifiedformula type="exists">
           <variable symbol="p"/>
           <quantifiedformula type="exists">
            <variable symbol="q"/>
            <conjunctiveformula type="and">
             <conjunctiveformula type="neg">
              <constantatomformula symbol="=">
               <variable symbol="p"/>
               <variable symbol="q"/>
              </constantatomformula>
             </conjunctiveformula>
             <constantatomformula symbol="=">
              <function symbol="f">
               <variable symbol="p"/>
              </function>
              <function symbol="f">
               <variable symbol="q"/>
              </function>
             </constantatomformula>
            </conjunctiveformula>
           </quantifiedformula>
          </quantifiedformula>
          <constantatomformula symbol="=">
           <function symbol="f">
            <function symbol="+">
             <variable symbol="x_{109}"/>
             <function symbol="+">
              <constant symbol="1"/>
              <function symbol="+">
               <variable symbol="x_{110}"/>
               <function symbol="+">
                <constant symbol="1"/>
                <function symbol="+">
                 <constant symbol="1"/>
                 <variable symbol="x_{111}"/>
                </function>
               </function>
              </function>
             </function>
            </function>
           </function>
           <constant symbol="1"/>
          </constantatomformula>
         </formulalist>
        </sequent>
        <rule type="permr" symbol="\pi:r" param="">
         <sequent>
          <formulalist>
           <constantatomformula symbol="A" definition="A"/>
          </formulalist>
          <formulalist>
           <quantifiedformula type="exists">
            <variable symbol="p"/>
            <quantifiedformula type="exists">
             <variable symbol="q"/>
             <conjunctiveformula type="and">
              <conjunctiveformula type="neg">
               <constantatomformula symbol="=">
                <variable symbol="p"/>
                <variable symbol="q"/>
               </constantatomformula>
              </conjunctiveformula>
              <constantatomformula symbol="=">
               <function symbol="f">
                <variable symbol="p"/>
               </function>
               <function symbol="f">
                <variable symbol="q"/>
               </function>
              </constantatomformula>
             </conjunctiveformula>
            </quantifiedformula>
           </quantifiedformula>
           <quantifiedformula type="exists">
            <variable symbol="p"/>
            <quantifiedformula type="exists">
             <variable symbol="q"/>
             <conjunctiveformula type="and">
              <conjunctiveformula type="neg">
               <constantatomformula symbol="=">
                <variable symbol="p"/>
                <variable symbol="q"/>
               </constantatomformula>
              </conjunctiveformula>
              <constantatomformula symbol="=">
               <function symbol="f">
                <variable symbol="p"/>
               </function>
               <function symbol="f">
                <variable symbol="q"/>
               </function>
              </constantatomformula>
             </conjunctiveformula>
            </quantifiedformula>
           </quantifiedformula>
           <constantatomformula symbol="=">
            <function symbol="f">
             <function symbol="+">
              <variable symbol="x_{109}"/>
              <function symbol="+">
               <constant symbol="1"/>
               <function symbol="+">
                <variable symbol="x_{110}"/>
                <function symbol="+">
                 <constant symbol="1"/>
                 <function symbol="+">
                  <constant symbol="1"/>
                  <variable symbol="x_{111}"/>
                 </function>
                </function>
               </function>
              </function>
             </function>
            </function>
            <constant symbol="1"/>
           </constantatomformula>
          </formulalist>
         </sequent>
         <rule type="contrl" symbol="c:l" param="2">
          <sequent>
           <formulalist>
            <constantatomformula symbol="A" definition="A"/>
           </formulalist>
           <formulalist>
            <quantifiedformula type="exists">
             <variable symbol="p"/>
             <quantifiedformula type="exists">
              <variable symbol="q"/>
              <conjunctiveformula type="and">
               <conjunctiveformula type="neg">
                <constantatomformula symbol="=">
                 <variable symbol="p"/>
                 <variable symbol="q"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="=">
                <function symbol="f">
                 <variable symbol="p"/>
                </function>
                <function symbol="f">
                 <variable symbol="q"/>
                </function>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </quantifiedformula>
            <quantifiedformula type="exists">
             <variable symbol="p"/>
             <quantifiedformula type="exists">
              <variable symbol="q"/>
              <conjunctiveformula type="and">
               <conjunctiveformula type="neg">
                <constantatomformula symbol="=">
                 <variable symbol="p"/>
                 <variable symbol="q"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="=">
                <function symbol="f">
                 <variable symbol="p"/>
                </function>
                <function symbol="f">
                 <variable symbol="q"/>
                </function>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </quantifiedformula>
            <constantatomformula symbol="=">
             <function symbol="f">
              <function symbol="+">
               <variable symbol="x_{109}"/>
               <function symbol="+">
                <constant symbol="1"/>
                <function symbol="+">
                 <variable symbol="x_{110}"/>
                 <function symbol="+">
                  <constant symbol="1"/>
                  <function symbol="+">
                   <constant symbol="1"/>
                   <variable symbol="x_{111}"/>
                  </function>
                 </function>
                </function>
               </function>
              </function>
             </function>
             <constant symbol="1"/>
            </constantatomformula>
           </formulalist>
          </sequent>
          <rule type="perml" symbol="\pi:l" param="">
           <sequent>
            <formulalist>
             <constantatomformula symbol="A" definition="A"/>
             <constantatomformula symbol="A" definition="A"/>
            </formulalist>
            <formulalist>
             <quantifiedformula type="exists">
              <variable symbol="p"/>
              <quantifiedformula type="exists">
               <variable symbol="q"/>
               <conjunctiveformula type="and">
                <conjunctiveformula type="neg">
                 <constantatomformula symbol="=">
                  <variable symbol="p"/>
                  <variable symbol="q"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="=">
                 <function symbol="f">
                  <variable symbol="p"/>
                 </function>
                 <function symbol="f">
                  <variable symbol="q"/>
                 </function>
                </constantatomformula>
               </conjunctiveformula>
              </quantifiedformula>
             </quantifiedformula>
             <quantifiedformula type="exists">
              <variable symbol="p"/>
              <quantifiedformula type="exists">
               <variable symbol="q"/>
               <conjunctiveformula type="and">
                <conjunctiveformula type="neg">
                 <constantatomformula symbol="=">
                  <variable symbol="p"/>
                  <variable symbol="q"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="=">
                 <function symbol="f">
                  <variable symbol="p"/>
                 </function>
                 <function symbol="f">
                  <variable symbol="q"/>
                 </function>
                </constantatomformula>
               </conjunctiveformula>
              </quantifiedformula>
             </quantifiedformula>
             <constantatomformula symbol="=">
              <function symbol="f">
               <function symbol="+">
                <variable symbol="x_{109}"/>
                <function symbol="+">
                 <constant symbol="1"/>
                 <function symbol="+">
                  <variable symbol="x_{110}"/>
                  <function symbol="+">
                   <constant symbol="1"/>
                   <function symbol="+">
                    <constant symbol="1"/>
                    <variable symbol="x_{111}"/>
                   </function>
                  </function>
                 </function>
                </function>
               </function>
              </function>
              <constant symbol="1"/>
             </constantatomformula>
            </formulalist>
           </sequent>
           <rule type="eqr1" symbol="=:r1" param="(1 1 2 2)">
            <sequent>
             <formulalist>
              <constantatomformula symbol="A" definition="A"/>
              <constantatomformula symbol="A" definition="A"/>
             </formulalist>
             <formulalist>
              <quantifiedformula type="exists">
               <variable symbol="p"/>
               <quantifiedformula type="exists">
                <variable symbol="q"/>
                <conjunctiveformula type="and">
                 <conjunctiveformula type="neg">
                  <constantatomformula symbol="=">
                   <variable symbol="p"/>
                   <variable symbol="q"/>
                  </constantatomformula>
                 </conjunctiveformula>
                 <constantatomformula symbol="=">
                  <function symbol="f">
                   <variable symbol="p"/>
                  </function>
                  <function symbol="f">
                   <variable symbol="q"/>
                  </function>
                 </constantatomformula>
                </conjunctiveformula>
               </quantifiedformula>
              </quantifiedformula>
              <quantifiedformula type="exists">
               <variable symbol="p"/>
               <quantifiedformula type="exists">
                <variable symbol="q"/>
                <conjunctiveformula type="and">
                 <conjunctiveformula type="neg">
                  <constantatomformula symbol="=">
                   <variable symbol="p"/>
                   <variable symbol="q"/>
                  </constantatomformula>
                 </conjunctiveformula>
                 <constantatomformula symbol="=">
                  <function symbol="f">
                   <variable symbol="p"/>
                  </function>
                  <function symbol="f">
                   <variable symbol="q"/>
                  </function>
                 </constantatomformula>
                </conjunctiveformula>
               </quantifiedformula>
              </quantifiedformula>
              <constantatomformula symbol="=">
               <function symbol="f">
                <function symbol="+">
                 <variable symbol="x_{109}"/>
                 <function symbol="+">
                  <constant symbol="1"/>
                  <function symbol="+">
                   <variable symbol="x_{110}"/>
                   <function symbol="+">
                    <constant symbol="1"/>
                    <function symbol="+">
                     <constant symbol="1"/>
                     <variable symbol="x_{111}"/>
                    </function>
                   </function>
                  </function>
                 </function>
                </function>
               </function>
               <constant symbol="1"/>
              </constantatomformula>
             </formulalist>
            </sequent>
            <rule type="contrr" symbol="c:r" param="2,1">
             <sequent>
              <formulalist>
               <constantatomformula symbol="A" definition="A"/>
              </formulalist>
              <formulalist>
               <quantifiedformula type="exists">
                <variable symbol="p"/>
                <quantifiedformula type="exists">
                 <variable symbol="q"/>
                 <conjunctiveformula type="and">
                  <conjunctiveformula type="neg">
                   <constantatomformula symbol="=">
                    <variable symbol="p"/>
                    <variable symbol="q"/>
                   </constantatomformula>
                  </conjunctiveformula>
                  <constantatomformula symbol="=">
                   <function symbol="f">
                    <variable symbol="p"/>
                   </function>
                   <function symbol="f">
                    <variable symbol="q"/>
                   </function>
                  </constantatomformula>
                 </conjunctiveformula>
                </quantifiedformula>
               </quantifiedformula>
               <constantatomformula symbol="=">
                <function symbol="+">
                 <function symbol="+">
                  <variable symbol="x_{110}"/>
                  <constant symbol="1"/>
                 </function>
                 <function symbol="+">
                  <constant symbol="1"/>
                  <variable symbol="x_{111}"/>
                 </function>
                </function>
                <function symbol="+">
                 <variable symbol="x_{110}"/>
                 <function symbol="+">
                  <constant symbol="1"/>
                  <function symbol="+">
                   <constant symbol="1"/>
                   <variable symbol="x_{111}"/>
                  </function>
                 </function>
                </function>
               </constantatomformula>
              </formulalist>
             </sequent>
             <rule type="permr" symbol="\pi:r" param="">
              <sequent>
               <formulalist>
                <constantatomformula symbol="A" definition="A"/>
               </formulalist>
               <formulalist>
                <quantifiedformula type="exists">
                 <variable symbol="p"/>
                 <quantifiedformula type="exists">
                  <variable symbol="q"/>
                  <conjunctiveformula type="and">
                   <conjunctiveformula type="neg">
                    <constantatomformula symbol="=">
                     <variable symbol="p"/>
                     <variable symbol="q"/>
                    </constantatomformula>
                   </conjunctiveformula>
                   <constantatomformula symbol="=">
                    <function symbol="f">
                     <variable symbol="p"/>
                    </function>
                    <function symbol="f">
                     <variable symbol="q"/>
                    </function>
                   </constantatomformula>
                  </conjunctiveformula>
                 </quantifiedformula>
                </quantifiedformula>
                <quantifiedformula type="exists">
                 <variable symbol="p"/>
                 <quantifiedformula type="exists">
                  <variable symbol="q"/>
                  <conjunctiveformula type="and">
                   <conjunctiveformula type="neg">
                    <constantatomformula symbol="=">
                     <variable symbol="p"/>
                     <variable symbol="q"/>
                    </constantatomformula>
                   </conjunctiveformula>
                   <constantatomformula symbol="=">
                    <function symbol="f">
                     <variable symbol="p"/>
                    </function>
                    <function symbol="f">
                     <variable symbol="q"/>
                    </function>
                   </constantatomformula>
                  </conjunctiveformula>
                 </quantifiedformula>
                </quantifiedformula>
                <constantatomformula symbol="=">
                 <function symbol="+">
                  <function symbol="+">
                   <variable symbol="x_{110}"/>
                   <constant symbol="1"/>
                  </function>
                  <function symbol="+">
                   <constant symbol="1"/>
                   <variable symbol="x_{111}"/>
                  </function>
                 </function>
                 <function symbol="+">
                  <variable symbol="x_{110}"/>
                  <function symbol="+">
                   <constant symbol="1"/>
                   <function symbol="+">
                    <constant symbol="1"/>
                    <variable symbol="x_{111}"/>
                   </function>
                  </function>
                 </function>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="contrl" symbol="c:l" param="2">
               <sequent>
                <formulalist>
                 <constantatomformula symbol="A" definition="A"/>
                </formulalist>
                <formulalist>
                 <quantifiedformula type="exists">
                  <variable symbol="p"/>
                  <quantifiedformula type="exists">
                   <variable symbol="q"/>
                   <conjunctiveformula type="and">
                    <conjunctiveformula type="neg">
                     <constantatomformula symbol="=">
                      <variable symbol="p"/>
                      <variable symbol="q"/>
                     </constantatomformula>
                    </conjunctiveformula>
                    <constantatomformula symbol="=">
                     <function symbol="f">
                      <variable symbol="p"/>
                     </function>
                     <function symbol="f">
                      <variable symbol="q"/>
                     </function>
                    </constantatomformula>
                   </conjunctiveformula>
                  </quantifiedformula>
                 </quantifiedformula>
                 <quantifiedformula type="exists">
                  <variable symbol="p"/>
                  <quantifiedformula type="exists">
                   <variable symbol="q"/>
                   <conjunctiveformula type="and">
                    <conjunctiveformula type="neg">
                     <constantatomformula symbol="=">
                      <variable symbol="p"/>
                      <variable symbol="q"/>
                     </constantatomformula>
                    </conjunctiveformula>
                    <constantatomformula symbol="=">
                     <function symbol="f">
                      <variable symbol="p"/>
                     </function>
                     <function symbol="f">
                      <variable symbol="q"/>
                     </function>
                    </constantatomformula>
                   </conjunctiveformula>
                  </quantifiedformula>
                 </quantifiedformula>
                 <constantatomformula symbol="=">
                  <function symbol="+">
                   <function symbol="+">
                    <variable symbol="x_{110}"/>
                    <constant symbol="1"/>
                   </function>
                   <function symbol="+">
                    <constant symbol="1"/>
                    <variable symbol="x_{111}"/>
                   </function>
                  </function>
                  <function symbol="+">
                   <variable symbol="x_{110}"/>
                   <function symbol="+">
                    <constant symbol="1"/>
                    <function symbol="+">
                     <constant symbol="1"/>
                     <variable symbol="x_{111}"/>
                    </function>
                   </function>
                  </function>
                 </constantatomformula>
                </formulalist>
               </sequent>
               <rule type="perml" symbol="\pi:l" param="">
                <sequent>
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                       <constantatomformula symbol="=">
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                      <function symbol="+">
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                      <function symbol="+">
                       <function symbol="+">
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                    <formulalist>
                     <constantatomformula symbol="A" definition="A"/>
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                    <formulalist>
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                      <quantifiedformula type="exists">
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                        <constantatomformula symbol="=">
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                        <constantatomformula symbol="=">
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                       <function symbol="+">
                        <constant symbol="1"/>
                        <variable symbol="x_{111}"/>
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                       <function symbol="+">
                        <function symbol="+">
                         <variable symbol="x_{110}"/>
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                        <function symbol="+">
                         <constant symbol="1"/>
                         <variable symbol="x_{111}"/>
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                      <constantatomformula symbol="A" definition="A"/>
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                     <formulalist>
                      <quantifiedformula type="exists">
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                          <constantatomformula symbol="=">
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                         <constantatomformula symbol="=">
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                           <variable symbol="p"/>
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                          <constantatomformula symbol="=">
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                           <variable symbol="q"/>
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                         <constantatomformula symbol="=">
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                           <variable symbol="p"/>
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                          <function symbol="f">
                           <variable symbol="q"/>
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                      <constantatomformula symbol="=">
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                         <function symbol="+">
                          <variable symbol="x_{110}"/>
                          <constant symbol="1"/>
                         </function>
                        </function>
                        <function symbol="+">
                         <constant symbol="1"/>
                         <variable symbol="x_{111}"/>
                        </function>
                       </function>
                       <function symbol="+">
                        <constant symbol="1"/>
                        <function symbol="+">
                         <function symbol="+">
                          <variable symbol="x_{110}"/>
                          <constant symbol="1"/>
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                         <function symbol="+">
                          <constant symbol="1"/>
                          <variable symbol="x_{111}"/>
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                       <constantatomformula symbol="A" definition="A"/>
                       <constantatomformula symbol="A" definition="A"/>
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                      <formulalist>
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                        <quantifiedformula type="exists">
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                          <constantatomformula symbol="=">
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                            <variable symbol="p"/>
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                           <function symbol="f">
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                        <quantifiedformula type="exists">
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                           <constantatomformula symbol="=">
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                          <constantatomformula symbol="=">
                           <function symbol="f">
                            <variable symbol="p"/>
                           </function>
                           <function symbol="f">
                            <variable symbol="q"/>
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                          </constantatomformula>
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                       <constantatomformula symbol="=">
                        <function symbol="+">
                         <function symbol="+">
                          <constant symbol="1"/>
                          <function symbol="+">
                           <variable symbol="x_{110}"/>
                           <constant symbol="1"/>
                          </function>
                         </function>
                         <function symbol="+">
                          <constant symbol="1"/>
                          <variable symbol="x_{111}"/>
                         </function>
                        </function>
                        <function symbol="+">
                         <constant symbol="1"/>
                         <function symbol="+">
                          <function symbol="+">
                           <variable symbol="x_{110}"/>
                           <constant symbol="1"/>
                          </function>
                          <function symbol="+">
                           <constant symbol="1"/>
                           <variable symbol="x_{111}"/>
                          </function>
                         </function>
                        </function>
                       </constantatomformula>
                      </formulalist>
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                       <formulalist>
                        <constantatomformula symbol="A" definition="A"/>
                        <constantatomformula symbol="A" definition="A"/>
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                       <formulalist>
                        <quantifiedformula type="exists">
                         <variable symbol="p"/>
                         <quantifiedformula type="exists">
                          <variable symbol="q"/>
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                            <constantatomformula symbol="=">
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                           <constantatomformula symbol="=">
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                             <variable symbol="p"/>
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                            <function symbol="f">
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                        <quantifiedformula type="exists">
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                           <constantatomformula symbol="=">
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                             <variable symbol="p"/>
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                            <function symbol="f">
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                        <constantatomformula symbol="=">
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                          <function symbol="+">
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                           <function symbol="+">
                            <variable symbol="x_{110}"/>
                            <constant symbol="1"/>
                           </function>
                          </function>
                          <function symbol="+">
                           <constant symbol="1"/>
                           <variable symbol="x_{111}"/>
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                         </function>
                         <function symbol="+">
                          <constant symbol="1"/>
                          <function symbol="+">
                           <function symbol="+">
                            <variable symbol="x_{110}"/>
                            <constant symbol="1"/>
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                           <function symbol="+">
                            <constant symbol="1"/>
                            <variable symbol="x_{111}"/>
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                         <constantatomformula symbol="A" definition="A"/>
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                         <quantifiedformula type="exists">
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                            <constantatomformula symbol="=">
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                              <variable symbol="p"/>
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                             <function symbol="f">
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                         <constantatomformula symbol="=">
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                            <function symbol="+">
                             <variable symbol="x_{110}"/>
                             <constant symbol="1"/>
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                            <function symbol="+">
                             <constant symbol="1"/>
                             <variable symbol="x_{111}"/>
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                           </function>
                          </function>
                          <function symbol="+">
                           <function symbol="+">
                            <constant symbol="1"/>
                            <function symbol="+">
                             <variable symbol="x_{110}"/>
                             <constant symbol="1"/>
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                           </function>
                           <function symbol="+">
                            <constant symbol="1"/>
                            <variable symbol="x_{111}"/>
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                          <constantatomformula symbol="A" definition="A"/>
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                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="+">
                            <constant symbol="1"/>
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                             <function symbol="+">
                              <variable symbol="x_{110}"/>
                              <constant symbol="1"/>
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                             <function symbol="+">
                              <constant symbol="1"/>
                              <variable symbol="x_{111}"/>
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                           </function>
                           <function symbol="+">
                            <function symbol="+">
                             <constant symbol="1"/>
                             <function symbol="+">
                              <variable symbol="x_{110}"/>
                              <constant symbol="1"/>
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                            <function symbol="+">
                             <constant symbol="1"/>
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                          <quantifiedformula type="exists">
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                             <constantatomformula symbol="=">
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                          <formulalist>
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                          <formulalist>
                           <constantatomformula symbol="=">
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                              <function symbol="+">
                               <variable symbol="x_{110}"/>
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                              <function symbol="+">
                               <constant symbol="1"/>
                               <variable symbol="x_{111}"/>
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                             </function>
                            </function>
                            <function symbol="+">
                             <function symbol="+">
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                              <function symbol="+">
                               <variable symbol="x_{110}"/>
                               <constant symbol="1"/>
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                             <function symbol="+">
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                           <formulalist>
                            <constantatomformula symbol="=">
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                               <function symbol="+">
                                <variable symbol="x_{110}"/>
                                <constant symbol="1"/>
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                               <function symbol="+">
                                <constant symbol="1"/>
                                <variable symbol="x_{111}"/>
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                              </function>
                             </function>
                             <function symbol="+">
                              <function symbol="+">
                               <constant symbol="1"/>
                               <function symbol="+">
                                <variable symbol="x_{110}"/>
                                <constant symbol="1"/>
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                              </function>
                              <function symbol="+">
                               <constant symbol="1"/>
                               <variable symbol="x_{111}"/>
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                          <rule type="axiom" symbol="axiom">
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                            <formulalist/>
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                             <constantatomformula symbol="=">
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                                <function symbol="+">
                                 <variable symbol="x_{110}"/>
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                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <variable symbol="x_{111}"/>
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                              </function>
                              <function symbol="+">
                               <function symbol="+">
                                <constant symbol="1"/>
                                <function symbol="+">
                                 <variable symbol="x_{110}"/>
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                               <function symbol="+">
                                <constant symbol="1"/>
                                <variable symbol="x_{111}"/>
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                         <constantatomformula symbol="=">
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                            <function symbol="+">
                             <constant symbol="1"/>
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                           </function>
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                          <function symbol="+">
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                           <function symbol="+">
                            <function symbol="+">
                             <variable symbol="x_{110}"/>
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                            <function symbol="+">
                             <constant symbol="1"/>
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                          <constantatomformula symbol="A" definition="A"/>
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                         <formulalist>
                          <constantatomformula symbol="=">
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                             <function symbol="+">
                              <variable symbol="x_{110}"/>
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                             <function symbol="+">
                              <constant symbol="1"/>
                              <variable symbol="x_{111}"/>
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                           <function symbol="+">
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                            <function symbol="+">
                             <function symbol="+">
                              <variable symbol="x_{110}"/>
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                             <function symbol="+">
                              <constant symbol="1"/>
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                             <constantatomformula symbol="=">
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                              <function symbol="f">
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                          <formulalist>
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                              <function symbol="+">
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                              <function symbol="+">
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                               <variable symbol="x_{111}"/>
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                            <function symbol="+">
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                             <function symbol="+">
                              <function symbol="+">
                               <variable symbol="x_{110}"/>
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                              <function symbol="+">
                               <constant symbol="1"/>
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                               <function symbol="+">
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                             <function symbol="+">
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                               <function symbol="+">
                                <variable symbol="x_{110}"/>
                                <constant symbol="1"/>
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                               <function symbol="+">
                                <constant symbol="1"/>
                                <variable symbol="x_{111}"/>
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                            <formulalist/>
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                             <constantatomformula symbol="=">
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                                <function symbol="+">
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                              <function symbol="+">
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                               <function symbol="+">
                                <function symbol="+">
                                 <variable symbol="x_{110}"/>
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                                <function symbol="+">
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                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
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                                         <function symbol="+">
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <function symbol="+">
                                            <variable symbol="x_{110}"/>
                                            <constant symbol="1"/>
                                           </function>
                                          </function>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <variable symbol="x_{111}"/>
                                          </function>
                                         </function>
                                         <variable symbol="x_{109}"/>
                                        </function>
                                       </function>
                                       <constant symbol="0"/>
                                      </constantatomformula>
                                      <constantatomformula symbol="=">
                                       <function symbol="f">
                                        <function symbol="+">
                                         <variable symbol="x_{109}"/>
                                         <function symbol="+">
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <function symbol="+">
                                            <variable symbol="x_{110}"/>
                                            <constant symbol="1"/>
                                           </function>
                                          </function>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <variable symbol="x_{111}"/>
                                          </function>
                                         </function>
                                        </function>
                                       </function>
                                       <constant symbol="1"/>
                                      </constantatomformula>
                                     </formulalist>
                                    </sequent>
                                    <rule type="foralll" symbol="\all:l">
                                     <sequent>
                                      <formulalist>
                                       <quantifiedformula type="all">
                                        <variable symbol="x"/>
                                        <conjunctiveformula type="or">
                                         <constantatomformula symbol="=">
                                          <function symbol="f">
                                           <variable symbol="x"/>
                                          </function>
                                          <constant symbol="0"/>
                                         </constantatomformula>
                                         <constantatomformula symbol="=">
                                          <function symbol="f">
                                           <variable symbol="x"/>
                                          </function>
                                          <constant symbol="1"/>
                                         </constantatomformula>
                                        </conjunctiveformula>
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                                      </formulalist>
                                      <formulalist>
                                       <constantatomformula symbol="=">
                                        <function symbol="f">
                                         <function symbol="+">
                                          <function symbol="+">
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
                                             <variable symbol="x_{110}"/>
                                             <constant symbol="1"/>
                                            </function>
                                           </function>
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <variable symbol="x_{111}"/>
                                           </function>
                                          </function>
                                          <variable symbol="x_{109}"/>
                                         </function>
                                        </function>
                                        <constant symbol="0"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="=">
                                        <function symbol="f">
                                         <function symbol="+">
                                          <variable symbol="x_{109}"/>
                                          <function symbol="+">
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
                                             <variable symbol="x_{110}"/>
                                             <constant symbol="1"/>
                                            </function>
                                           </function>
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <variable symbol="x_{111}"/>
                                           </function>
                                          </function>
                                         </function>
                                        </function>
                                        <constant symbol="1"/>
                                       </constantatomformula>
                                      </formulalist>
                                     </sequent>
                                     <rule type="orl" symbol="\or:l">
                                      <sequent>
                                       <formulalist>
                                        <conjunctiveformula type="or">
                                         <constantatomformula symbol="=">
                                          <function symbol="f">
                                           <function symbol="+">
                                            <function symbol="+">
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <variable symbol="x_{111}"/>
                                             </function>
                                            </function>
                                            <variable symbol="x_{109}"/>
                                           </function>
                                          </function>
                                          <constant symbol="0"/>
                                         </constantatomformula>
                                         <constantatomformula symbol="=">
                                          <function symbol="f">
                                           <function symbol="+">
                                            <function symbol="+">
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <variable symbol="x_{111}"/>
                                             </function>
                                            </function>
                                            <variable symbol="x_{109}"/>
                                           </function>
                                          </function>
                                          <constant symbol="1"/>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                       </formulalist>
                                       <formulalist>
                                        <constantatomformula symbol="=">
                                         <function symbol="f">
                                          <function symbol="+">
                                           <function symbol="+">
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <variable symbol="x_{110}"/>
                                              <constant symbol="1"/>
                                             </function>
                                            </function>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <variable symbol="x_{111}"/>
                                            </function>
                                           </function>
                                           <variable symbol="x_{109}"/>
                                          </function>
                                         </function>
                                         <constant symbol="0"/>
                                        </constantatomformula>
                                        <constantatomformula symbol="=">
                                         <function symbol="f">
                                          <function symbol="+">
                                           <variable symbol="x_{109}"/>
                                           <function symbol="+">
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <variable symbol="x_{110}"/>
                                              <constant symbol="1"/>
                                             </function>
                                            </function>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <variable symbol="x_{111}"/>
                                            </function>
                                           </function>
                                          </function>
                                         </function>
                                         <constant symbol="1"/>
                                        </constantatomformula>
                                       </formulalist>
                                      </sequent>
                                      <rule type="axiom" symbol="axiom">
                                       <sequent>
                                        <formulalist>
                                         <constantatomformula symbol="=">
                                          <function symbol="f">
                                           <function symbol="+">
                                            <function symbol="+">
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <variable symbol="x_{111}"/>
                                             </function>
                                            </function>
                                            <variable symbol="x_{109}"/>
                                           </function>
                                          </function>
                                          <constant symbol="0"/>
                                         </constantatomformula>
                                        </formulalist>
                                        <formulalist>
                                         <constantatomformula symbol="=">
                                          <function symbol="f">
                                           <function symbol="+">
                                            <function symbol="+">
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <variable symbol="x_{111}"/>
                                             </function>
                                            </function>
                                            <variable symbol="x_{109}"/>
                                           </function>
                                          </function>
                                          <constant symbol="0"/>
                                         </constantatomformula>
                                        </formulalist>
                                       </sequent>
                                      </rule>
                                      <rule type="eql2" symbol="=:l2" param="(1 1)">
                                       <sequent>
                                        <formulalist>
                                         <constantatomformula symbol="=">
                                          <function symbol="f">
                                           <function symbol="+">
                                            <function symbol="+">
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <variable symbol="x_{111}"/>
                                             </function>
                                            </function>
                                            <variable symbol="x_{109}"/>
                                           </function>
                                          </function>
                                          <constant symbol="1"/>
                                         </constantatomformula>
                                        </formulalist>
                                        <formulalist>
                                         <constantatomformula symbol="=">
                                          <function symbol="f">
                                           <function symbol="+">
                                            <variable symbol="x_{109}"/>
                                            <function symbol="+">
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <variable symbol="x_{111}"/>
                                             </function>
                                            </function>
                                           </function>
                                          </function>
                                          <constant symbol="1"/>
                                         </constantatomformula>
                                        </formulalist>
                                       </sequent>
                                       <rule type="axiom" symbol="axiom">
                                        <sequent>
                                         <formulalist/>
                                         <formulalist>
                                          <constantatomformula symbol="=">
                                           <function symbol="+">
                                            <function symbol="+">
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <variable symbol="x_{111}"/>
                                             </function>
                                            </function>
                                            <variable symbol="x_{109}"/>
                                           </function>
                                           <function symbol="+">
                                            <variable symbol="x_{109}"/>
                                            <function symbol="+">
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <variable symbol="x_{111}"/>
                                             </function>
                                            </function>
                                           </function>
                                          </constantatomformula>
                                         </formulalist>
                                        </sequent>
                                       </rule>
                                       <rule type="axiom" symbol="axiom">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="=">
                                           <function symbol="f">
                                            <function symbol="+">
                                             <variable symbol="x_{109}"/>
                                             <function symbol="+">
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <variable symbol="x_{110}"/>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <variable symbol="x_{111}"/>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                           <constant symbol="1"/>
                                          </constantatomformula>
                                         </formulalist>
                                         <formulalist>
                                          <constantatomformula symbol="=">
                                           <function symbol="f">
                                            <function symbol="+">
                                             <variable symbol="x_{109}"/>
                                             <function symbol="+">
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <variable symbol="x_{110}"/>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <variable symbol="x_{111}"/>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                           <constant symbol="1"/>
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                                         </formulalist>
                                        </sequent>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                          </rule>
                         </rule>
                        </rule>
                       </rule>
                      </rule>
                      <rule type="contrr" symbol="c:r" param="2">
                       <sequent>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="f">
                           <function symbol="+">
                            <variable symbol="x_{109}"/>
                            <function symbol="+">
                             <function symbol="+">
                              <constant symbol="1"/>
                              <function symbol="+">
                               <variable symbol="x_{110}"/>
                               <constant symbol="1"/>
                              </function>
                             </function>
                             <function symbol="+">
                              <constant symbol="1"/>
                              <variable symbol="x_{111}"/>
                             </function>
                            </function>
                           </function>
                          </function>
                          <constant symbol="0"/>
                         </constantatomformula>
                         <constantatomformula symbol="A" definition="A"/>
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                        <formulalist>
                         <quantifiedformula type="exists">
                          <variable symbol="p"/>
                          <quantifiedformula type="exists">
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                             <constantatomformula symbol="=">
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                            <constantatomformula symbol="=">
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                             <function symbol="f">
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                            </constantatomformula>
                           </conjunctiveformula>
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                       <rule type="permr" symbol="\pi:r" param="">
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                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="f">
                            <function symbol="+">
                             <variable symbol="x_{109}"/>
                             <function symbol="+">
                              <function symbol="+">
                               <constant symbol="1"/>
                               <function symbol="+">
                                <variable symbol="x_{110}"/>
                                <constant symbol="1"/>
                               </function>
                              </function>
                              <function symbol="+">
                               <constant symbol="1"/>
                               <variable symbol="x_{111}"/>
                              </function>
                             </function>
                            </function>
                           </function>
                           <constant symbol="0"/>
                          </constantatomformula>
                          <constantatomformula symbol="A" definition="A"/>
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                         <formulalist>
                          <quantifiedformula type="exists">
                           <variable symbol="p"/>
                           <quantifiedformula type="exists">
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                               <variable symbol="q"/>
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                             <constantatomformula symbol="=">
                              <function symbol="f">
                               <variable symbol="p"/>
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                              <function symbol="f">
                               <variable symbol="q"/>
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                             </constantatomformula>
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                           </quantifiedformula>
                          </quantifiedformula>
                          <quantifiedformula type="exists">
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                              <constantatomformula symbol="=">
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                               <variable symbol="q"/>
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                             </conjunctiveformula>
                             <constantatomformula symbol="=">
                              <function symbol="f">
                               <variable symbol="p"/>
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                              <function symbol="f">
                               <variable symbol="q"/>
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                             </constantatomformula>
                            </conjunctiveformula>
                           </quantifiedformula>
                          </quantifiedformula>
                         </formulalist>
                        </sequent>
                        <rule type="contrl" symbol="c:l" param="1,2">
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                          <formulalist>
                           <constantatomformula symbol="=">
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                             <function symbol="+">
                              <variable symbol="x_{109}"/>
                              <function symbol="+">
                               <function symbol="+">
                                <constant symbol="1"/>
                                <function symbol="+">
                                 <variable symbol="x_{110}"/>
                                 <constant symbol="1"/>
                                </function>
                               </function>
                               <function symbol="+">
                                <constant symbol="1"/>
                                <variable symbol="x_{111}"/>
                               </function>
                              </function>
                             </function>
                            </function>
                            <constant symbol="0"/>
                           </constantatomformula>
                           <constantatomformula symbol="A" definition="A"/>
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                          <formulalist>
                           <quantifiedformula type="exists">
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                            <quantifiedformula type="exists">
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                               <constantatomformula symbol="=">
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                              <constantatomformula symbol="=">
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                                <variable symbol="p"/>
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                               <function symbol="f">
                                <variable symbol="q"/>
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                              </constantatomformula>
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                           </quantifiedformula>
                           <quantifiedformula type="exists">
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                            <quantifiedformula type="exists">
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                               <constantatomformula symbol="=">
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                              </conjunctiveformula>
                              <constantatomformula symbol="=">
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                                <variable symbol="p"/>
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                               <function symbol="f">
                                <variable symbol="q"/>
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                              </constantatomformula>
                             </conjunctiveformula>
                            </quantifiedformula>
                           </quantifiedformula>
                          </formulalist>
                         </sequent>
                         <rule type="perml" symbol="\pi:l" param="(1 2)">
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                           <formulalist>
                            <constantatomformula symbol="=">
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                              <function symbol="+">
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                               <function symbol="+">
                                <function symbol="+">
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                                 <function symbol="+">
                                  <variable symbol="x_{110}"/>
                                  <constant symbol="1"/>
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                                </function>
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <variable symbol="x_{111}"/>
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                               </function>
                              </function>
                             </function>
                             <constant symbol="0"/>
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                            <constantatomformula symbol="A" definition="A"/>
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                           <formulalist>
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                               <constantatomformula symbol="=">
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                            </quantifiedformula>
                            <quantifiedformula type="exists">
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                                <constantatomformula symbol="=">
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                               <constantatomformula symbol="=">
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                                 <variable symbol="p"/>
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                                <function symbol="f">
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                               </constantatomformula>
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                             </quantifiedformula>
                            </quantifiedformula>
                           </formulalist>
                          </sequent>
                          <rule type="cut" symbol="cut">
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                            <formulalist>
                             <constantatomformula symbol="A" definition="A"/>
                             <constantatomformula symbol="=">
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                               <function symbol="+">
                                <variable symbol="x_{109}"/>
                                <function symbol="+">
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <function symbol="+">
                                   <variable symbol="x_{110}"/>
                                   <constant symbol="1"/>
                                  </function>
                                 </function>
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <variable symbol="x_{111}"/>
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                                </function>
                               </function>
                              </function>
                              <constant symbol="0"/>
                             </constantatomformula>
                             <constantatomformula symbol="A" definition="A"/>
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                            <formulalist>
                             <quantifiedformula type="exists">
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                              <quantifiedformula type="exists">
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                                <constantatomformula symbol="=">
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                             </quantifiedformula>
                             <quantifiedformula type="exists">
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                                <constantatomformula symbol="=">
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                                  <variable symbol="p"/>
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                                 <function symbol="f">
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                                </constantatomformula>
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                             </quantifiedformula>
                            </formulalist>
                           </sequent>
                           <rule type="contrr" symbol="c:r" param="1,2">
                            <sequent>
                             <formulalist>
                              <constantatomformula symbol="A" definition="A"/>
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                             <formulalist>
                              <quantifiedformula type="exists">
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                                 <constantatomformula symbol="=">
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                                   <variable symbol="p"/>
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                                  <function symbol="f">
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                              </quantifiedformula>
                              <constantatomformula symbol="=">
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                                 <variable symbol="x_{109}"/>
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <function symbol="+">
                                   <variable symbol="x_{110}"/>
                                   <constant symbol="1"/>
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                                </function>
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                               <constant symbol="0"/>
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                            <rule type="permr" symbol="\pi:r" param="(2 3)">
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                              <formulalist>
                               <constantatomformula symbol="A" definition="A"/>
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                              <formulalist>
                               <quantifiedformula type="exists">
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                                   <constantatomformula symbol="=">
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                                    <variable symbol="q"/>
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                                  <constantatomformula symbol="=">
                                   <function symbol="f">
                                    <variable symbol="p"/>
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                                   <function symbol="f">
                                    <variable symbol="q"/>
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                                  </constantatomformula>
                                 </conjunctiveformula>
                                </quantifiedformula>
                               </quantifiedformula>
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                                             <function symbol="+">
                                              <variable symbol="x_{110}"/>
                                              <constant symbol="1"/>
                                             </function>
                                            </function>
                                           </function>
                                          </function>
                                          <constant symbol="0"/>
                                         </constantatomformula>
                                         <constantatomformula symbol="=">
                                          <function symbol="f">
                                           <function symbol="+">
                                            <variable symbol="x_{109}"/>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <variable symbol="x_{110}"/>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                          </function>
                                          <constant symbol="0"/>
                                         </constantatomformula>
                                        </formulalist>
                                       </sequent>
                                       <rule type="contrr" symbol="c:r" param="2,1">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="A" definition="A"/>
                                         </formulalist>
                                         <formulalist>
                                          <quantifiedformula type="exists">
                                           <variable symbol="p"/>
                                           <quantifiedformula type="exists">
                                            <variable symbol="q"/>
                                            <conjunctiveformula type="and">
                                             <conjunctiveformula type="neg">
                                              <constantatomformula symbol="=">
                                               <variable symbol="p"/>
                                               <variable symbol="q"/>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <variable symbol="p"/>
                                              </function>
                                              <function symbol="f">
                                               <variable symbol="q"/>
                                              </function>
                                             </constantatomformula>
                                            </conjunctiveformula>
                                           </quantifiedformula>
                                          </quantifiedformula>
                                          <constantatomformula symbol="=">
                                           <function symbol="+">
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <constant symbol="1"/>
                                            </function>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <variable symbol="x_{110}"/>
                                              <constant symbol="1"/>
                                             </function>
                                            </function>
                                           </function>
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                          </constantatomformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="permr" symbol="\pi:r" param="">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="A" definition="A"/>
                                          </formulalist>
                                          <formulalist>
                                           <quantifiedformula type="exists">
                                            <variable symbol="p"/>
                                            <quantifiedformula type="exists">
                                             <variable symbol="q"/>
                                             <conjunctiveformula type="and">
                                              <conjunctiveformula type="neg">
                                               <constantatomformula symbol="=">
                                                <variable symbol="p"/>
                                                <variable symbol="q"/>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <variable symbol="p"/>
                                               </function>
                                               <function symbol="f">
                                                <variable symbol="q"/>
                                               </function>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                            </quantifiedformula>
                                           </quantifiedformula>
                                           <quantifiedformula type="exists">
                                            <variable symbol="p"/>
                                            <quantifiedformula type="exists">
                                             <variable symbol="q"/>
                                             <conjunctiveformula type="and">
                                              <conjunctiveformula type="neg">
                                               <constantatomformula symbol="=">
                                                <variable symbol="p"/>
                                                <variable symbol="q"/>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <variable symbol="p"/>
                                               </function>
                                               <function symbol="f">
                                                <variable symbol="q"/>
                                               </function>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                            </quantifiedformula>
                                           </quantifiedformula>
                                           <constantatomformula symbol="=">
                                            <function symbol="+">
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <constant symbol="1"/>
                                             </function>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                            </function>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <variable symbol="x_{110}"/>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="contrl" symbol="c:l" param="2">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="A" definition="A"/>
                                           </formulalist>
                                           <formulalist>
                                            <quantifiedformula type="exists">
                                             <variable symbol="p"/>
                                             <quantifiedformula type="exists">
                                              <variable symbol="q"/>
                                              <conjunctiveformula type="and">
                                               <conjunctiveformula type="neg">
                                                <constantatomformula symbol="=">
                                                 <variable symbol="p"/>
                                                 <variable symbol="q"/>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <variable symbol="p"/>
                                                </function>
                                                <function symbol="f">
                                                 <variable symbol="q"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                             </quantifiedformula>
                                            </quantifiedformula>
                                            <quantifiedformula type="exists">
                                             <variable symbol="p"/>
                                             <quantifiedformula type="exists">
                                              <variable symbol="q"/>
                                              <conjunctiveformula type="and">
                                               <conjunctiveformula type="neg">
                                                <constantatomformula symbol="=">
                                                 <variable symbol="p"/>
                                                 <variable symbol="q"/>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <variable symbol="p"/>
                                                </function>
                                                <function symbol="f">
                                                 <variable symbol="q"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                             </quantifiedformula>
                                            </quantifiedformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="+">
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <constant symbol="1"/>
                                              </function>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <variable symbol="x_{110}"/>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                             </function>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <variable symbol="x_{110}"/>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="perml" symbol="\pi:l" param="">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="A" definition="A"/>
                                             <constantatomformula symbol="A" definition="A"/>
                                            </formulalist>
                                            <formulalist>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="+">
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <constant symbol="1"/>
                                               </function>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <variable symbol="x_{110}"/>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                              </function>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="eqr1" symbol="=:r1" param="(1)">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="A" definition="A"/>
                                              <constantatomformula symbol="A" definition="A"/>
                                             </formulalist>
                                             <formulalist>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <constant symbol="1"/>
                                                </function>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                               </function>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="permr" symbol="\pi:r" param="(1 2)">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="A" definition="A"/>
                                              </formulalist>
                                              <formulalist>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="weakr" symbol="w:r">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </constantatomformula>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="perml" symbol="\pi:l" param="">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="weakl" symbol="w:l">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="axiom" symbol="axiom">
                                                 <sequent>
                                                  <formulalist/>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                            <rule type="permr" symbol="\pi:r" param="(1 2)">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="A" definition="A"/>
                                              </formulalist>
                                              <formulalist>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="weakr" symbol="w:r">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </constantatomformula>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="perml" symbol="\pi:l" param="">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="weakl" symbol="w:l">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="axiom" symbol="axiom">
                                                 <sequent>
                                                  <formulalist/>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                       <rule type="contrr" symbol="c:r" param="2,1,1">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="A" definition="A"/>
                                         </formulalist>
                                         <formulalist>
                                          <quantifiedformula type="exists">
                                           <variable symbol="p"/>
                                           <quantifiedformula type="exists">
                                            <variable symbol="q"/>
                                            <conjunctiveformula type="and">
                                             <conjunctiveformula type="neg">
                                              <constantatomformula symbol="=">
                                               <variable symbol="p"/>
                                               <variable symbol="q"/>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <variable symbol="p"/>
                                              </function>
                                              <function symbol="f">
                                               <variable symbol="q"/>
                                              </function>
                                             </constantatomformula>
                                            </conjunctiveformula>
                                           </quantifiedformula>
                                          </quantifiedformula>
                                          <constantatomformula symbol="=">
                                           <function symbol="f">
                                            <function symbol="+">
                                             <variable symbol="x_{109}"/>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                           <constant symbol="0"/>
                                          </constantatomformula>
                                          <constantatomformula symbol="=">
                                           <function symbol="f">
                                            <function symbol="+">
                                             <variable symbol="x_{109}"/>
                                             <function symbol="+">
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <constant symbol="1"/>
                                              </function>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <variable symbol="x_{110}"/>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                           <constant symbol="0"/>
                                          </constantatomformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="permr" symbol="\pi:r" param="">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="A" definition="A"/>
                                          </formulalist>
                                          <formulalist>
                                           <quantifiedformula type="exists">
                                            <variable symbol="p"/>
                                            <quantifiedformula type="exists">
                                             <variable symbol="q"/>
                                             <conjunctiveformula type="and">
                                              <conjunctiveformula type="neg">
                                               <constantatomformula symbol="=">
                                                <variable symbol="p"/>
                                                <variable symbol="q"/>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <variable symbol="p"/>
                                               </function>
                                               <function symbol="f">
                                                <variable symbol="q"/>
                                               </function>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                            </quantifiedformula>
                                           </quantifiedformula>
                                           <quantifiedformula type="exists">
                                            <variable symbol="p"/>
                                            <quantifiedformula type="exists">
                                             <variable symbol="q"/>
                                             <conjunctiveformula type="and">
                                              <conjunctiveformula type="neg">
                                               <constantatomformula symbol="=">
                                                <variable symbol="p"/>
                                                <variable symbol="q"/>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <variable symbol="p"/>
                                               </function>
                                               <function symbol="f">
                                                <variable symbol="q"/>
                                               </function>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                            </quantifiedformula>
                                           </quantifiedformula>
                                           <constantatomformula symbol="=">
                                            <function symbol="f">
                                             <function symbol="+">
                                              <variable symbol="x_{109}"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <variable symbol="x_{110}"/>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                            <constant symbol="0"/>
                                           </constantatomformula>
                                           <constantatomformula symbol="=">
                                            <function symbol="f">
                                             <function symbol="+">
                                              <variable symbol="x_{109}"/>
                                              <function symbol="+">
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <constant symbol="1"/>
                                               </function>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <variable symbol="x_{110}"/>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                            <constant symbol="0"/>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="contrl" symbol="c:l" param="2">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="A" definition="A"/>
                                           </formulalist>
                                           <formulalist>
                                            <quantifiedformula type="exists">
                                             <variable symbol="p"/>
                                             <quantifiedformula type="exists">
                                              <variable symbol="q"/>
                                              <conjunctiveformula type="and">
                                               <conjunctiveformula type="neg">
                                                <constantatomformula symbol="=">
                                                 <variable symbol="p"/>
                                                 <variable symbol="q"/>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <variable symbol="p"/>
                                                </function>
                                                <function symbol="f">
                                                 <variable symbol="q"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                             </quantifiedformula>
                                            </quantifiedformula>
                                            <quantifiedformula type="exists">
                                             <variable symbol="p"/>
                                             <quantifiedformula type="exists">
                                              <variable symbol="q"/>
                                              <conjunctiveformula type="and">
                                               <conjunctiveformula type="neg">
                                                <constantatomformula symbol="=">
                                                 <variable symbol="p"/>
                                                 <variable symbol="q"/>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <variable symbol="p"/>
                                                </function>
                                                <function symbol="f">
                                                 <variable symbol="q"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                             </quantifiedformula>
                                            </quantifiedformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="f">
                                              <function symbol="+">
                                               <variable symbol="x_{109}"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <variable symbol="x_{110}"/>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                             <constant symbol="0"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="f">
                                              <function symbol="+">
                                               <variable symbol="x_{109}"/>
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <constant symbol="1"/>
                                                </function>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                             <constant symbol="0"/>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="perml" symbol="\pi:l" param="">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="A" definition="A"/>
                                             <constantatomformula symbol="A" definition="A"/>
                                            </formulalist>
                                            <formulalist>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <function symbol="+">
                                                <variable symbol="x_{109}"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                              <constant symbol="0"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <function symbol="+">
                                                <variable symbol="x_{109}"/>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                              <constant symbol="0"/>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="eqr1" symbol="=:r1" param="(1 1)">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="A" definition="A"/>
                                              <constantatomformula symbol="A" definition="A"/>
                                             </formulalist>
                                             <formulalist>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <variable symbol="x_{109}"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                               <constant symbol="0"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <variable symbol="x_{109}"/>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                               <constant symbol="0"/>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="contrr" symbol="c:r" param="2,1">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="A" definition="A"/>
                                              </formulalist>
                                              <formulalist>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <variable symbol="x_{109}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                </function>
                                                <function symbol="+">
                                                 <variable symbol="x_{109}"/>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="permr" symbol="\pi:r" param="">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <variable symbol="x_{109}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="contrl" symbol="c:l" param="2">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="perml" symbol="\pi:l" param="">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="eqr1" symbol="=:r1" param="(1)">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="weakr" symbol="w:r">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="perml" symbol="\pi:l" param="">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="weakl" symbol="w:l">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="axiom" symbol="axiom">
                                                      <sequent>
                                                       <formulalist/>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                 <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="weakr" symbol="w:r">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="perml" symbol="\pi:l" param="">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="weakl" symbol="w:l">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="axiom" symbol="axiom">
                                                      <sequent>
                                                       <formulalist/>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                            <rule type="permr" symbol="\pi:r" param="(2 3)">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="A" definition="A"/>
                                              </formulalist>
                                              <formulalist>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <variable symbol="x_{109}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <constant symbol="0"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <constant symbol="0"/>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="contrr" symbol="c:r" param="2,1,1">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="permr" symbol="\pi:r" param="(2 3)">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="contrl" symbol="c:l" param="2">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="perml" symbol="\pi:l" param="">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="cut" symbol="cut">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="permr" symbol="\pi:r" param="(1 3 2)">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
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                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
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                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
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                                                         <function symbol="f">
                                                          <variable symbol="q"/>
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                                                        </constantatomformula>
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                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
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                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="perml" symbol="\pi:l" param="">
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                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
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                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
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                                                          <constantatomformula symbol="=">
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                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
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                                                          <function symbol="f">
                                                           <variable symbol="q"/>
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                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="contrr" symbol="c:r" param="1,1,2">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="0"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </constantatomformula>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
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                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
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                                                           <function symbol="f">
                                                            <variable symbol="q"/>
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                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="weakr" symbol="w:r">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="0"/>
                                                        </constantatomformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </constantatomformula>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
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                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
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                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="weakr" symbol="w:r">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </constantatomformula>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
                                                            <conjunctiveformula type="neg">
                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="defl" symbol="d:l" param="();[()];()">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="0"/>
                                                          </constantatomformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="foralll" symbol="\all:l">
                                                         <sequent>
                                                          <formulalist>
                                                           <quantifiedformula type="all">
                                                            <variable symbol="x"/>
                                                            <conjunctiveformula type="or">
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="x"/>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="x"/>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </formulalist>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="0"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="orl" symbol="\or:l">
                                                          <sequent>
                                                           <formulalist>
                                                            <conjunctiveformula type="or">
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </formulalist>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="0"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="axiom" symbol="axiom">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                            </formulalist>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                          </rule>
                                                          <rule type="eql2" symbol="=:l2" param="(1 1)">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </constantatomformula>
                                                            </formulalist>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="axiom" symbol="axiom">
                                                            <sequent>
                                                             <formulalist/>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                           </rule>
                                                           <rule type="axiom" symbol="axiom">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </constantatomformula>
                                                             </formulalist>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                  <rule type="contrr" symbol="c:r" param="2,1">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="permr" symbol="\pi:r" param="(2 3)">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="contrl" symbol="c:l" param="1,2">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="0"/>
                                                       </constantatomformula>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </constantatomformula>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="0"/>
                                                        </constantatomformula>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="cut" symbol="cut">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </constantatomformula>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
                                                            <conjunctiveformula type="neg">
                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
                                                            <conjunctiveformula type="neg">
                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="permr" symbol="\pi:r" param="(1 3 2)">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
                                                            <conjunctiveformula type="and">
                                                             <conjunctiveformula type="neg">
                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="0"/>
                                                          </constantatomformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="perml" symbol="\pi:l" param="">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="0"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </constantatomformula>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="p"/>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="q"/>
                                                             <conjunctiveformula type="and">
                                                              <conjunctiveformula type="neg">
                                                               <constantatomformula symbol="=">
                                                                <variable symbol="p"/>
                                                                <variable symbol="q"/>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="contrr" symbol="c:r" param="1,1,2">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="0"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </constantatomformula>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="p"/>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="q"/>
                                                              <conjunctiveformula type="and">
                                                               <conjunctiveformula type="neg">
                                                                <constantatomformula symbol="=">
                                                                 <variable symbol="p"/>
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                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <variable symbol="p"/>
                                                                </function>
                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </quantifiedformula>
                                                            </quantifiedformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="weakr" symbol="w:r">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="A" definition="A"/>
                                                            </formulalist>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </constantatomformula>
                                                             <quantifiedformula type="exists">
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                                                              <quantifiedformula type="exists">
                                                               <variable symbol="q"/>
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                                                                 <constantatomformula symbol="=">
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                                                                <constantatomformula symbol="=">
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                                                                  <variable symbol="p"/>
                                                                 </function>
                                                                 <function symbol="f">
                                                                  <variable symbol="q"/>
                                                                 </function>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                              </quantifiedformula>
                                                             </quantifiedformula>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="p"/>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="q"/>
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                                                                 <constantatomformula symbol="=">
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                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <variable symbol="p"/>
                                                                 </function>
                                                                 <function symbol="f">
                                                                  <variable symbol="q"/>
                                                                 </function>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                              </quantifiedformula>
                                                             </quantifiedformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="weakr" symbol="w:r">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="A" definition="A"/>
                                                             </formulalist>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="0"/>
                                                              </constantatomformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
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                                                              <quantifiedformula type="exists">
                                                               <variable symbol="p"/>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="q"/>
                                                                <conjunctiveformula type="and">
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                                                                  <constantatomformula symbol="=">
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                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                 <constantatomformula symbol="=">
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                                                                  </function>
                                                                  <function symbol="f">
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                                                                  </function>
                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                               </quantifiedformula>
                                                              </quantifiedformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="defl" symbol="d:l" param="();[()];()">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="A" definition="A"/>
                                                              </formulalist>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="0"/>
                                                               </constantatomformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
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                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <variable symbol="x_{109}"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </constantatomformula>
                                                              </formulalist>
                                                             </sequent>
                                                             <rule type="foralll" symbol="\all:l">
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                                                               <formulalist>
                                                                <quantifiedformula type="all">
                                                                 <variable symbol="x"/>
                                                                 <conjunctiveformula type="or">
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <variable symbol="x"/>
                                                                   </function>
                                                                   <constant symbol="0"/>
                                                                  </constantatomformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <variable symbol="x"/>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                </quantifiedformula>
                                                               </formulalist>
                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{109}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="0"/>
                                                                </constantatomformula>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </constantatomformula>
                                                               </formulalist>
                                                              </sequent>
                                                              <rule type="orl" symbol="\or:l">
                                                               <sequent>
                                                                <formulalist>
                                                                 <conjunctiveformula type="or">
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{109}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="0"/>
                                                                  </constantatomformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{109}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                </formulalist>
                                                                <formulalist>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{109}"/>
                                                                    <function symbol="+">
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                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="0"/>
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                                                                 <constantatomformula symbol="=">
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                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
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                                                                </formulalist>
                                                               </sequent>
                                                               <rule type="axiom" symbol="axiom">
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                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
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                                                                     <variable symbol="x_{109}"/>
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                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="0"/>
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                                                                 </formulalist>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
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                                                                     <function symbol="+">
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                                                                       <variable symbol="x_{110}"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="0"/>
                                                                  </constantatomformula>
                                                                 </formulalist>
                                                                </sequent>
                                                               </rule>
                                                               <rule type="eql2" symbol="=:l2" param="(1 1)">
                                                                <sequent>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
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                                                                     <variable symbol="x_{109}"/>
                                                                     <function symbol="+">
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                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
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                                                                 </formulalist>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
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                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </constantatomformula>
                                                                 </formulalist>
                                                                </sequent>
                                                                <rule type="axiom" symbol="axiom">
                                                                 <sequent>
                                                                  <formulalist/>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{109}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                </rule>
                                                                <rule type="axiom" symbol="axiom">
                                                                 <sequent>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
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                                                                  </formulalist>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
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                                                                      <function symbol="+">
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                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
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                                                                  </formulalist>
                                                                 </sequent>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                       <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
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                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </constantatomformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
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                                                             <function symbol="+">
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                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </constantatomformula>
                                                          <constantatomformula symbol="A" definition="A"/>
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                                                         <formulalist>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
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                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
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                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="contrr" symbol="c:r" param="2">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="p"/>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="q"/>
                                                             <conjunctiveformula type="and">
                                                              <conjunctiveformula type="neg">
                                                               <constantatomformula symbol="=">
                                                                <variable symbol="p"/>
                                                                <variable symbol="q"/>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="permr" symbol="\pi:r" param="">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="A" definition="A"/>
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                                                           <formulalist>
                                                            <quantifiedformula type="exists">
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                                                             <quantifiedformula type="exists">
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                                                                <constantatomformula symbol="=">
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                                                               </conjunctiveformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <variable symbol="p"/>
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                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
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                                                               </constantatomformula>
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                                                            <quantifiedformula type="exists">
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                                                             <quantifiedformula type="exists">
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                                                                <constantatomformula symbol="=">
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                                                                 <variable symbol="q"/>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <variable symbol="p"/>
                                                                </function>
                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </quantifiedformula>
                                                            </quantifiedformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="contrl" symbol="c:l" param="1,1,2">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </constantatomformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </constantatomformula>
                                                             <constantatomformula symbol="A" definition="A"/>
                                                            </formulalist>
                                                            <formulalist>
                                                             <quantifiedformula type="exists">
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                                                              <quantifiedformula type="exists">
                                                               <variable symbol="q"/>
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                                                                <constantatomformula symbol="=">
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                                                                 <function symbol="f">
                                                                  <variable symbol="q"/>
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                                                                </constantatomformula>
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                                                                <constantatomformula symbol="=">
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                                                                 <function symbol="f">
                                                                  <variable symbol="q"/>
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                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                              </quantifiedformula>
                                                             </quantifiedformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="perml" symbol="\pi:l" param="(2 3)">
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                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
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                                                                   <constant symbol="1"/>
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                                                                 </function>
                                                                 <function symbol="+">
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                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
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                                                               </function>
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                                                              <constantatomformula symbol="=">
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                                                                 <function symbol="+">
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                                                             <formulalist>
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                                                             </formulalist>
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                                                            <rule type="eql1" symbol="=:l1" param="(1 1 2)">
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                                                               <constantatomformula symbol="=">
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                                                                  <function symbol="+">
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                                                                    <constant symbol="1"/>
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                                                                   <constant symbol="1"/>
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                                                              </sequent>
                                                              <rule type="permr" symbol="\pi:r" param="">
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                                                               </sequent>
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                                                                  <constantatomformula symbol="=">
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                                                                </sequent>
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                                                                  </formulalist>
                                                                 </sequent>
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                                                                   <formulalist>
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                                                                   <formulalist>
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                                                                     </quantifiedformula>
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                                                                        <constantatomformula symbol="=">
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                                                                       <constantatomformula symbol="=">
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                                                                       </constantatomformula>
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                                                                    </quantifiedformula>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{109}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                    </constantatomformula>
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                                                                  </sequent>
                                                                  <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                   <sequent>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="A" definition="A"/>
                                                                    </formulalist>
                                                                    <formulalist>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="p"/>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="q"/>
                                                                       <conjunctiveformula type="and">
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                                                                         <constantatomformula symbol="=">
                                                                          <variable symbol="p"/>
                                                                          <variable symbol="q"/>
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                                                                        </conjunctiveformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <variable symbol="p"/>
                                                                         </function>
                                                                         <function symbol="f">
                                                                          <variable symbol="q"/>
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                                                                        </constantatomformula>
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                                                                     </quantifiedformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <constant symbol="1"/>
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                                                                      </function>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <constant symbol="1"/>
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                                                                       <constant symbol="1"/>
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                                                                     </constantatomformula>
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                                                                   </sequent>
                                                                   <rule type="weakr" symbol="w:r">
                                                                    <sequent>
                                                                     <formulalist>
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                                                                     </formulalist>
                                                                     <formulalist>
                                                                      <constantatomformula symbol="=">
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                                                                        <variable symbol="x_{109}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <constant symbol="1"/>
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                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{109}"/>
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                                                                          <constantatomformula symbol="=">
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                                                                         <constantatomformula symbol="=">
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                                                                           <variable symbol="p"/>
                                                                          </function>
                                                                          <function symbol="f">
                                                                           <variable symbol="q"/>
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                                                                         </constantatomformula>
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                                                                       </quantifiedformula>
                                                                      </quantifiedformula>
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                                                                    <rule type="perml" symbol="\pi:l" param="">
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                                                                      <formulalist>
                                                                       <constantatomformula symbol="A" definition="A"/>
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                                                                      <formulalist>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{109}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <constant symbol="1"/>
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                                                                        </function>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
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                                                                          <constant symbol="1"/>
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                                                                         <constant symbol="1"/>
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                                                                     <rule type="weakl" symbol="w:l">
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                                                                        <constantatomformula symbol="A" definition="A"/>
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                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <constant symbol="1"/>
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                                                                         </function>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
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                                                                          <constant symbol="1"/>
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                                                                      <rule type="axiom" symbol="axiom">
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                                                                        <formulalist/>
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                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <constant symbol="1"/>
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                                                                         </constantatomformula>
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                                                                       </sequent>
                                                                      </rule>
                                                                     </rule>
                                                                    </rule>
                                                                   </rule>
                                                                  </rule>
                                                                  <rule type="permr" symbol="\pi:r" param="(1 2)">
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                                                                     <constantatomformula symbol="A" definition="A"/>
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                                                                    <formulalist>
                                                                     <quantifiedformula type="exists">
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                                                                        <constantatomformula symbol="=">
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                                                                         <function symbol="f">
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                                                                        </constantatomformula>
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                                                                     <constantatomformula symbol="=">
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                                                                       <variable symbol="x_{109}"/>
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                                                                      </function>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <constant symbol="1"/>
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                                                                   </sequent>
                                                                   <rule type="weakr" symbol="w:r">
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                                                                      <constantatomformula symbol="A" definition="A"/>
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                                                                     <formulalist>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <constant symbol="1"/>
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                                                                       </function>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <constant symbol="1"/>
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                                                                         <constantatomformula symbol="=">
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                                                                          <function symbol="f">
                                                                           <variable symbol="q"/>
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                                                                    </sequent>
                                                                    <rule type="perml" symbol="\pi:l" param="">
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                                                                      <formulalist>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{109}"/>
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                                                                          <constant symbol="1"/>
                                                                          <constant symbol="1"/>
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                                                                        <function symbol="+">
                                                                         <variable symbol="x_{109}"/>
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                                                                          <constant symbol="1"/>
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                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
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                                                                           <constant symbol="1"/>
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                                                                           <constant symbol="1"/>
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                                                                      <rule type="axiom" symbol="axiom">
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                                                                         <constantatomformula symbol="=">
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                                                                           <function symbol="+">
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                                                                         <variable symbol="p"/>
                                                                         <variable symbol="q"/>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <variable symbol="p"/>
                                                                        </function>
                                                                        <function symbol="f">
                                                                         <variable symbol="q"/>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </conjunctiveformula>
                                                                     </quantifiedformula>
                                                                    </quantifiedformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                  <rule type="perml" symbol="\pi:l" param="(2 3)">
                                                                   <sequent>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="A" definition="A"/>
                                                                     <constantatomformula symbol="A" definition="A"/>
                                                                    </formulalist>
                                                                    <formulalist>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="p"/>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="q"/>
                                                                       <conjunctiveformula type="and">
                                                                        <conjunctiveformula type="neg">
                                                                         <constantatomformula symbol="=">
                                                                          <variable symbol="p"/>
                                                                          <variable symbol="q"/>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <variable symbol="p"/>
                                                                         </function>
                                                                         <function symbol="f">
                                                                          <variable symbol="q"/>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                      </quantifiedformula>
                                                                     </quantifiedformula>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="p"/>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="q"/>
                                                                       <conjunctiveformula type="and">
                                                                        <conjunctiveformula type="neg">
                                                                         <constantatomformula symbol="=">
                                                                          <variable symbol="p"/>
                                                                          <variable symbol="q"/>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <variable symbol="p"/>
                                                                         </function>
                                                                         <function symbol="f">
                                                                          <variable symbol="q"/>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                      </quantifiedformula>
                                                                     </quantifiedformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                   <rule type="eql1" symbol="=:l1" param="(1 1)">
                                                                    <sequent>
                                                                     <formulalist>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="f">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </constantatomformula>
                                                                      <constantatomformula symbol="A" definition="A"/>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="f">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </constantatomformula>
                                                                      <constantatomformula symbol="A" definition="A"/>
                                                                     </formulalist>
                                                                     <formulalist>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="p"/>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="q"/>
                                                                        <conjunctiveformula type="and">
                                                                         <conjunctiveformula type="neg">
                                                                          <constantatomformula symbol="=">
                                                                           <variable symbol="p"/>
                                                                           <variable symbol="q"/>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <variable symbol="p"/>
                                                                          </function>
                                                                          <function symbol="f">
                                                                           <variable symbol="q"/>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                       </quantifiedformula>
                                                                      </quantifiedformula>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="p"/>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="q"/>
                                                                        <conjunctiveformula type="and">
                                                                         <conjunctiveformula type="neg">
                                                                          <constantatomformula symbol="=">
                                                                           <variable symbol="p"/>
                                                                           <variable symbol="q"/>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <variable symbol="p"/>
                                                                          </function>
                                                                          <function symbol="f">
                                                                           <variable symbol="q"/>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                       </quantifiedformula>
                                                                      </quantifiedformula>
                                                                     </formulalist>
                                                                    </sequent>
                                                                    <rule type="contrr" symbol="c:r" param="2,1">
                                                                     <sequent>
                                                                      <formulalist>
                                                                       <constantatomformula symbol="A" definition="A"/>
                                                                      </formulalist>
                                                                      <formulalist>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="p"/>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="q"/>
                                                                         <conjunctiveformula type="and">
                                                                          <conjunctiveformula type="neg">
                                                                           <constantatomformula symbol="=">
                                                                            <variable symbol="p"/>
                                                                            <variable symbol="q"/>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <variable symbol="p"/>
                                                                           </function>
                                                                           <function symbol="f">
                                                                            <variable symbol="q"/>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                        </quantifiedformula>
                                                                       </quantifiedformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </formulalist>
                                                                     </sequent>
                                                                     <rule type="permr" symbol="\pi:r" param="">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="A" definition="A"/>
                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="p"/>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="q"/>
                                                                          <conjunctiveformula type="and">
                                                                           <conjunctiveformula type="neg">
                                                                            <constantatomformula symbol="=">
                                                                             <variable symbol="p"/>
                                                                             <variable symbol="q"/>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <variable symbol="p"/>
                                                                            </function>
                                                                            <function symbol="f">
                                                                             <variable symbol="q"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                         </quantifiedformula>
                                                                        </quantifiedformula>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="p"/>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="q"/>
                                                                          <conjunctiveformula type="and">
                                                                           <conjunctiveformula type="neg">
                                                                            <constantatomformula symbol="=">
                                                                             <variable symbol="p"/>
                                                                             <variable symbol="q"/>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <variable symbol="p"/>
                                                                            </function>
                                                                            <function symbol="f">
                                                                             <variable symbol="q"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                         </quantifiedformula>
                                                                        </quantifiedformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                      <rule type="contrl" symbol="c:l" param="2">
                                                                       <sequent>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="A" definition="A"/>
                                                                        </formulalist>
                                                                        <formulalist>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="p"/>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="q"/>
                                                                           <conjunctiveformula type="and">
                                                                            <conjunctiveformula type="neg">
                                                                             <constantatomformula symbol="=">
                                                                              <variable symbol="p"/>
                                                                              <variable symbol="q"/>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <variable symbol="p"/>
                                                                             </function>
                                                                             <function symbol="f">
                                                                              <variable symbol="q"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                          </quantifiedformula>
                                                                         </quantifiedformula>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="p"/>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="q"/>
                                                                           <conjunctiveformula type="and">
                                                                            <conjunctiveformula type="neg">
                                                                             <constantatomformula symbol="=">
                                                                              <variable symbol="p"/>
                                                                              <variable symbol="q"/>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <variable symbol="p"/>
                                                                             </function>
                                                                             <function symbol="f">
                                                                              <variable symbol="q"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                          </quantifiedformula>
                                                                         </quantifiedformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{109}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{109}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                       <rule type="perml" symbol="\pi:l" param="">
                                                                        <sequent>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                         </formulalist>
                                                                         <formulalist>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="p"/>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="q"/>
                                                                            <conjunctiveformula type="and">
                                                                             <conjunctiveformula type="neg">
                                                                              <constantatomformula symbol="=">
                                                                               <variable symbol="p"/>
                                                                               <variable symbol="q"/>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <variable symbol="p"/>
                                                                              </function>
                                                                              <function symbol="f">
                                                                               <variable symbol="q"/>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                           </quantifiedformula>
                                                                          </quantifiedformula>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="p"/>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="q"/>
                                                                            <conjunctiveformula type="and">
                                                                             <conjunctiveformula type="neg">
                                                                              <constantatomformula symbol="=">
                                                                               <variable symbol="p"/>
                                                                               <variable symbol="q"/>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <variable symbol="p"/>
                                                                              </function>
                                                                              <function symbol="f">
                                                                               <variable symbol="q"/>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                           </quantifiedformula>
                                                                          </quantifiedformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{110}"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{109}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{109}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </formulalist>
                                                                        </sequent>
                                                                        <rule type="eqr1" symbol="=:r1" param="(1)">
                                                                         <sequent>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="A" definition="A"/>
                                                                           <constantatomformula symbol="A" definition="A"/>
                                                                          </formulalist>
                                                                          <formulalist>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="p"/>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="q"/>
                                                                             <conjunctiveformula type="and">
                                                                              <conjunctiveformula type="neg">
                                                                               <constantatomformula symbol="=">
                                                                                <variable symbol="p"/>
                                                                                <variable symbol="q"/>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <variable symbol="p"/>
                                                                               </function>
                                                                               <function symbol="f">
                                                                                <variable symbol="q"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                            </quantifiedformula>
                                                                           </quantifiedformula>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="p"/>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="q"/>
                                                                             <conjunctiveformula type="and">
                                                                              <conjunctiveformula type="neg">
                                                                               <constantatomformula symbol="=">
                                                                                <variable symbol="p"/>
                                                                                <variable symbol="q"/>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <variable symbol="p"/>
                                                                               </function>
                                                                               <function symbol="f">
                                                                                <variable symbol="q"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                            </quantifiedformula>
                                                                           </quantifiedformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{109}"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{110}"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{109}"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </formulalist>
                                                                         </sequent>
                                                                         <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                          <sequent>
                                                                           <formulalist>
                                                                            <constantatomformula symbol="A" definition="A"/>
                                                                           </formulalist>
                                                                           <formulalist>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="p"/>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="q"/>
                                                                              <conjunctiveformula type="and">
                                                                               <conjunctiveformula type="neg">
                                                                                <constantatomformula symbol="=">
                                                                                 <variable symbol="p"/>
                                                                                 <variable symbol="q"/>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <variable symbol="p"/>
                                                                                </function>
                                                                                <function symbol="f">
                                                                                 <variable symbol="q"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                             </quantifiedformula>
                                                                            </quantifiedformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{109}"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{109}"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </formulalist>
                                                                          </sequent>
                                                                          <rule type="weakr" symbol="w:r">
                                                                           <sequent>
                                                                            <formulalist>
                                                                             <constantatomformula symbol="A" definition="A"/>
                                                                            </formulalist>
                                                                            <formulalist>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{109}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{109}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </constantatomformula>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="p"/>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="q"/>
                                                                               <conjunctiveformula type="and">
                                                                                <conjunctiveformula type="neg">
                                                                                 <constantatomformula symbol="=">
                                                                                  <variable symbol="p"/>
                                                                                  <variable symbol="q"/>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <variable symbol="p"/>
                                                                                 </function>
                                                                                 <function symbol="f">
                                                                                  <variable symbol="q"/>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                              </quantifiedformula>
                                                                             </quantifiedformula>
                                                                            </formulalist>
                                                                           </sequent>
                                                                           <rule type="perml" symbol="\pi:l" param="">
                                                                            <sequent>
                                                                             <formulalist>
                                                                              <constantatomformula symbol="A" definition="A"/>
                                                                             </formulalist>
                                                                             <formulalist>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{109}"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{109}"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </formulalist>
                                                                            </sequent>
                                                                            <rule type="weakl" symbol="w:l">
                                                                             <sequent>
                                                                              <formulalist>
                                                                               <constantatomformula symbol="A" definition="A"/>
                                                                              </formulalist>
                                                                              <formulalist>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{109}"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{109}"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </formulalist>
                                                                             </sequent>
                                                                             <rule type="axiom" symbol="axiom">
                                                                              <sequent>
                                                                               <formulalist/>
                                                                               <formulalist>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{109}"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{110}"/>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{109}"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </formulalist>
                                                                              </sequent>
                                                                             </rule>
                                                                            </rule>
                                                                           </rule>
                                                                          </rule>
                                                                         </rule>
                                                                         <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                          <sequent>
                                                                           <formulalist>
                                                                            <constantatomformula symbol="A" definition="A"/>
                                                                           </formulalist>
                                                                           <formulalist>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="p"/>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="q"/>
                                                                              <conjunctiveformula type="and">
                                                                               <conjunctiveformula type="neg">
                                                                                <constantatomformula symbol="=">
                                                                                 <variable symbol="p"/>
                                                                                 <variable symbol="q"/>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <variable symbol="p"/>
                                                                                </function>
                                                                                <function symbol="f">
                                                                                 <variable symbol="q"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                             </quantifiedformula>
                                                                            </quantifiedformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{109}"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{109}"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </formulalist>
                                                                          </sequent>
                                                                          <rule type="weakr" symbol="w:r">
                                                                           <sequent>
                                                                            <formulalist>
                                                                             <constantatomformula symbol="A" definition="A"/>
                                                                            </formulalist>
                                                                            <formulalist>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{109}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{109}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                             </constantatomformula>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="p"/>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="q"/>
                                                                               <conjunctiveformula type="and">
                                                                                <conjunctiveformula type="neg">
                                                                                 <constantatomformula symbol="=">
                                                                                  <variable symbol="p"/>
                                                                                  <variable symbol="q"/>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <variable symbol="p"/>
                                                                                 </function>
                                                                                 <function symbol="f">
                                                                                  <variable symbol="q"/>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                              </quantifiedformula>
                                                                             </quantifiedformula>
                                                                            </formulalist>
                                                                           </sequent>
                                                                           <rule type="perml" symbol="\pi:l" param="">
                                                                            <sequent>
                                                                             <formulalist>
                                                                              <constantatomformula symbol="A" definition="A"/>
                                                                             </formulalist>
                                                                             <formulalist>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{109}"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{109}"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </formulalist>
                                                                            </sequent>
                                                                            <rule type="weakl" symbol="w:l">
                                                                             <sequent>
                                                                              <formulalist>
                                                                               <constantatomformula symbol="A" definition="A"/>
                                                                              </formulalist>
                                                                              <formulalist>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{109}"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{109}"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </formulalist>
                                                                             </sequent>
                                                                             <rule type="axiom" symbol="axiom">
                                                                              <sequent>
                                                                               <formulalist/>
                                                                               <formulalist>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{109}"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{109}"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </formulalist>
                                                                              </sequent>
                                                                             </rule>
                                                                            </rule>
                                                                           </rule>
                                                                          </rule>
                                                                         </rule>
                                                                        </rule>
                                                                       </rule>
                                                                      </rule>
                                                                     </rule>
                                                                    </rule>
                                                                    <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                                     <sequent>
                                                                      <formulalist>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </constantatomformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </constantatomformula>
                                                                       <constantatomformula symbol="A" definition="A"/>
                                                                      </formulalist>
                                                                      <formulalist>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="p"/>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="q"/>
                                                                         <conjunctiveformula type="and">
                                                                          <conjunctiveformula type="neg">
                                                                           <constantatomformula symbol="=">
                                                                            <variable symbol="p"/>
                                                                            <variable symbol="q"/>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <variable symbol="p"/>
                                                                           </function>
                                                                           <function symbol="f">
                                                                            <variable symbol="q"/>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                        </quantifiedformula>
                                                                       </quantifiedformula>
                                                                      </formulalist>
                                                                     </sequent>
                                                                     <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </constantatomformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{109}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </constantatomformula>
                                                                        <constantatomformula symbol="A" definition="A"/>
                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="p"/>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="q"/>
                                                                          <conjunctiveformula type="and">
                                                                           <conjunctiveformula type="neg">
                                                                            <constantatomformula symbol="=">
                                                                             <variable symbol="p"/>
                                                                             <variable symbol="q"/>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <variable symbol="p"/>
                                                                            </function>
                                                                            <function symbol="f">
                                                                             <variable symbol="q"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                         </quantifiedformula>
                                                                        </quantifiedformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                      <rule type="contrr" symbol="c:r" param="2">
                                                                       <sequent>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{110}"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{109}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </constantatomformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </constantatomformula>
                                                                         <constantatomformula symbol="A" definition="A"/>
                                                                        </formulalist>
                                                                        <formulalist>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="p"/>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="q"/>
                                                                           <conjunctiveformula type="and">
                                                                            <conjunctiveformula type="neg">
                                                                             <constantatomformula symbol="=">
                                                                              <variable symbol="p"/>
                                                                              <variable symbol="q"/>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <variable symbol="p"/>
                                                                             </function>
                                                                             <function symbol="f">
                                                                              <variable symbol="q"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                          </quantifiedformula>
                                                                         </quantifiedformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                       <rule type="permr" symbol="\pi:r" param="">
                                                                        <sequent>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{109}"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </constantatomformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{110}"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <variable symbol="x_{109}"/>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </constantatomformula>
                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                         </formulalist>
                                                                         <formulalist>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="p"/>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="q"/>
                                                                            <conjunctiveformula type="and">
                                                                             <conjunctiveformula type="neg">
                                                                              <constantatomformula symbol="=">
                                                                               <variable symbol="p"/>
                                                                               <variable symbol="q"/>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <variable symbol="p"/>
                                                                              </function>
                                                                              <function symbol="f">
                                                                               <variable symbol="q"/>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                           </quantifiedformula>
                                                                          </quantifiedformula>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="p"/>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="q"/>
                                                                            <conjunctiveformula type="and">
                                                                             <conjunctiveformula type="neg">
                                                                              <constantatomformula symbol="=">
                                                                               <variable symbol="p"/>
                                                                               <variable symbol="q"/>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <variable symbol="p"/>
                                                                              </function>
                                                                              <function symbol="f">
                                                                               <variable symbol="q"/>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                           </quantifiedformula>
                                                                          </quantifiedformula>
                                                                         </formulalist>
                                                                        </sequent>
                                                                        <rule type="contrl" symbol="c:l" param="1,1,2">
                                                                         <sequent>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{109}"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </constantatomformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <variable symbol="x_{109}"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </constantatomformula>
                                                                           <constantatomformula symbol="A" definition="A"/>
                                                                          </formulalist>
                                                                          <formulalist>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="p"/>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="q"/>
                                                                             <conjunctiveformula type="and">
                                                                              <conjunctiveformula type="neg">
                                                                               <constantatomformula symbol="=">
                                                                                <variable symbol="p"/>
                                                                                <variable symbol="q"/>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <variable symbol="p"/>
                                                                               </function>
                                                                               <function symbol="f">
                                                                                <variable symbol="q"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                            </quantifiedformula>
                                                                           </quantifiedformula>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="p"/>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="q"/>
                                                                             <conjunctiveformula type="and">
                                                                              <conjunctiveformula type="neg">
                                                                               <constantatomformula symbol="=">
                                                                                <variable symbol="p"/>
                                                                                <variable symbol="q"/>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <variable symbol="p"/>
                                                                               </function>
                                                                               <function symbol="f">
                                                                                <variable symbol="q"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                            </quantifiedformula>
                                                                           </quantifiedformula>
                                                                          </formulalist>
                                                                         </sequent>
                                                                         <rule type="perml" symbol="\pi:l" param="(2 3)">
                                                                          <sequent>
                                                                           <formulalist>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{109}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </constantatomformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </constantatomformula>
                                                                            <constantatomformula symbol="A" definition="A"/>
                                                                            <constantatomformula symbol="A" definition="A"/>
                                                                           </formulalist>
                                                                           <formulalist>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="p"/>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="q"/>
                                                                              <conjunctiveformula type="and">
                                                                               <conjunctiveformula type="neg">
                                                                                <constantatomformula symbol="=">
                                                                                 <variable symbol="p"/>
                                                                                 <variable symbol="q"/>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <variable symbol="p"/>
                                                                                </function>
                                                                                <function symbol="f">
                                                                                 <variable symbol="q"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                             </quantifiedformula>
                                                                            </quantifiedformula>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="p"/>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="q"/>
                                                                              <conjunctiveformula type="and">
                                                                               <conjunctiveformula type="neg">
                                                                                <constantatomformula symbol="=">
                                                                                 <variable symbol="p"/>
                                                                                 <variable symbol="q"/>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <variable symbol="p"/>
                                                                                </function>
                                                                                <function symbol="f">
                                                                                 <variable symbol="q"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                             </quantifiedformula>
                                                                            </quantifiedformula>
                                                                           </formulalist>
                                                                          </sequent>
                                                                          <rule type="eql1" symbol="=:l1" param="(1 1 1)">
                                                                           <sequent>
                                                                            <formulalist>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{109}"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </constantatomformula>
                                                                             <constantatomformula symbol="A" definition="A"/>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <variable symbol="x_{109}"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </constantatomformula>
                                                                             <constantatomformula symbol="A" definition="A"/>
                                                                            </formulalist>
                                                                            <formulalist>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="p"/>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="q"/>
                                                                               <conjunctiveformula type="and">
                                                                                <conjunctiveformula type="neg">
                                                                                 <constantatomformula symbol="=">
                                                                                  <variable symbol="p"/>
                                                                                  <variable symbol="q"/>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <variable symbol="p"/>
                                                                                 </function>
                                                                                 <function symbol="f">
                                                                                  <variable symbol="q"/>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                              </quantifiedformula>
                                                                             </quantifiedformula>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="p"/>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="q"/>
                                                                               <conjunctiveformula type="and">
                                                                                <conjunctiveformula type="neg">
                                                                                 <constantatomformula symbol="=">
                                                                                  <variable symbol="p"/>
                                                                                  <variable symbol="q"/>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <variable symbol="p"/>
                                                                                 </function>
                                                                                 <function symbol="f">
                                                                                  <variable symbol="q"/>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                              </quantifiedformula>
                                                                             </quantifiedformula>
                                                                            </formulalist>
                                                                           </sequent>
                                                                           <rule type="contrr" symbol="c:r" param="2,1">
                                                                            <sequent>
                                                                             <formulalist>
                                                                              <constantatomformula symbol="A" definition="A"/>
                                                                             </formulalist>
                                                                             <formulalist>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="p"/>
                                                                               <quantifiedformula type="exists">
                                                                                <variable symbol="q"/>
                                                                                <conjunctiveformula type="and">
                                                                                 <conjunctiveformula type="neg">
                                                                                  <constantatomformula symbol="=">
                                                                                   <variable symbol="p"/>
                                                                                   <variable symbol="q"/>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="f">
                                                                                   <variable symbol="p"/>
                                                                                  </function>
                                                                                  <function symbol="f">
                                                                                   <variable symbol="q"/>
                                                                                  </function>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                               </quantifiedformula>
                                                                              </quantifiedformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <variable symbol="x_{109}"/>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{109}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </formulalist>
                                                                            </sequent>
                                                                            <rule type="permr" symbol="\pi:r" param="">
                                                                             <sequent>
                                                                              <formulalist>
                                                                               <constantatomformula symbol="A" definition="A"/>
                                                                              </formulalist>
                                                                              <formulalist>
                                                                               <quantifiedformula type="exists">
                                                                                <variable symbol="p"/>
                                                                                <quantifiedformula type="exists">
                                                                                 <variable symbol="q"/>
                                                                                 <conjunctiveformula type="and">
                                                                                  <conjunctiveformula type="neg">
                                                                                   <constantatomformula symbol="=">
                                                                                    <variable symbol="p"/>
                                                                                    <variable symbol="q"/>
                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="f">
                                                                                    <variable symbol="p"/>
                                                                                   </function>
                                                                                   <function symbol="f">
                                                                                    <variable symbol="q"/>
                                                                                   </function>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                </quantifiedformula>
                                                                               </quantifiedformula>
                                                                               <quantifiedformula type="exists">
                                                                                <variable symbol="p"/>
                                                                                <quantifiedformula type="exists">
                                                                                 <variable symbol="q"/>
                                                                                 <conjunctiveformula type="and">
                                                                                  <conjunctiveformula type="neg">
                                                                                   <constantatomformula symbol="=">
                                                                                    <variable symbol="p"/>
                                                                                    <variable symbol="q"/>
                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="f">
                                                                                    <variable symbol="p"/>
                                                                                   </function>
                                                                                   <function symbol="f">
                                                                                    <variable symbol="q"/>
                                                                                   </function>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                </quantifiedformula>
                                                                               </quantifiedformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{109}"/>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{109}"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </formulalist>
                                                                             </sequent>
                                                                             <rule type="contrl" symbol="c:l" param="2">
                                                                              <sequent>
                                                                               <formulalist>
                                                                                <constantatomformula symbol="A" definition="A"/>
                                                                               </formulalist>
                                                                               <formulalist>
                                                                                <quantifiedformula type="exists">
                                                                                 <variable symbol="p"/>
                                                                                 <quantifiedformula type="exists">
                                                                                  <variable symbol="q"/>
                                                                                  <conjunctiveformula type="and">
                                                                                   <conjunctiveformula type="neg">
                                                                                    <constantatomformula symbol="=">
                                                                                     <variable symbol="p"/>
                                                                                     <variable symbol="q"/>
                                                                                    </constantatomformula>
                                                                                   </conjunctiveformula>
                                                                                   <constantatomformula symbol="=">
                                                                                    <function symbol="f">
                                                                                     <variable symbol="p"/>
                                                                                    </function>
                                                                                    <function symbol="f">
                                                                                     <variable symbol="q"/>
                                                                                    </function>
                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                 </quantifiedformula>
                                                                                </quantifiedformula>
                                                                                <quantifiedformula type="exists">
                                                                                 <variable symbol="p"/>
                                                                                 <quantifiedformula type="exists">
                                                                                  <variable symbol="q"/>
                                                                                  <conjunctiveformula type="and">
                                                                                   <conjunctiveformula type="neg">
                                                                                    <constantatomformula symbol="=">
                                                                                     <variable symbol="p"/>
                                                                                     <variable symbol="q"/>
                                                                                    </constantatomformula>
                                                                                   </conjunctiveformula>
                                                                                   <constantatomformula symbol="=">
                                                                                    <function symbol="f">
                                                                                     <variable symbol="p"/>
                                                                                    </function>
                                                                                    <function symbol="f">
                                                                                     <variable symbol="q"/>
                                                                                    </function>
                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                 </quantifiedformula>
                                                                                </quantifiedformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{110}"/>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <variable symbol="x_{109}"/>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{109}"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </formulalist>
                                                                              </sequent>
                                                                              <rule type="perml" symbol="\pi:l" param="">
                                                                               <sequent>
                                                                                <formulalist>
                                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                                </formulalist>
                                                                                <formulalist>
                                                                                 <quantifiedformula type="exists">
                                                                                  <variable symbol="p"/>
                                                                                  <quantifiedformula type="exists">
                                                                                   <variable symbol="q"/>
                                                                                   <conjunctiveformula type="and">
                                                                                    <conjunctiveformula type="neg">
                                                                                     <constantatomformula symbol="=">
                                                                                      <variable symbol="p"/>
                                                                                      <variable symbol="q"/>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
                                                                                      <variable symbol="p"/>
                                                                                     </function>
                                                                                     <function symbol="f">
                                                                                      <variable symbol="q"/>
                                                                                     </function>
                                                                                    </constantatomformula>
                                                                                   </conjunctiveformula>
                                                                                  </quantifiedformula>
                                                                                 </quantifiedformula>
                                                                                 <quantifiedformula type="exists">
                                                                                  <variable symbol="p"/>
                                                                                  <quantifiedformula type="exists">
                                                                                   <variable symbol="q"/>
                                                                                   <conjunctiveformula type="and">
                                                                                    <conjunctiveformula type="neg">
                                                                                     <constantatomformula symbol="=">
                                                                                      <variable symbol="p"/>
                                                                                      <variable symbol="q"/>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
                                                                                      <variable symbol="p"/>
                                                                                     </function>
                                                                                     <function symbol="f">
                                                                                      <variable symbol="q"/>
                                                                                     </function>
                                                                                    </constantatomformula>
                                                                                   </conjunctiveformula>
                                                                                  </quantifiedformula>
                                                                                 </quantifiedformula>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{110}"/>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <variable symbol="x_{109}"/>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{110}"/>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{109}"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </constantatomformula>
                                                                                </formulalist>
                                                                               </sequent>
                                                                               <rule type="eqr1" symbol="=:r1" param="(1)">
                                                                                <sequent>
                                                                                 <formulalist>
                                                                                  <constantatomformula symbol="A" definition="A"/>
                                                                                  <constantatomformula symbol="A" definition="A"/>
                                                                                 </formulalist>
                                                                                 <formulalist>
                                                                                  <quantifiedformula type="exists">
                                                                                   <variable symbol="p"/>
                                                                                   <quantifiedformula type="exists">
                                                                                    <variable symbol="q"/>
                                                                                    <conjunctiveformula type="and">
                                                                                     <conjunctiveformula type="neg">
                                                                                      <constantatomformula symbol="=">
                                                                                       <variable symbol="p"/>
                                                                                       <variable symbol="q"/>
                                                                                      </constantatomformula>
                                                                                     </conjunctiveformula>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="f">
                                                                                       <variable symbol="p"/>
                                                                                      </function>
                                                                                      <function symbol="f">
                                                                                       <variable symbol="q"/>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                   </quantifiedformula>
                                                                                  </quantifiedformula>
                                                                                  <quantifiedformula type="exists">
                                                                                   <variable symbol="p"/>
                                                                                   <quantifiedformula type="exists">
                                                                                    <variable symbol="q"/>
                                                                                    <conjunctiveformula type="and">
                                                                                     <conjunctiveformula type="neg">
                                                                                      <constantatomformula symbol="=">
                                                                                       <variable symbol="p"/>
                                                                                       <variable symbol="q"/>
                                                                                      </constantatomformula>
                                                                                     </conjunctiveformula>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="f">
                                                                                       <variable symbol="p"/>
                                                                                      </function>
                                                                                      <function symbol="f">
                                                                                       <variable symbol="q"/>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                   </quantifiedformula>
                                                                                  </quantifiedformula>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{110}"/>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <variable symbol="x_{109}"/>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{110}"/>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{109}"/>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </constantatomformula>
                                                                                 </formulalist>
                                                                                </sequent>
                                                                                <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                                 <sequent>
                                                                                  <formulalist>
                                                                                   <constantatomformula symbol="A" definition="A"/>
                                                                                  </formulalist>
                                                                                  <formulalist>
                                                                                   <quantifiedformula type="exists">
                                                                                    <variable symbol="p"/>
                                                                                    <quantifiedformula type="exists">
                                                                                     <variable symbol="q"/>
                                                                                     <conjunctiveformula type="and">
                                                                                      <conjunctiveformula type="neg">
                                                                                       <constantatomformula symbol="=">
                                                                                        <variable symbol="p"/>
                                                                                        <variable symbol="q"/>
                                                                                       </constantatomformula>
                                                                                      </conjunctiveformula>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="f">
                                                                                        <variable symbol="p"/>
                                                                                       </function>
                                                                                       <function symbol="f">
                                                                                        <variable symbol="q"/>
                                                                                       </function>
                                                                                      </constantatomformula>
                                                                                     </conjunctiveformula>
                                                                                    </quantifiedformula>
                                                                                   </quantifiedformula>
                                                                                   <constantatomformula symbol="=">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{110}"/>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{109}"/>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{110}"/>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="x_{109}"/>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </constantatomformula>
                                                                                  </formulalist>
                                                                                 </sequent>
                                                                                 <rule type="weakr" symbol="w:r">
                                                                                  <sequent>
                                                                                   <formulalist>
                                                                                    <constantatomformula symbol="A" definition="A"/>
                                                                                   </formulalist>
                                                                                   <formulalist>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{110}"/>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{109}"/>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{110}"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <variable symbol="x_{109}"/>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </constantatomformula>
                                                                                    <quantifiedformula type="exists">
                                                                                     <variable symbol="p"/>
                                                                                     <quantifiedformula type="exists">
                                                                                      <variable symbol="q"/>
                                                                                      <conjunctiveformula type="and">
                                                                                       <conjunctiveformula type="neg">
                                                                                        <constantatomformula symbol="=">
                                                                                         <variable symbol="p"/>
                                                                                         <variable symbol="q"/>
                                                                                        </constantatomformula>
                                                                                       </conjunctiveformula>
                                                                                       <constantatomformula symbol="=">
                                                                                        <function symbol="f">
                                                                                         <variable symbol="p"/>
                                                                                        </function>
                                                                                        <function symbol="f">
                                                                                         <variable symbol="q"/>
                                                                                        </function>
                                                                                       </constantatomformula>
                                                                                      </conjunctiveformula>
                                                                                     </quantifiedformula>
                                                                                    </quantifiedformula>
                                                                                   </formulalist>
                                                                                  </sequent>
                                                                                  <rule type="perml" symbol="\pi:l" param="">
                                                                                   <sequent>
                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="A" definition="A"/>
                                                                                    </formulalist>
                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{110}"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{109}"/>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{110}"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <variable symbol="x_{109}"/>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </formulalist>
                                                                                   </sequent>
                                                                                   <rule type="weakl" symbol="w:l">
                                                                                    <sequent>
                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="A" definition="A"/>
                                                                                     </formulalist>
                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{110}"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{109}"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{110}"/>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <variable symbol="x_{109}"/>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </constantatomformula>
                                                                                     </formulalist>
                                                                                    </sequent>
                                                                                    <rule type="axiom" symbol="axiom">
                                                                                     <sequent>
                                                                                      <formulalist/>
                                                                                      <formulalist>
                                                                                       <constantatomformula symbol="=">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{110}"/>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{109}"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{110}"/>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <variable symbol="x_{109}"/>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </constantatomformula>
                                                                                      </formulalist>
                                                                                     </sequent>
                                                                                    </rule>
                                                                                   </rule>
                                                                                  </rule>
                                                                                 </rule>
                                                                                </rule>
                                                                                <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                                 <sequent>
                                                                                  <formulalist>
                                                                                   <constantatomformula symbol="A" definition="A"/>
                                                                                  </formulalist>
                                                                                  <formulalist>
                                                                                   <quantifiedformula type="exists">
                                                                                    <variable symbol="p"/>
                                                                                    <quantifiedformula type="exists">
                                                                                     <variable symbol="q"/>
                                                                                     <conjunctiveformula type="and">
                                                                                      <conjunctiveformula type="neg">
                                                                                       <constantatomformula symbol="=">
                                                                                        <variable symbol="p"/>
                                                                                        <variable symbol="q"/>
                                                                                       </constantatomformula>
                                                                                      </conjunctiveformula>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="f">
                                                                                        <variable symbol="p"/>
                                                                                       </function>
                                                                                       <function symbol="f">
                                                                                        <variable symbol="q"/>
                                                                                       </function>
                                                                                      </constantatomformula>
                                                                                     </conjunctiveformula>
                                                                                    </quantifiedformula>
                                                                                   </quantifiedformula>
                                                                                   <constantatomformula symbol="=">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{110}"/>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{109}"/>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{110}"/>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{109}"/>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </constantatomformula>
                                                                                  </formulalist>
                                                                                 </sequent>
                                                                                 <rule type="weakr" symbol="w:r">
                                                                                  <sequent>
                                                                                   <formulalist>
                                                                                    <constantatomformula symbol="A" definition="A"/>
                                                                                   </formulalist>
                                                                                   <formulalist>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{110}"/>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{109}"/>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{110}"/>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{109}"/>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </constantatomformula>
                                                                                    <quantifiedformula type="exists">
                                                                                     <variable symbol="p"/>
                                                                                     <quantifiedformula type="exists">
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                                                                                      <conjunctiveformula type="and">
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                                                                                        <constantatomformula symbol="=">
                                                                                         <variable symbol="p"/>
                                                                                         <variable symbol="q"/>
                                                                                        </constantatomformula>
                                                                                       </conjunctiveformula>
                                                                                       <constantatomformula symbol="=">
                                                                                        <function symbol="f">
                                                                                         <variable symbol="p"/>
                                                                                        </function>
                                                                                        <function symbol="f">
                                                                                         <variable symbol="q"/>
                                                                                        </function>
                                                                                       </constantatomformula>
                                                                                      </conjunctiveformula>
                                                                                     </quantifiedformula>
                                                                                    </quantifiedformula>
                                                                                   </formulalist>
                                                                                  </sequent>
                                                                                  <rule type="perml" symbol="\pi:l" param="">
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                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="A" definition="A"/>
                                                                                    </formulalist>
                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{110}"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{109}"/>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{110}"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{109}"/>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </formulalist>
                                                                                   </sequent>
                                                                                   <rule type="weakl" symbol="w:l">
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                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="A" definition="A"/>
                                                                                     </formulalist>
                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{110}"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{109}"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{110}"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{109}"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </constantatomformula>
                                                                                     </formulalist>
                                                                                    </sequent>
                                                                                    <rule type="axiom" symbol="axiom">
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                                                                                      <formulalist/>
                                                                                      <formulalist>
                                                                                       <constantatomformula symbol="=">
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                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{110}"/>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{109}"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
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                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{110}"/>
                                                                                           <constant symbol="1"/>
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                                                                                         </function>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{109}"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </constantatomformula>
                                                                                      </formulalist>
                                                                                     </sequent>
                                                                                    </rule>
                                                                                   </rule>
                                                                                  </rule>
                                                                                 </rule>
                                                                                </rule>
                                                                               </rule>
                                                                              </rule>
                                                                             </rule>
                                                                            </rule>
                                                                           </rule>
                                                                           <rule type="perml" symbol="\pi:l" param="(1 2)">
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                                                                             <formulalist>
                                                                              <constantatomformula symbol="=">
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                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{110}"/>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <variable symbol="x_{109}"/>
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                                                                                  <constant symbol="1"/>
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                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
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                                                                              <constantatomformula symbol="=">
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                                                                                   <variable symbol="x_{110}"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <variable symbol="x_{109}"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </constantatomformula>
                                                                              <constantatomformula symbol="A" definition="A"/>
                                                                             </formulalist>
                                                                             <formulalist>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="p"/>
                                                                               <quantifiedformula type="exists">
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                                                                                   <variable symbol="p"/>
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                                                                                 </conjunctiveformula>
                                                                                 <constantatomformula symbol="=">
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                                                                                   <variable symbol="p"/>
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                                                                                  <function symbol="f">
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                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                               </quantifiedformula>
                                                                              </quantifiedformula>
                                                                             </formulalist>
                                                                            </sequent>
                                                                            <rule type="permr" symbol="\pi:r" param="">
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                                                                              <formulalist>
                                                                               <constantatomformula symbol="=">
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                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
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                                                                                    <variable symbol="x_{110}"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{109}"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </constantatomformula>
                                                                               <constantatomformula symbol="=">
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                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{110}"/>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <variable symbol="x_{109}"/>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </constantatomformula>
                                                                               <constantatomformula symbol="A" definition="A"/>
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                                                                              <formulalist>
                                                                               <quantifiedformula type="exists">
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                                                                                  <constantatomformula symbol="=">
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                                                                                   </function>
                                                                                   <function symbol="f">
                                                                                    <variable symbol="q"/>
                                                                                   </function>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                </quantifiedformula>
                                                                               </quantifiedformula>
                                                                              </formulalist>
                                                                             </sequent>
                                                                             <rule type="perml" symbol="\pi:l" param="(1 2 3)">
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                                                                               <formulalist>
                                                                                <constantatomformula symbol="=">
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                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{110}"/>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <variable symbol="x_{109}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </constantatomformula>
                                                                                <constantatomformula symbol="=">
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                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{110}"/>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <variable symbol="x_{109}"/>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </constantatomformula>
                                                                                <constantatomformula symbol="A" definition="A"/>
                                                                               </formulalist>
                                                                               <formulalist>
                                                                                <quantifiedformula type="exists">
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                                                                                   </conjunctiveformula>
                                                                                   <constantatomformula symbol="=">
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                                                                                     <variable symbol="p"/>
                                                                                    </function>
                                                                                    <function symbol="f">
                                                                                     <variable symbol="q"/>
                                                                                    </function>
                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                 </quantifiedformula>
                                                                                </quantifiedformula>
                                                                               </formulalist>
                                                                              </sequent>
                                                                              <rule type="contrr" symbol="c:r" param="2">
                                                                               <sequent>
                                                                                <formulalist>
                                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                                 <constantatomformula symbol="=">
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                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{110}"/>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <variable symbol="x_{109}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </constantatomformula>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="f">
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{110}"/>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="x_{109}"/>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </constantatomformula>
                                                                                </formulalist>
                                                                                <formulalist>
                                                                                 <quantifiedformula type="exists">
                                                                                  <variable symbol="p"/>
                                                                                  <quantifiedformula type="exists">
                                                                                   <variable symbol="q"/>
                                                                                   <conjunctiveformula type="and">
                                                                                    <conjunctiveformula type="neg">
                                                                                     <constantatomformula symbol="=">
                                                                                      <variable symbol="p"/>
                                                                                      <variable symbol="q"/>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
                                                                                      <variable symbol="p"/>
                                                                                     </function>
                                                                                     <function symbol="f">
                                                                                      <variable symbol="q"/>
                                                                                     </function>
                                                                                    </constantatomformula>
                                                                                   </conjunctiveformula>
                                                                                  </quantifiedformula>
                                                                                 </quantifiedformula>
                                                                                </formulalist>
                                                                               </sequent>
                                                                               <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                                <sequent>
                                                                                 <formulalist>
                                                                                  <constantatomformula symbol="A" definition="A"/>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="f">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{110}"/>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <variable symbol="x_{109}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </constantatomformula>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="f">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{110}"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <variable symbol="x_{109}"/>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </constantatomformula>
                                                                                 </formulalist>
                                                                                 <formulalist>
                                                                                  <quantifiedformula type="exists">
                                                                                   <variable symbol="p"/>
                                                                                   <quantifiedformula type="exists">
                                                                                    <variable symbol="q"/>
                                                                                    <conjunctiveformula type="and">
                                                                                     <conjunctiveformula type="neg">
                                                                                      <constantatomformula symbol="=">
                                                                                       <variable symbol="p"/>
                                                                                       <variable symbol="q"/>
                                                                                      </constantatomformula>
                                                                                     </conjunctiveformula>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="f">
                                                                                       <variable symbol="p"/>
                                                                                      </function>
                                                                                      <function symbol="f">
                                                                                       <variable symbol="q"/>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                   </quantifiedformula>
                                                                                  </quantifiedformula>
                                                                                  <quantifiedformula type="exists">
                                                                                   <variable symbol="p"/>
                                                                                   <quantifiedformula type="exists">
                                                                                    <variable symbol="q"/>
                                                                                    <conjunctiveformula type="and">
                                                                                     <conjunctiveformula type="neg">
                                                                                      <constantatomformula symbol="=">
                                                                                       <variable symbol="p"/>
                                                                                       <variable symbol="q"/>
                                                                                      </constantatomformula>
                                                                                     </conjunctiveformula>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="f">
                                                                                       <variable symbol="p"/>
                                                                                      </function>
                                                                                      <function symbol="f">
                                                                                       <variable symbol="q"/>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                   </quantifiedformula>
                                                                                  </quantifiedformula>
                                                                                 </formulalist>
                                                                                </sequent>
                                                                                <rule type="weakr" symbol="w:r">
                                                                                 <sequent>
                                                                                  <formulalist>
                                                                                   <constantatomformula symbol="A" definition="A"/>
                                                                                   <constantatomformula symbol="=">
                                                                                    <function symbol="f">
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{110}"/>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="x_{109}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </constantatomformula>
                                                                                   <constantatomformula symbol="=">
                                                                                    <function symbol="f">
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{110}"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <variable symbol="x_{109}"/>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </constantatomformula>
                                                                                  </formulalist>
                                                                                  <formulalist>
                                                                                   <quantifiedformula type="exists">
                                                                                    <variable symbol="p"/>
                                                                                    <quantifiedformula type="exists">
                                                                                     <variable symbol="q"/>
                                                                                     <conjunctiveformula type="and">
                                                                                      <conjunctiveformula type="neg">
                                                                                       <constantatomformula symbol="=">
                                                                                        <variable symbol="p"/>
                                                                                        <variable symbol="q"/>
                                                                                       </constantatomformula>
                                                                                      </conjunctiveformula>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="f">
                                                                                        <variable symbol="p"/>
                                                                                       </function>
                                                                                       <function symbol="f">
                                                                                        <variable symbol="q"/>
                                                                                       </function>
                                                                                      </constantatomformula>
                                                                                     </conjunctiveformula>
                                                                                    </quantifiedformula>
                                                                                   </quantifiedformula>
                                                                                   <quantifiedformula type="exists">
                                                                                    <variable symbol="p"/>
                                                                                    <quantifiedformula type="exists">
                                                                                     <variable symbol="q"/>
                                                                                     <conjunctiveformula type="and">
                                                                                      <conjunctiveformula type="neg">
                                                                                       <constantatomformula symbol="=">
                                                                                        <variable symbol="p"/>
                                                                                        <variable symbol="q"/>
                                                                                       </constantatomformula>
                                                                                      </conjunctiveformula>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="f">
                                                                                        <variable symbol="p"/>
                                                                                       </function>
                                                                                       <function symbol="f">
                                                                                        <variable symbol="q"/>
                                                                                       </function>
                                                                                      </constantatomformula>
                                                                                     </conjunctiveformula>
                                                                                    </quantifiedformula>
                                                                                   </quantifiedformula>
                                                                                  </formulalist>
                                                                                 </sequent>
                                                                                 <rule type="weakl" symbol="w:l">
                                                                                  <sequent>
                                                                                   <formulalist>
                                                                                    <constantatomformula symbol="A" definition="A"/>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{110}"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <variable symbol="x_{109}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </constantatomformula>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{110}"/>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <variable symbol="x_{109}"/>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </constantatomformula>
                                                                                   </formulalist>
                                                                                   <formulalist>
                                                                                    <quantifiedformula type="exists">
                                                                                     <variable symbol="p"/>
                                                                                     <quantifiedformula type="exists">
                                                                                      <variable symbol="q"/>
                                                                                      <conjunctiveformula type="and">
                                                                                       <conjunctiveformula type="neg">
                                                                                        <constantatomformula symbol="=">
                                                                                         <variable symbol="p"/>
                                                                                         <variable symbol="q"/>
                                                                                        </constantatomformula>
                                                                                       </conjunctiveformula>
                                                                                       <constantatomformula symbol="=">
                                                                                        <function symbol="f">
                                                                                         <variable symbol="p"/>
                                                                                        </function>
                                                                                        <function symbol="f">
                                                                                         <variable symbol="q"/>
                                                                                        </function>
                                                                                       </constantatomformula>
                                                                                      </conjunctiveformula>
                                                                                     </quantifiedformula>
                                                                                    </quantifiedformula>
                                                                                   </formulalist>
                                                                                  </sequent>
                                                                                  <rule type="existsr" symbol="\ex:r">
                                                                                   <sequent>
                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="f">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{110}"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <variable symbol="x_{109}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </constantatomformula>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="f">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{110}"/>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <variable symbol="x_{109}"/>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </constantatomformula>
                                                                                    </formulalist>
                                                                                    <formulalist>
                                                                                     <quantifiedformula type="exists">
                                                                                      <variable symbol="p"/>
                                                                                      <quantifiedformula type="exists">
                                                                                       <variable symbol="q"/>
                                                                                       <conjunctiveformula type="and">
                                                                                        <conjunctiveformula type="neg">
                                                                                         <constantatomformula symbol="=">
                                                                                          <variable symbol="p"/>
                                                                                          <variable symbol="q"/>
                                                                                         </constantatomformula>
                                                                                        </conjunctiveformula>
                                                                                        <constantatomformula symbol="=">
                                                                                         <function symbol="f">
                                                                                          <variable symbol="p"/>
                                                                                         </function>
                                                                                         <function symbol="f">
                                                                                          <variable symbol="q"/>
                                                                                         </function>
                                                                                        </constantatomformula>
                                                                                       </conjunctiveformula>
                                                                                      </quantifiedformula>
                                                                                     </quantifiedformula>
                                                                                    </formulalist>
                                                                                   </sequent>
                                                                                   <rule type="existsr" symbol="\ex:r">
                                                                                    <sequent>
                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="f">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{110}"/>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <variable symbol="x_{109}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </constantatomformula>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="f">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </constantatomformula>
                                                                                     </formulalist>
                                                                                     <formulalist>
                                                                                      <quantifiedformula type="exists">
                                                                                       <variable symbol="q"/>
                                                                                       <conjunctiveformula type="and">
                                                                                        <conjunctiveformula type="neg">
                                                                                         <constantatomformula symbol="=">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                          <variable symbol="q"/>
                                                                                         </constantatomformula>
                                                                                        </conjunctiveformula>
                                                                                        <constantatomformula symbol="=">
                                                                                         <function symbol="f">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <function symbol="f">
                                                                                          <variable symbol="q"/>
                                                                                         </function>
                                                                                        </constantatomformula>
                                                                                       </conjunctiveformula>
                                                                                      </quantifiedformula>
                                                                                     </formulalist>
                                                                                    </sequent>
                                                                                    <rule type="andr" symbol="\and:r">
                                                                                     <sequent>
                                                                                      <formulalist>
                                                                                       <constantatomformula symbol="=">
                                                                                        <function symbol="f">
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{110}"/>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <variable symbol="x_{109}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </constantatomformula>
                                                                                       <constantatomformula symbol="=">
                                                                                        <function symbol="f">
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{110}"/>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                            </function>
                                                                                            <variable symbol="x_{109}"/>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </constantatomformula>
                                                                                      </formulalist>
                                                                                      <formulalist>
                                                                                       <conjunctiveformula type="and">
                                                                                        <conjunctiveformula type="neg">
                                                                                         <constantatomformula symbol="=">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </constantatomformula>
                                                                                        </conjunctiveformula>
                                                                                        <constantatomformula symbol="=">
                                                                                         <function symbol="f">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <function symbol="f">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </constantatomformula>
                                                                                       </conjunctiveformula>
                                                                                      </formulalist>
                                                                                     </sequent>
                                                                                     <rule type="negr" symbol="\neg:r">
                                                                                      <sequent>
                                                                                       <formulalist/>
                                                                                       <formulalist>
                                                                                        <conjunctiveformula type="neg">
                                                                                         <constantatomformula symbol="=">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </constantatomformula>
                                                                                        </conjunctiveformula>
                                                                                       </formulalist>
                                                                                      </sequent>
                                                                                      <rule type="eql1" symbol="=:l1" param="(2)">
                                                                                       <sequent>
                                                                                        <formulalist>
                                                                                         <constantatomformula symbol="=">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </constantatomformula>
                                                                                        </formulalist>
                                                                                        <formulalist/>
                                                                                       </sequent>
                                                                                       <rule type="axiom" symbol="axiom">
                                                                                        <sequent>
                                                                                         <formulalist/>
                                                                                         <formulalist>
                                                                                          <constantatomformula symbol="=">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{110}"/>
                                                                                                <constant symbol="1"/>
                                                                                               </function>
                                                                                              </function>
                                                                                              <variable symbol="x_{109}"/>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </constantatomformula>
                                                                                         </formulalist>
                                                                                        </sequent>
                                                                                       </rule>
                                                                                       <rule type="axiom" symbol="axiom">
                                                                                        <sequent>
                                                                                         <formulalist>
                                                                                          <constantatomformula symbol="=">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{110}"/>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                            </function>
                                                                                            <variable symbol="x_{109}"/>
                                                                                           </function>
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </constantatomformula>
                                                                                         </formulalist>
                                                                                         <formulalist/>
                                                                                        </sequent>
                                                                                       </rule>
                                                                                      </rule>
                                                                                     </rule>
                                                                                     <rule type="eqr2" symbol="=:r2" param="(1)">
                                                                                      <sequent>
                                                                                       <formulalist>
                                                                                        <constantatomformula symbol="=">
                                                                                         <function symbol="f">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </constantatomformula>
                                                                                        <constantatomformula symbol="=">
                                                                                         <function symbol="f">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </constantatomformula>
                                                                                       </formulalist>
                                                                                       <formulalist>
                                                                                        <constantatomformula symbol="=">
                                                                                         <function symbol="f">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <function symbol="f">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </constantatomformula>
                                                                                       </formulalist>
                                                                                      </sequent>
                                                                                      <rule type="axiom" symbol="axiom">
                                                                                       <sequent>
                                                                                        <formulalist>
                                                                                         <constantatomformula symbol="=">
                                                                                          <function symbol="f">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{110}"/>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                            </function>
                                                                                            <variable symbol="x_{109}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <constant symbol="1"/>
                                                                                         </constantatomformula>
                                                                                        </formulalist>
                                                                                        <formulalist>
                                                                                         <constantatomformula symbol="=">
                                                                                          <function symbol="f">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{110}"/>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                            </function>
                                                                                            <variable symbol="x_{109}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <constant symbol="1"/>
                                                                                         </constantatomformula>
                                                                                        </formulalist>
                                                                                       </sequent>
                                                                                      </rule>
                                                                                      <rule type="eqr2" symbol="=:r2" param="(2)">
                                                                                       <sequent>
                                                                                        <formulalist>
                                                                                         <constantatomformula symbol="=">
                                                                                          <function symbol="f">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{110}"/>
                                                                                                <constant symbol="1"/>
                                                                                               </function>
                                                                                              </function>
                                                                                              <variable symbol="x_{109}"/>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <constant symbol="1"/>
                                                                                         </constantatomformula>
                                                                                        </formulalist>
                                                                                        <formulalist>
                                                                                         <constantatomformula symbol="=">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="f">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{110}"/>
                                                                                                <constant symbol="1"/>
                                                                                               </function>
                                                                                              </function>
                                                                                              <variable symbol="x_{109}"/>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </constantatomformula>
                                                                                        </formulalist>
                                                                                       </sequent>
                                                                                       <rule type="axiom" symbol="axiom">
                                                                                        <sequent>
                                                                                         <formulalist>
                                                                                          <constantatomformula symbol="=">
                                                                                           <function symbol="f">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                 <constant symbol="1"/>
                                                                                                </function>
                                                                                               </function>
                                                                                               <variable symbol="x_{109}"/>
                                                                                              </function>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </constantatomformula>
                                                                                         </formulalist>
                                                                                         <formulalist>
                                                                                          <constantatomformula symbol="=">
                                                                                           <function symbol="f">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                 <constant symbol="1"/>
                                                                                                </function>
                                                                                               </function>
                                                                                               <variable symbol="x_{109}"/>
                                                                                              </function>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </constantatomformula>
                                                                                         </formulalist>
                                                                                        </sequent>
                                                                                       </rule>
                                                                                       <rule type="axiom" symbol="axiom">
                                                                                        <sequent>
                                                                                         <formulalist/>
                                                                                         <formulalist>
                                                                                          <constantatomformula symbol="=">
                                                                                           <constant symbol="1"/>
                                                                                           <constant symbol="1"/>
                                                                                          </constantatomformula>
                                                                                         </formulalist>
                                                                                        </sequent>
                                                                                       </rule>
                                                                                      </rule>
                                                                                     </rule>
                                                                                    </rule>
                                                                                   </rule>
                                                                                  </rule>
                                                                                 </rule>
                                                                                </rule>
                                                                               </rule>
                                                                              </rule>
                                                                             </rule>
                                                                            </rule>
                                                                           </rule>
                                                                          </rule>
                                                                         </rule>
                                                                        </rule>
                                                                       </rule>
                                                                      </rule>
                                                                     </rule>
                                                                    </rule>
                                                                   </rule>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                  <rule type="contrr" symbol="c:r" param="2,1">
                                   <sequent>
                                    <formulalist>
                                     <constantatomformula symbol="=">
                                      <function symbol="f">
                                       <function symbol="+">
                                        <variable symbol="x_{109}"/>
                                        <function symbol="+">
                                         <constant symbol="1"/>
                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <function symbol="+">
                                            <variable symbol="x_{110}"/>
                                            <constant symbol="1"/>
                                           </function>
                                          </function>
                                         </function>
                                        </function>
                                       </function>
                                      </function>
                                      <constant symbol="0"/>
                                     </constantatomformula>
                                     <constantatomformula symbol="A" definition="A"/>
                                    </formulalist>
                                    <formulalist>
                                     <quantifiedformula type="exists">
                                      <variable symbol="p"/>
                                      <quantifiedformula type="exists">
                                       <variable symbol="q"/>
                                       <conjunctiveformula type="and">
                                        <conjunctiveformula type="neg">
                                         <constantatomformula symbol="=">
                                          <variable symbol="p"/>
                                          <variable symbol="q"/>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                        <constantatomformula symbol="=">
                                         <function symbol="f">
                                          <variable symbol="p"/>
                                         </function>
                                         <function symbol="f">
                                          <variable symbol="q"/>
                                         </function>
                                        </constantatomformula>
                                       </conjunctiveformula>
                                      </quantifiedformula>
                                     </quantifiedformula>
                                     <constantatomformula symbol="=">
                                      <function symbol="f">
                                       <function symbol="+">
                                        <variable symbol="x_{109}"/>
                                        <function symbol="+">
                                         <constant symbol="1"/>
                                         <function symbol="+">
                                          <variable symbol="x_{110}"/>
                                          <constant symbol="1"/>
                                         </function>
                                        </function>
                                       </function>
                                      </function>
                                      <constant symbol="0"/>
                                     </constantatomformula>
                                    </formulalist>
                                   </sequent>
                                   <rule type="permr" symbol="\pi:r" param="(2 3)">
                                    <sequent>
                                     <formulalist>
                                      <constantatomformula symbol="=">
                                       <function symbol="f">
                                        <function symbol="+">
                                         <variable symbol="x_{109}"/>
                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
                                             <variable symbol="x_{110}"/>
                                             <constant symbol="1"/>
                                            </function>
                                           </function>
                                          </function>
                                         </function>
                                        </function>
                                       </function>
                                       <constant symbol="0"/>
                                      </constantatomformula>
                                      <constantatomformula symbol="A" definition="A"/>
                                     </formulalist>
                                     <formulalist>
                                      <quantifiedformula type="exists">
                                       <variable symbol="p"/>
                                       <quantifiedformula type="exists">
                                        <variable symbol="q"/>
                                        <conjunctiveformula type="and">
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                                          <constantatomformula symbol="=">
                                           <variable symbol="p"/>
                                           <variable symbol="q"/>
                                          </constantatomformula>
                                         </conjunctiveformula>
                                         <constantatomformula symbol="=">
                                          <function symbol="f">
                                           <variable symbol="p"/>
                                          </function>
                                          <function symbol="f">
                                           <variable symbol="q"/>
                                          </function>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                       </quantifiedformula>
                                      </quantifiedformula>
                                      <quantifiedformula type="exists">
                                       <variable symbol="p"/>
                                       <quantifiedformula type="exists">
                                        <variable symbol="q"/>
                                        <conjunctiveformula type="and">
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                                          <constantatomformula symbol="=">
                                           <variable symbol="p"/>
                                           <variable symbol="q"/>
                                          </constantatomformula>
                                         </conjunctiveformula>
                                         <constantatomformula symbol="=">
                                          <function symbol="f">
                                           <variable symbol="p"/>
                                          </function>
                                          <function symbol="f">
                                           <variable symbol="q"/>
                                          </function>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                       </quantifiedformula>
                                      </quantifiedformula>
                                      <constantatomformula symbol="=">
                                       <function symbol="f">
                                        <function symbol="+">
                                         <variable symbol="x_{109}"/>
                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <function symbol="+">
                                           <variable symbol="x_{110}"/>
                                           <constant symbol="1"/>
                                          </function>
                                         </function>
                                        </function>
                                       </function>
                                       <constant symbol="0"/>
                                      </constantatomformula>
                                     </formulalist>
                                    </sequent>
                                    <rule type="contrl" symbol="c:l" param="1,2">
                                     <sequent>
                                      <formulalist>
                                       <constantatomformula symbol="=">
                                        <function symbol="f">
                                         <function symbol="+">
                                          <variable symbol="x_{109}"/>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <variable symbol="x_{110}"/>
                                              <constant symbol="1"/>
                                             </function>
                                            </function>
                                           </function>
                                          </function>
                                         </function>
                                        </function>
                                        <constant symbol="0"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="A" definition="A"/>
                                      </formulalist>
                                      <formulalist>
                                       <quantifiedformula type="exists">
                                        <variable symbol="p"/>
                                        <quantifiedformula type="exists">
                                         <variable symbol="q"/>
                                         <conjunctiveformula type="and">
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                                           <constantatomformula symbol="=">
                                            <variable symbol="p"/>
                                            <variable symbol="q"/>
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                                          </conjunctiveformula>
                                          <constantatomformula symbol="=">
                                           <function symbol="f">
                                            <variable symbol="p"/>
                                           </function>
                                           <function symbol="f">
                                            <variable symbol="q"/>
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                                          </constantatomformula>
                                         </conjunctiveformula>
                                        </quantifiedformula>
                                       </quantifiedformula>
                                       <constantatomformula symbol="=">
                                        <function symbol="f">
                                         <function symbol="+">
                                          <variable symbol="x_{109}"/>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <function symbol="+">
                                            <variable symbol="x_{110}"/>
                                            <constant symbol="1"/>
                                           </function>
                                          </function>
                                         </function>
                                        </function>
                                        <constant symbol="0"/>
                                       </constantatomformula>
                                       <quantifiedformula type="exists">
                                        <variable symbol="p"/>
                                        <quantifiedformula type="exists">
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                                           <constantatomformula symbol="=">
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                                          </conjunctiveformula>
                                          <constantatomformula symbol="=">
                                           <function symbol="f">
                                            <variable symbol="p"/>
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                                           <function symbol="f">
                                            <variable symbol="q"/>
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                                          </constantatomformula>
                                         </conjunctiveformula>
                                        </quantifiedformula>
                                       </quantifiedformula>
                                      </formulalist>
                                     </sequent>
                                     <rule type="perml" symbol="\pi:l" param="(1 2)">
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                                       <formulalist>
                                        <constantatomformula symbol="=">
                                         <function symbol="f">
                                          <function symbol="+">
                                           <variable symbol="x_{109}"/>
                                           <function symbol="+">
                                            <constant symbol="1"/>
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                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                          </function>
                                         </function>
                                         <constant symbol="0"/>
                                        </constantatomformula>
                                        <constantatomformula symbol="A" definition="A"/>
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                                       </formulalist>
                                       <formulalist>
                                        <quantifiedformula type="exists">
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                                           <constantatomformula symbol="=">
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                                            <function symbol="f">
                                             <variable symbol="q"/>
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                                           </constantatomformula>
                                          </conjunctiveformula>
                                         </quantifiedformula>
                                        </quantifiedformula>
                                        <constantatomformula symbol="=">
                                         <function symbol="f">
                                          <function symbol="+">
                                           <variable symbol="x_{109}"/>
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
                                             <variable symbol="x_{110}"/>
                                             <constant symbol="1"/>
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                                           </function>
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                                         </function>
                                         <constant symbol="0"/>
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                                        <quantifiedformula type="exists">
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                                           <constantatomformula symbol="=">
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                                             <variable symbol="p"/>
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                                            <function symbol="f">
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                                           </constantatomformula>
                                          </conjunctiveformula>
                                         </quantifiedformula>
                                        </quantifiedformula>
                                       </formulalist>
                                      </sequent>
                                      <rule type="cut" symbol="cut">
                                       <sequent>
                                        <formulalist>
                                         <constantatomformula symbol="A" definition="A"/>
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                                           <function symbol="+">
                                            <variable symbol="x_{109}"/>
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                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <variable symbol="x_{110}"/>
                                                <constant symbol="1"/>
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                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                          </function>
                                          <constant symbol="0"/>
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                                        <formulalist>
                                         <quantifiedformula type="exists">
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                                            <constantatomformula symbol="=">
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                                             <function symbol="f">
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                                            </constantatomformula>
                                           </conjunctiveformula>
                                          </quantifiedformula>
                                         </quantifiedformula>
                                         <constantatomformula symbol="=">
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                                            <function symbol="+">
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                                             <function symbol="+">
                                              <variable symbol="x_{110}"/>
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                                          <constant symbol="0"/>
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                                         <quantifiedformula type="exists">
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                                          <quantifiedformula type="exists">
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                                             <constantatomformula symbol="=">
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                                            <constantatomformula symbol="=">
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                                             <function symbol="f">
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                                            </constantatomformula>
                                           </conjunctiveformula>
                                          </quantifiedformula>
                                         </quantifiedformula>
                                        </formulalist>
                                       </sequent>
                                       <rule type="permr" symbol="\pi:r" param="(2 3)">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="A" definition="A"/>
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                                         <formulalist>
                                          <quantifiedformula type="exists">
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                                          <constantatomformula symbol="=">
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                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
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                                         </formulalist>
                                        </sequent>
                                        <rule type="contrr" symbol="c:r" param="2,1,1">
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                                          <formulalist>
                                           <constantatomformula symbol="A" definition="A"/>
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                                               <function symbol="+">
                                                <variable symbol="x_{110}"/>
                                                <constant symbol="1"/>
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                                             </function>
                                            </function>
                                            <constant symbol="0"/>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="permr" symbol="\pi:r" param="">
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                                           <formulalist>
                                            <constantatomformula symbol="A" definition="A"/>
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                                           <formulalist>
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                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{110}"/>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="axiom" symbol="axiom">
                                                       <sequent>
                                                        <formulalist/>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{110}"/>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                  <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="weakr" symbol="w:r">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </constantatomformula>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="perml" symbol="\pi:l" param="">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="weakl" symbol="w:l">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="axiom" symbol="axiom">
                                                       <sequent>
                                                        <formulalist/>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                             <rule type="contrr" symbol="c:r" param="2,1,1">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{110}"/>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="permr" symbol="\pi:r" param="">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{110}"/>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="contrl" symbol="c:l" param="2">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{110}"/>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="perml" symbol="\pi:l" param="">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{110}"/>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="eqr1" symbol="=:r1" param="(1 1)">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{110}"/>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="contrr" symbol="c:r" param="2,1">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{110}"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{110}"/>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="permr" symbol="\pi:r" param="">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{110}"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{110}"/>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="contrl" symbol="c:l" param="2">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{110}"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{110}"/>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="perml" symbol="\pi:l" param="">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{110}"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{110}"/>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="eqr1" symbol="=:r1" param="(1)">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
                                                            <conjunctiveformula type="neg">
                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
                                                            <conjunctiveformula type="neg">
                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{110}"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{110}"/>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
                                                            <conjunctiveformula type="and">
                                                             <conjunctiveformula type="neg">
                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{110}"/>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{110}"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="weakr" symbol="w:r">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{110}"/>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{110}"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </constantatomformula>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="p"/>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="q"/>
                                                             <conjunctiveformula type="and">
                                                              <conjunctiveformula type="neg">
                                                               <constantatomformula symbol="=">
                                                                <variable symbol="p"/>
                                                                <variable symbol="q"/>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="perml" symbol="\pi:l" param="">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{110}"/>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{110}"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="weakl" symbol="w:l">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="A" definition="A"/>
                                                            </formulalist>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{110}"/>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{110}"/>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="axiom" symbol="axiom">
                                                            <sequent>
                                                             <formulalist/>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{110}"/>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{110}"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                       <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
                                                            <conjunctiveformula type="and">
                                                             <conjunctiveformula type="neg">
                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{110}"/>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{110}"/>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="weakr" symbol="w:r">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{110}"/>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{110}"/>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </constantatomformula>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="p"/>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="q"/>
                                                             <conjunctiveformula type="and">
                                                              <conjunctiveformula type="neg">
                                                               <constantatomformula symbol="=">
                                                                <variable symbol="p"/>
                                                                <variable symbol="q"/>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="perml" symbol="\pi:l" param="">
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                                                           <formulalist>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{110}"/>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{110}"/>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="weakl" symbol="w:l">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="A" definition="A"/>
                                                            </formulalist>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{110}"/>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{110}"/>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="axiom" symbol="axiom">
                                                            <sequent>
                                                             <formulalist/>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{110}"/>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{110}"/>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                  <rule type="permr" symbol="\pi:r" param="(2 3)">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
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                                                    <formulalist>
                                                     <quantifiedformula type="exists">
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                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
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                                                         <constantatomformula symbol="=">
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                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{110}"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="contrr" symbol="c:r" param="2,1,1">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
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                                                          <constantatomformula symbol="=">
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                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{110}"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
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                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <constant symbol="1"/>
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                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="permr" symbol="\pi:r" param="(2 3)">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
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                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
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                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{110}"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="0"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="0"/>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="contrl" symbol="c:l" param="2">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{110}"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="0"/>
                                                        </constantatomformula>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="0"/>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="perml" symbol="\pi:l" param="">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
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                                                             <constantatomformula symbol="=">
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                                                             </constantatomformula>
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                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{110}"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
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                                                             <constantatomformula symbol="=">
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                                                              <variable symbol="q"/>
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                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="cut" symbol="cut">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
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                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
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                                                          </quantifiedformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{110}"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="0"/>
                                                          </constantatomformula>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
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                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
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                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
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                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="0"/>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="permr" symbol="\pi:r" param="(1 3 2)">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <quantifiedformula type="exists">
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                                                            <quantifiedformula type="exists">
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                                                               <constantatomformula symbol="=">
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                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
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                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
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                                                              </constantatomformula>
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                                                           </quantifiedformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{110}"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
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                                                            </function>
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                                                           </constantatomformula>
                                                           <constantatomformula symbol="=">
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                                                             <function symbol="+">
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                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{110}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
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                                                         </sequent>
                                                         <rule type="perml" symbol="\pi:l" param="">
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                                                           <formulalist>
                                                            <constantatomformula symbol="A" definition="A"/>
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                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
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                                                                 <variable symbol="x_{110}"/>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
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                                                             </function>
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                                                            <constantatomformula symbol="=">
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                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{110}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
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                                                               <constantatomformula symbol="=">
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                                                               </constantatomformula>
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                                                             </quantifiedformula>
                                                            </quantifiedformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="contrr" symbol="c:r" param="1,1,2">
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                                                            <formulalist>
                                                             <constantatomformula symbol="A" definition="A"/>
                                                            </formulalist>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
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                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{110}"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
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                                                              </function>
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                                                             </constantatomformula>
                                                             <constantatomformula symbol="=">
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                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{110}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
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                                                                </constantatomformula>
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                                                           </sequent>
                                                           <rule type="weakr" symbol="w:r">
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                                                             <formulalist>
                                                              <constantatomformula symbol="A" definition="A"/>
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                                                             <formulalist>
                                                              <constantatomformula symbol="=">
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                                                                  <variable symbol="x_{109}"/>
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                                                               </function>
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                                                              <constantatomformula symbol="=">
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                                                                   <variable symbol="x_{110}"/>
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                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
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                                                              <quantifiedformula type="exists">
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                                                                 </constantatomformula>
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                                                              </quantifiedformula>
                                                              <quantifiedformula type="exists">
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                                                                 </constantatomformula>
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                                                               </quantifiedformula>
                                                              </quantifiedformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="weakr" symbol="w:r">
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                                                              <formulalist>
                                                               <constantatomformula symbol="A" definition="A"/>
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                                                              <formulalist>
                                                               <constantatomformula symbol="=">
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                                                                    <variable symbol="x_{110}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
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                                                               <quantifiedformula type="exists">
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                                                                  </constantatomformula>
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                                                                </quantifiedformula>
                                                               </quantifiedformula>
                                                              </formulalist>
                                                             </sequent>
                                                             <rule type="defl" symbol="d:l" param="();[()];()">
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                                                               <formulalist>
                                                                <constantatomformula symbol="A" definition="A"/>
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                                                               <formulalist>
                                                                <constantatomformula symbol="=">
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                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
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                                                               </formulalist>
                                                              </sequent>
                                                              <rule type="foralll" symbol="\all:l">
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                                                                <formulalist>
                                                                 <quantifiedformula type="all">
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                                                                </formulalist>
                                                                <formulalist>
                                                                 <constantatomformula symbol="=">
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                                                                  </function>
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                                                                      <variable symbol="x_{110}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
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                                                                </formulalist>
                                                               </sequent>
                                                               <rule type="orl" symbol="\or:l">
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                                                                      </function>
                                                                      <constant symbol="1"/>
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                                                                    </function>
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                                                                  </conjunctiveformula>
                                                                 </formulalist>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
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                                                                       <variable symbol="x_{110}"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
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                                                                   </function>
                                                                   <constant symbol="0"/>
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                                                                  <constantatomformula symbol="=">
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                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{110}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </constantatomformula>
                                                                 </formulalist>
                                                                </sequent>
                                                                <rule type="axiom" symbol="axiom">
                                                                 <sequent>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{110}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="0"/>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{110}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="0"/>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                </rule>
                                                                <rule type="eql2" symbol="=:l2" param="(1 1)">
                                                                 <sequent>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{110}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
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                                                                  </formulalist>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{110}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                 <rule type="axiom" symbol="axiom">
                                                                  <sequent>
                                                                   <formulalist/>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{110}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{110}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                 </rule>
                                                                 <rule type="axiom" symbol="axiom">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{110}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
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                                                                   </formulalist>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
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                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{110}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
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                                                                   </formulalist>
                                                                  </sequent>
                                                                 </rule>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                        <rule type="contrr" symbol="c:r" param="2,1">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{110}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="p"/>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="q"/>
                                                             <conjunctiveformula type="and">
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                                                               <constantatomformula symbol="=">
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                                                               </constantatomformula>
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                                                              <constantatomformula symbol="=">
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                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
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                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="0"/>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="permr" symbol="\pi:r" param="(2 3)">
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                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{110}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <quantifiedformula type="exists">
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                                                             <quantifiedformula type="exists">
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                                                               <constantatomformula symbol="=">
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                                                                 <variable symbol="p"/>
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                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
                                                                </function>
                                                               </constantatomformula>
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                                                               <constantatomformula symbol="=">
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                                                                 <variable symbol="q"/>
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                                                               </constantatomformula>
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                                                            <constantatomformula symbol="=">
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                                                               <variable symbol="x_{109}"/>
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                                                                 <function symbol="+">
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                                                                  <variable symbol="x_{110}"/>
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                                                                </function>
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                                                            <formulalist>
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                                                             <constantatomformula symbol="=">
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                                                              </sequent>
                                                              <rule type="perml" symbol="\pi:l" param="">
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                                                                <formulalist>
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                                                               </sequent>
                                                               <rule type="contrr" symbol="c:r" param="1,1,2">
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                                                                 <formulalist>
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                                                                 <formulalist>
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                                                                </sequent>
                                                                <rule type="weakr" symbol="w:r">
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                                                                  <formulalist>
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                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
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                                                                      </constantatomformula>
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                                                                  </formulalist>
                                                                 </sequent>
                                                                 <rule type="weakr" symbol="w:r">
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                                                                   <formulalist>
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                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
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                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{109}"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </constantatomformula>
                                                                    <quantifiedformula type="exists">
                                                                     <variable symbol="p"/>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="q"/>
                                                                      <conjunctiveformula type="and">
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                                                                        <constantatomformula symbol="=">
                                                                         <variable symbol="p"/>
                                                                         <variable symbol="q"/>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <variable symbol="p"/>
                                                                        </function>
                                                                        <function symbol="f">
                                                                         <variable symbol="q"/>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </conjunctiveformula>
                                                                     </quantifiedformula>
                                                                    </quantifiedformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                  <rule type="defl" symbol="d:l" param="();[()];()">
                                                                   <sequent>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="A" definition="A"/>
                                                                    </formulalist>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="0"/>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                   <rule type="foralll" symbol="\all:l">
                                                                    <sequent>
                                                                     <formulalist>
                                                                      <quantifiedformula type="all">
                                                                       <variable symbol="x"/>
                                                                       <conjunctiveformula type="or">
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <variable symbol="x"/>
                                                                         </function>
                                                                         <constant symbol="0"/>
                                                                        </constantatomformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <variable symbol="x"/>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </constantatomformula>
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                                                                      </quantifiedformula>
                                                                     </formulalist>
                                                                     <formulalist>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="f">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{109}"/>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="0"/>
                                                                      </constantatomformula>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="f">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
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                                                                    </sequent>
                                                                    <rule type="orl" symbol="\or:l">
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                                                                       <conjunctiveformula type="or">
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="0"/>
                                                                        </constantatomformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
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                                                                      <formulalist>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="0"/>
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                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
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                                                                      </formulalist>
                                                                     </sequent>
                                                                     <rule type="axiom" symbol="axiom">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="0"/>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="0"/>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                     </rule>
                                                                     <rule type="eql2" symbol="=:l2" param="(1 1)">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                      <rule type="axiom" symbol="axiom">
                                                                       <sequent>
                                                                        <formulalist/>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                      </rule>
                                                                      <rule type="axiom" symbol="axiom">
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                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </constantatomformula>
                                                                        </formulalist>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
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                                                                        </formulalist>
                                                                       </sequent>
                                                                      </rule>
                                                                     </rule>
                                                                    </rule>
                                                                   </rule>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                             <rule type="perml" symbol="\pi:l" param="(1 2)">
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                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{109}"/>
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                                                                <constantatomformula symbol="=">
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                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{109}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{110}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </constantatomformula>
                                                                <constantatomformula symbol="A" definition="A"/>
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                                                               <formulalist>
                                                                <quantifiedformula type="exists">
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                                                                 <quantifiedformula type="exists">
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                                                                   <constantatomformula symbol="=">
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                                                                    <function symbol="f">
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                                                                   </constantatomformula>
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                                                                 </quantifiedformula>
                                                                </quantifiedformula>
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                                                              <rule type="contrr" symbol="c:r" param="2">
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                                                                 <constantatomformula symbol="=">
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                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{109}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{110}"/>
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                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
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                                                                 <constantatomformula symbol="=">
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                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{109}"/>
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                                                                  <constant symbol="1"/>
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                                                                 <constantatomformula symbol="A" definition="A"/>
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                                                                <formulalist>
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                                                                    </constantatomformula>
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                                                                      <function symbol="+">
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                                                                       <variable symbol="x_{110}"/>
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                                                                 <formulalist>
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                                                                    <formulalist>
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                                                                   </sequent>
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                                                                         </constantatomformula>
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                                                                     </formulalist>
                                                                    </sequent>
                                                                    <rule type="permr" symbol="\pi:r" param="">
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                                                                      <formulalist>
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                                                                      <formulalist>
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                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <variable symbol="p"/>
                                                                           </function>
                                                                           <function symbol="f">
                                                                            <variable symbol="q"/>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                        </quantifiedformula>
                                                                       </quantifiedformula>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="p"/>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="q"/>
                                                                         <conjunctiveformula type="and">
                                                                          <conjunctiveformula type="neg">
                                                                           <constantatomformula symbol="=">
                                                                            <variable symbol="p"/>
                                                                            <variable symbol="q"/>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <variable symbol="p"/>
                                                                           </function>
                                                                           <function symbol="f">
                                                                            <variable symbol="q"/>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                        </quantifiedformula>
                                                                       </quantifiedformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                         <variable symbol="x_{110}"/>
                                                                        </function>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{109}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{110}"/>
                                                                         </function>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </formulalist>
                                                                     </sequent>
                                                                     <rule type="contrl" symbol="c:l" param="2">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="A" definition="A"/>
                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="p"/>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="q"/>
                                                                          <conjunctiveformula type="and">
                                                                           <conjunctiveformula type="neg">
                                                                            <constantatomformula symbol="=">
                                                                             <variable symbol="p"/>
                                                                             <variable symbol="q"/>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <variable symbol="p"/>
                                                                            </function>
                                                                            <function symbol="f">
                                                                             <variable symbol="q"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                         </quantifiedformula>
                                                                        </quantifiedformula>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="p"/>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="q"/>
                                                                          <conjunctiveformula type="and">
                                                                           <conjunctiveformula type="neg">
                                                                            <constantatomformula symbol="=">
                                                                             <variable symbol="p"/>
                                                                             <variable symbol="q"/>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <variable symbol="p"/>
                                                                            </function>
                                                                            <function symbol="f">
                                                                             <variable symbol="q"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                         </quantifiedformula>
                                                                        </quantifiedformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <variable symbol="x_{110}"/>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{110}"/>
                                                                          </function>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                      <rule type="perml" symbol="\pi:l" param="">
                                                                       <sequent>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="A" definition="A"/>
                                                                         <constantatomformula symbol="A" definition="A"/>
                                                                        </formulalist>
                                                                        <formulalist>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="p"/>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="q"/>
                                                                           <conjunctiveformula type="and">
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                                                                             <constantatomformula symbol="=">
                                                                              <variable symbol="p"/>
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                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <variable symbol="p"/>
                                                                             </function>
                                                                             <function symbol="f">
                                                                              <variable symbol="q"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                          </quantifiedformula>
                                                                         </quantifiedformula>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="p"/>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="q"/>
                                                                           <conjunctiveformula type="and">
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                                                                             <constantatomformula symbol="=">
                                                                              <variable symbol="p"/>
                                                                              <variable symbol="q"/>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <variable symbol="p"/>
                                                                             </function>
                                                                             <function symbol="f">
                                                                              <variable symbol="q"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                          </quantifiedformula>
                                                                         </quantifiedformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <variable symbol="x_{110}"/>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{110}"/>
                                                                           </function>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                       <rule type="eqr1" symbol="=:r1" param="(1)">
                                                                        <sequent>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                         </formulalist>
                                                                         <formulalist>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="p"/>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="q"/>
                                                                            <conjunctiveformula type="and">
                                                                             <conjunctiveformula type="neg">
                                                                              <constantatomformula symbol="=">
                                                                               <variable symbol="p"/>
                                                                               <variable symbol="q"/>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <variable symbol="p"/>
                                                                              </function>
                                                                              <function symbol="f">
                                                                               <variable symbol="q"/>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                           </quantifiedformula>
                                                                          </quantifiedformula>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="p"/>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="q"/>
                                                                            <conjunctiveformula type="and">
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                                                                              <constantatomformula symbol="=">
                                                                               <variable symbol="p"/>
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                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <variable symbol="p"/>
                                                                              </function>
                                                                              <function symbol="f">
                                                                               <variable symbol="q"/>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                           </quantifiedformula>
                                                                          </quantifiedformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{109}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <variable symbol="x_{110}"/>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{110}"/>
                                                                            </function>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </formulalist>
                                                                        </sequent>
                                                                        <rule type="permr" symbol="\pi:r" param="(1 2)">
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                                                                          <formulalist>
                                                                           <constantatomformula symbol="A" definition="A"/>
                                                                          </formulalist>
                                                                          <formulalist>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="p"/>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="q"/>
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                                                                               <constantatomformula symbol="=">
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                                                                               </constantatomformula>
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                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <variable symbol="p"/>
                                                                               </function>
                                                                               <function symbol="f">
                                                                                <variable symbol="q"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                            </quantifiedformula>
                                                                           </quantifiedformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{109}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{110}"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{109}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                             <variable symbol="x_{110}"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </formulalist>
                                                                         </sequent>
                                                                         <rule type="weakr" symbol="w:r">
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                                                                           <formulalist>
                                                                            <constantatomformula symbol="A" definition="A"/>
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                                                                           <formulalist>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{109}"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{110}"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{109}"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                              <variable symbol="x_{110}"/>
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                                                                            <quantifiedformula type="exists">
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                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <variable symbol="p"/>
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                                                                                <function symbol="f">
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                                                                               </constantatomformula>
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                                                                             </quantifiedformula>
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                                                                           </formulalist>
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                                                                          <rule type="perml" symbol="\pi:l" param="">
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                                                                            <formulalist>
                                                                             <constantatomformula symbol="A" definition="A"/>
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                                                                            <formulalist>
                                                                             <constantatomformula symbol="=">
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                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <variable symbol="x_{110}"/>
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                                                                              </function>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{109}"/>
                                                                                <constant symbol="1"/>
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                                                                               <variable symbol="x_{110}"/>
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                                                                             </constantatomformula>
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                                                                           <rule type="weakl" symbol="w:l">
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                                                                             <formulalist>
                                                                              <constantatomformula symbol="=">
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                                                                                <variable symbol="x_{109}"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <variable symbol="x_{110}"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{109}"/>
                                                                                 <constant symbol="1"/>
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                                                                                <variable symbol="x_{110}"/>
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                                                                              </constantatomformula>
                                                                             </formulalist>
                                                                            </sequent>
                                                                            <rule type="axiom" symbol="axiom">
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                                                                              <formulalist/>
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                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{109}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{110}"/>
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                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{109}"/>
                                                                                  <constant symbol="1"/>
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                                                                                 <variable symbol="x_{110}"/>
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                                                                             </sequent>
                                                                            </rule>
                                                                           </rule>
                                                                          </rule>
                                                                         </rule>
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                                                                        <rule type="permr" symbol="\pi:r" param="(1 2)">
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                                                                          <formulalist>
                                                                           <constantatomformula symbol="A" definition="A"/>
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                                                                          <formulalist>
                                                                           <quantifiedformula type="exists">
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                                                                              <constantatomformula symbol="=">
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                                                                                <variable symbol="p"/>
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                                                                               <function symbol="f">
                                                                                <variable symbol="q"/>
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                                                                              </constantatomformula>
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                                                                            </quantifiedformula>
                                                                           </quantifiedformula>
                                                                           <constantatomformula symbol="=">
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                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{110}"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{109}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{110}"/>
                                                                             </function>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </formulalist>
                                                                         </sequent>
                                                                         <rule type="weakr" symbol="w:r">
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                                                                           <formulalist>
                                                                            <constantatomformula symbol="A" definition="A"/>
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                                                                           <formulalist>
                                                                            <constantatomformula symbol="=">
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                                                                              <variable symbol="x_{109}"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{110}"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{109}"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{110}"/>
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                                                                             </function>
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                                                                            <quantifiedformula type="exists">
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                                                                               <constantatomformula symbol="=">
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                                                                                 <variable symbol="p"/>
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                                                                                <function symbol="f">
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                                                                               </constantatomformula>
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                                                                             </quantifiedformula>
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                                                                           </formulalist>
                                                                          </sequent>
                                                                          <rule type="perml" symbol="\pi:l" param="">
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                                                                            <formulalist>
                                                                             <constantatomformula symbol="A" definition="A"/>
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                                                                            <formulalist>
                                                                             <constantatomformula symbol="=">
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                                                                               <variable symbol="x_{109}"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <variable symbol="x_{110}"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{109}"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <variable symbol="x_{110}"/>
                                                                               </function>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </formulalist>
                                                                           </sequent>
                                                                           <rule type="weakl" symbol="w:l">
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                                                                             <formulalist>
                                                                              <constantatomformula symbol="A" definition="A"/>
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                                                                             <formulalist>
                                                                              <constantatomformula symbol="=">
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                                                                                <variable symbol="x_{109}"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <variable symbol="x_{110}"/>
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                                                                               </function>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{109}"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <variable symbol="x_{110}"/>
                                                                                </function>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </formulalist>
                                                                            </sequent>
                                                                            <rule type="axiom" symbol="axiom">
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                                                                              <formulalist/>
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                                                                               <constantatomformula symbol="=">
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                                                                                 <variable symbol="x_{109}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{110}"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{109}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{110}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </formulalist>
                                                                             </sequent>
                                                                            </rule>
                                                                           </rule>
                                                                          </rule>
                                                                         </rule>
                                                                        </rule>
                                                                       </rule>
                                                                      </rule>
                                                                     </rule>
                                                                    </rule>
                                                                   </rule>
                                                                   <rule type="perml" symbol="\pi:l" param="(1 2)">
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                                                                     <formulalist>
                                                                      <constantatomformula symbol="=">
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                                                                          <variable symbol="x_{110}"/>
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                                                                      <constantatomformula symbol="=">
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                                                                      <constantatomformula symbol="A" definition="A"/>
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                                                                     <formulalist>
                                                                      <quantifiedformula type="exists">
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                                                                       <quantifiedformula type="exists">
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                                                                          </function>
                                                                          <function symbol="f">
                                                                           <variable symbol="q"/>
                                                                          </function>
                                                                         </constantatomformula>
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                                                                       </quantifiedformula>
                                                                      </quantifiedformula>
                                                                     </formulalist>
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                                                                    <rule type="perml" symbol="\pi:l" param="(1 2)">
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                                                                      <formulalist>
                                                                       <constantatomformula symbol="=">
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                                                                          <variable symbol="x_{109}"/>
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                                                                        <constant symbol="1"/>
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                                                                       <constantatomformula symbol="=">
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                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <variable symbol="x_{110}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </constantatomformula>
                                                                       <constantatomformula symbol="A" definition="A"/>
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                                                                      <formulalist>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="p"/>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="q"/>
                                                                         <conjunctiveformula type="and">
                                                                          <conjunctiveformula type="neg">
                                                                           <constantatomformula symbol="=">
                                                                            <variable symbol="p"/>
                                                                            <variable symbol="q"/>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <variable symbol="p"/>
                                                                           </function>
                                                                           <function symbol="f">
                                                                            <variable symbol="q"/>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                        </quantifiedformula>
                                                                       </quantifiedformula>
                                                                      </formulalist>
                                                                     </sequent>
                                                                     <rule type="contrr" symbol="c:r" param="2">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{109}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <variable symbol="x_{110}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </constantatomformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </constantatomformula>
                                                                        <constantatomformula symbol="A" definition="A"/>
                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="p"/>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="q"/>
                                                                          <conjunctiveformula type="and">
                                                                           <conjunctiveformula type="neg">
                                                                            <constantatomformula symbol="=">
                                                                             <variable symbol="p"/>
                                                                             <variable symbol="q"/>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <variable symbol="p"/>
                                                                            </function>
                                                                            <function symbol="f">
                                                                             <variable symbol="q"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                         </quantifiedformula>
                                                                        </quantifiedformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                      <rule type="permr" symbol="\pi:r" param="">
                                                                       <sequent>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{109}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                             <variable symbol="x_{110}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </constantatomformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </constantatomformula>
                                                                         <constantatomformula symbol="A" definition="A"/>
                                                                        </formulalist>
                                                                        <formulalist>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="p"/>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="q"/>
                                                                           <conjunctiveformula type="and">
                                                                            <conjunctiveformula type="neg">
                                                                             <constantatomformula symbol="=">
                                                                              <variable symbol="p"/>
                                                                              <variable symbol="q"/>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <variable symbol="p"/>
                                                                             </function>
                                                                             <function symbol="f">
                                                                              <variable symbol="q"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                          </quantifiedformula>
                                                                         </quantifiedformula>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="p"/>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="q"/>
                                                                           <conjunctiveformula type="and">
                                                                            <conjunctiveformula type="neg">
                                                                             <constantatomformula symbol="=">
                                                                              <variable symbol="p"/>
                                                                              <variable symbol="q"/>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <variable symbol="p"/>
                                                                             </function>
                                                                             <function symbol="f">
                                                                              <variable symbol="q"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                          </quantifiedformula>
                                                                         </quantifiedformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                       <rule type="contrl" symbol="c:l" param="1,1,2">
                                                                        <sequent>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{109}"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                              <variable symbol="x_{110}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </constantatomformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{109}"/>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </constantatomformula>
                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                         </formulalist>
                                                                         <formulalist>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="p"/>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="q"/>
                                                                            <conjunctiveformula type="and">
                                                                             <conjunctiveformula type="neg">
                                                                              <constantatomformula symbol="=">
                                                                               <variable symbol="p"/>
                                                                               <variable symbol="q"/>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <variable symbol="p"/>
                                                                              </function>
                                                                              <function symbol="f">
                                                                               <variable symbol="q"/>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                           </quantifiedformula>
                                                                          </quantifiedformula>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="p"/>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="q"/>
                                                                            <conjunctiveformula type="and">
                                                                             <conjunctiveformula type="neg">
                                                                              <constantatomformula symbol="=">
                                                                               <variable symbol="p"/>
                                                                               <variable symbol="q"/>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <variable symbol="p"/>
                                                                              </function>
                                                                              <function symbol="f">
                                                                               <variable symbol="q"/>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                           </quantifiedformula>
                                                                          </quantifiedformula>
                                                                         </formulalist>
                                                                        </sequent>
                                                                        <rule type="perml" symbol="\pi:l" param="(2 3)">
                                                                         <sequent>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{109}"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                               <variable symbol="x_{110}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </constantatomformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{109}"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </constantatomformula>
                                                                           <constantatomformula symbol="A" definition="A"/>
                                                                           <constantatomformula symbol="A" definition="A"/>
                                                                          </formulalist>
                                                                          <formulalist>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="p"/>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="q"/>
                                                                             <conjunctiveformula type="and">
                                                                              <conjunctiveformula type="neg">
                                                                               <constantatomformula symbol="=">
                                                                                <variable symbol="p"/>
                                                                                <variable symbol="q"/>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <variable symbol="p"/>
                                                                               </function>
                                                                               <function symbol="f">
                                                                                <variable symbol="q"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                            </quantifiedformula>
                                                                           </quantifiedformula>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="p"/>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="q"/>
                                                                             <conjunctiveformula type="and">
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                                                                               <constantatomformula symbol="=">
                                                                                <variable symbol="p"/>
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                                                                               </constantatomformula>
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                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <variable symbol="p"/>
                                                                               </function>
                                                                               <function symbol="f">
                                                                                <variable symbol="q"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                            </quantifiedformula>
                                                                           </quantifiedformula>
                                                                          </formulalist>
                                                                         </sequent>
                                                                         <rule type="eql1" symbol="=:l1" param="(1 1)">
                                                                          <sequent>
                                                                           <formulalist>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{109}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                                <variable symbol="x_{110}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </constantatomformula>
                                                                            <constantatomformula symbol="A" definition="A"/>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </constantatomformula>
                                                                            <constantatomformula symbol="A" definition="A"/>
                                                                           </formulalist>
                                                                           <formulalist>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="p"/>
                                                                             <quantifiedformula type="exists">
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                                                                                <constantatomformula symbol="=">
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                                                                               </conjunctiveformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <variable symbol="p"/>
                                                                                </function>
                                                                                <function symbol="f">
                                                                                 <variable symbol="q"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                             </quantifiedformula>
                                                                            </quantifiedformula>
                                                                            <quantifiedformula type="exists">
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                                                                                <constantatomformula symbol="=">
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                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <variable symbol="p"/>
                                                                                </function>
                                                                                <function symbol="f">
                                                                                 <variable symbol="q"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                             </quantifiedformula>
                                                                            </quantifiedformula>
                                                                           </formulalist>
                                                                          </sequent>
                                                                          <rule type="contrr" symbol="c:r" param="2,1">
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                                                                            <formulalist>
                                                                             <constantatomformula symbol="A" definition="A"/>
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                                                                            <formulalist>
                                                                             <quantifiedformula type="exists">
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                                                                              <quantifiedformula type="exists">
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                                                                                 <constantatomformula symbol="=">
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                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <variable symbol="p"/>
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                                                                                 <function symbol="f">
                                                                                  <variable symbol="q"/>
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                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                              </quantifiedformula>
                                                                             </quantifiedformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{109}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{110}"/>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{109}"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                                <variable symbol="x_{110}"/>
                                                                               </function>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </formulalist>
                                                                           </sequent>
                                                                           <rule type="permr" symbol="\pi:r" param="">
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                                                                             <formulalist>
                                                                              <constantatomformula symbol="A" definition="A"/>
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                                                                             <formulalist>
                                                                              <quantifiedformula type="exists">
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                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="f">
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                                                                                  <function symbol="f">
                                                                                   <variable symbol="q"/>
                                                                                  </function>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                               </quantifiedformula>
                                                                              </quantifiedformula>
                                                                              <quantifiedformula type="exists">
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                                                                               <quantifiedformula type="exists">
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                                                                                  <constantatomformula symbol="=">
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                                                                                 <constantatomformula symbol="=">
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                                                                                  <function symbol="f">
                                                                                   <variable symbol="q"/>
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                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                               </quantifiedformula>
                                                                              </quantifiedformula>
                                                                              <constantatomformula symbol="=">
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                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{109}"/>
                                                                                  <constant symbol="1"/>
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                                                                                </function>
                                                                                <variable symbol="x_{110}"/>
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                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{109}"/>
                                                                                  <constant symbol="1"/>
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                                                                                 <variable symbol="x_{110}"/>
                                                                                </function>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </formulalist>
                                                                            </sequent>
                                                                            <rule type="contrl" symbol="c:l" param="2">
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                                                                              <formulalist>
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                                                                              <formulalist>
                                                                               <quantifiedformula type="exists">
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                                                                                  <constantatomformula symbol="=">
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                                                                                    <variable symbol="q"/>
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                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
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                                                                               </quantifiedformula>
                                                                               <quantifiedformula type="exists">
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                                                                                    <variable symbol="q"/>
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                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                </quantifiedformula>
                                                                               </quantifiedformula>
                                                                               <constantatomformula symbol="=">
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                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{109}"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
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                                                                                <function symbol="+">
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                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
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                                                                                   <constant symbol="1"/>
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                                                                                  <variable symbol="x_{110}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </formulalist>
                                                                             </sequent>
                                                                             <rule type="perml" symbol="\pi:l" param="">
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                                                                               <formulalist>
                                                                                <constantatomformula symbol="A" definition="A"/>
                                                                                <constantatomformula symbol="A" definition="A"/>
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                                                                               <formulalist>
                                                                                <quantifiedformula type="exists">
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                                                                                    <function symbol="f">
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                                                                                    </function>
                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                 </quantifiedformula>
                                                                                </quantifiedformula>
                                                                                <quantifiedformula type="exists">
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                                                                                   <constantatomformula symbol="=">
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                                                                                    <function symbol="f">
                                                                                     <variable symbol="q"/>
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                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                 </quantifiedformula>
                                                                                </quantifiedformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{109}"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{110}"/>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{109}"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                   <variable symbol="x_{110}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </formulalist>
                                                                              </sequent>
                                                                              <rule type="eqr1" symbol="=:r1" param="(1)">
                                                                               <sequent>
                                                                                <formulalist>
                                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                                </formulalist>
                                                                                <formulalist>
                                                                                 <quantifiedformula type="exists">
                                                                                  <variable symbol="p"/>
                                                                                  <quantifiedformula type="exists">
                                                                                   <variable symbol="q"/>
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                                                                                     <constantatomformula symbol="=">
                                                                                      <variable symbol="p"/>
                                                                                      <variable symbol="q"/>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
                                                                                      <variable symbol="p"/>
                                                                                     </function>
                                                                                     <function symbol="f">
                                                                                      <variable symbol="q"/>
                                                                                     </function>
                                                                                    </constantatomformula>
                                                                                   </conjunctiveformula>
                                                                                  </quantifiedformula>
                                                                                 </quantifiedformula>
                                                                                 <quantifiedformula type="exists">
                                                                                  <variable symbol="p"/>
                                                                                  <quantifiedformula type="exists">
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                                                                                   <conjunctiveformula type="and">
                                                                                    <conjunctiveformula type="neg">
                                                                                     <constantatomformula symbol="=">
                                                                                      <variable symbol="p"/>
                                                                                      <variable symbol="q"/>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
                                                                                      <variable symbol="p"/>
                                                                                     </function>
                                                                                     <function symbol="f">
                                                                                      <variable symbol="q"/>
                                                                                     </function>
                                                                                    </constantatomformula>
                                                                                   </conjunctiveformula>
                                                                                  </quantifiedformula>
                                                                                 </quantifiedformula>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{109}"/>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <variable symbol="x_{110}"/>
                                                                                  </function>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{109}"/>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                    <variable symbol="x_{110}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </constantatomformula>
                                                                                </formulalist>
                                                                               </sequent>
                                                                               <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                                <sequent>
                                                                                 <formulalist>
                                                                                  <constantatomformula symbol="A" definition="A"/>
                                                                                 </formulalist>
                                                                                 <formulalist>
                                                                                  <quantifiedformula type="exists">
                                                                                   <variable symbol="p"/>
                                                                                   <quantifiedformula type="exists">
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                                                                                     <conjunctiveformula type="neg">
                                                                                      <constantatomformula symbol="=">
                                                                                       <variable symbol="p"/>
                                                                                       <variable symbol="q"/>
                                                                                      </constantatomformula>
                                                                                     </conjunctiveformula>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="f">
                                                                                       <variable symbol="p"/>
                                                                                      </function>
                                                                                      <function symbol="f">
                                                                                       <variable symbol="q"/>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                   </quantifiedformula>
                                                                                  </quantifiedformula>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{109}"/>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                     <variable symbol="x_{110}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{109}"/>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <variable symbol="x_{110}"/>
                                                                                   </function>
                                                                                  </constantatomformula>
                                                                                 </formulalist>
                                                                                </sequent>
                                                                                <rule type="weakr" symbol="w:r">
                                                                                 <sequent>
                                                                                  <formulalist>
                                                                                   <constantatomformula symbol="A" definition="A"/>
                                                                                  </formulalist>
                                                                                  <formulalist>
                                                                                   <constantatomformula symbol="=">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{109}"/>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                      <variable symbol="x_{110}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{109}"/>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <variable symbol="x_{110}"/>
                                                                                    </function>
                                                                                   </constantatomformula>
                                                                                   <quantifiedformula type="exists">
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                                                                                    <quantifiedformula type="exists">
                                                                                     <variable symbol="q"/>
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                                                                                       <constantatomformula symbol="=">
                                                                                        <variable symbol="p"/>
                                                                                        <variable symbol="q"/>
                                                                                       </constantatomformula>
                                                                                      </conjunctiveformula>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="f">
                                                                                        <variable symbol="p"/>
                                                                                       </function>
                                                                                       <function symbol="f">
                                                                                        <variable symbol="q"/>
                                                                                       </function>
                                                                                      </constantatomformula>
                                                                                     </conjunctiveformula>
                                                                                    </quantifiedformula>
                                                                                   </quantifiedformula>
                                                                                  </formulalist>
                                                                                 </sequent>
                                                                                 <rule type="perml" symbol="\pi:l" param="">
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                                                                                   <formulalist>
                                                                                    <constantatomformula symbol="A" definition="A"/>
                                                                                   </formulalist>
                                                                                   <formulalist>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{109}"/>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                       <variable symbol="x_{110}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{109}"/>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="x_{110}"/>
                                                                                     </function>
                                                                                    </constantatomformula>
                                                                                   </formulalist>
                                                                                  </sequent>
                                                                                  <rule type="weakl" symbol="w:l">
                                                                                   <sequent>
                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="A" definition="A"/>
                                                                                    </formulalist>
                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{109}"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                        <variable symbol="x_{110}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{109}"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <variable symbol="x_{110}"/>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </formulalist>
                                                                                   </sequent>
                                                                                   <rule type="axiom" symbol="axiom">
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                                                                                     <formulalist/>
                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{109}"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                         <variable symbol="x_{110}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{109}"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <variable symbol="x_{110}"/>
                                                                                       </function>
                                                                                      </constantatomformula>
                                                                                     </formulalist>
                                                                                    </sequent>
                                                                                   </rule>
                                                                                  </rule>
                                                                                 </rule>
                                                                                </rule>
                                                                               </rule>
                                                                               <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                                <sequent>
                                                                                 <formulalist>
                                                                                  <constantatomformula symbol="A" definition="A"/>
                                                                                 </formulalist>
                                                                                 <formulalist>
                                                                                  <quantifiedformula type="exists">
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                                                                                      <constantatomformula symbol="=">
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                                                                                     </conjunctiveformula>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="f">
                                                                                       <variable symbol="p"/>
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                                                                                      <function symbol="f">
                                                                                       <variable symbol="q"/>
                                                                                      </function>
                                                                                     </constantatomformula>
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                                                                                   </quantifiedformula>
                                                                                  </quantifiedformula>
                                                                                  <constantatomformula symbol="=">
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                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{109}"/>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                     <variable symbol="x_{110}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{109}"/>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                     <variable symbol="x_{110}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </constantatomformula>
                                                                                 </formulalist>
                                                                                </sequent>
                                                                                <rule type="weakr" symbol="w:r">
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                                                                                  <formulalist>
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                                                                                  <formulalist>
                                                                                   <constantatomformula symbol="=">
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                                                                                     <function symbol="+">
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                                                                                       <constant symbol="1"/>
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                                                                                      <variable symbol="x_{110}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{109}"/>
                                                                                       <constant symbol="1"/>
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                                                                                      <variable symbol="x_{110}"/>
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                                                                                    </function>
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                                                                                      </constantatomformula>
                                                                                     </conjunctiveformula>
                                                                                    </quantifiedformula>
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                                                                                 <rule type="perml" symbol="\pi:l" param="">
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                                                                                   <formulalist>
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                                                                                       <variable symbol="x_{110}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <function symbol="+">
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                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
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                                                                                        <constant symbol="1"/>
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                                                                                     </function>
                                                                                    </constantatomformula>
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                                                                                  </sequent>
                                                                                  <rule type="weakl" symbol="w:l">
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                                                                                    <formulalist>
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                                                                                        <variable symbol="x_{110}"/>
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                                                                                      </function>
                                                                                      <function symbol="+">
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                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
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                                                                                      </function>
                                                                                     </constantatomformula>
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                                                                                   </sequent>
                                                                                   <rule type="axiom" symbol="axiom">
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                                                                                       </function>
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                                                                                    </sequent>
                                                                                   </rule>
                                                                                  </rule>
                                                                                 </rule>
                                                                                </rule>
                                                                               </rule>
                                                                              </rule>
                                                                             </rule>
                                                                            </rule>
                                                                           </rule>
                                                                          </rule>
                                                                          <rule type="perml" symbol="\pi:l" param="(1 2)">
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                                                                            <formulalist>
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                                                                             <formulalist>
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                                                                               </quantifiedformula>
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                                                                              <formulalist>
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                                                                                  </constantatomformula>
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                                                                                </quantifiedformula>
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                                                                              </formulalist>
                                                                             </sequent>
                                                                             <rule type="permr" symbol="\pi:r" param="">
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                                                                                <constantatomformula symbol="=">
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                                                                               <formulalist>
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                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                 </quantifiedformula>
                                                                                </quantifiedformula>
                                                                                <quantifiedformula type="exists">
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                                                                                   </constantatomformula>
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                                                                                 </quantifiedformula>
                                                                                </quantifiedformula>
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                                                                                   </function>
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                                                                                  <constant symbol="1"/>
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                                                                                <formulalist>
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                                                                                    </constantatomformula>
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                                                                               </sequent>
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                                                                                  </quantifiedformula>
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                                                                                </sequent>
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                                                                                   <constantatomformula symbol="A" definition="A"/>
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                                                                                             </constantatomformula>
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                                                                                           </quantifiedformula>
                                                                                          </quantifiedformula>
                                                                                         </formulalist>
                                                                                        </sequent>
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                                                                                         <sequent>
                                                                                          <formulalist>
                                                                                           <constantatomformula symbol="A" definition="A"/>
                                                                                          </formulalist>
                                                                                          <formulalist>
                                                                                           <constantatomformula symbol="=">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{109}"/>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                            </function>
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <variable symbol="x_{109}"/>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </constantatomformula>
                                                                                          </formulalist>
                                                                                         </sequent>
                                                                                         <rule type="weakl" symbol="w:l">
                                                                                          <sequent>
                                                                                           <formulalist>
                                                                                            <constantatomformula symbol="A" definition="A"/>
                                                                                           </formulalist>
                                                                                           <formulalist>
                                                                                            <constantatomformula symbol="=">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{109}"/>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <variable symbol="x_{109}"/>
                                                                                              </function>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                            </constantatomformula>
                                                                                           </formulalist>
                                                                                          </sequent>
                                                                                          <rule type="axiom" symbol="axiom">
                                                                                           <sequent>
                                                                                            <formulalist/>
                                                                                            <formulalist>
                                                                                             <constantatomformula symbol="=">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{109}"/>
                                                                                                <constant symbol="1"/>
                                                                                               </function>
                                                                                              </function>
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <variable symbol="x_{109}"/>
                                                                                               </function>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </constantatomformula>
                                                                                            </formulalist>
                                                                                           </sequent>
                                                                                          </rule>
                                                                                         </rule>
                                                                                        </rule>
                                                                                       </rule>
                                                                                      </rule>
                                                                                      <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                                       <sequent>
                                                                                        <formulalist>
                                                                                         <constantatomformula symbol="A" definition="A"/>
                                                                                        </formulalist>
                                                                                        <formulalist>
                                                                                         <quantifiedformula type="exists">
                                                                                          <variable symbol="p"/>
                                                                                          <quantifiedformula type="exists">
                                                                                           <variable symbol="q"/>
                                                                                           <conjunctiveformula type="and">
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                                                                                             <constantatomformula symbol="=">
                                                                                              <variable symbol="p"/>
                                                                                              <variable symbol="q"/>
                                                                                             </constantatomformula>
                                                                                            </conjunctiveformula>
                                                                                            <constantatomformula symbol="=">
                                                                                             <function symbol="f">
                                                                                              <variable symbol="p"/>
                                                                                             </function>
                                                                                             <function symbol="f">
                                                                                              <variable symbol="q"/>
                                                                                             </function>
                                                                                            </constantatomformula>
                                                                                           </conjunctiveformula>
                                                                                          </quantifiedformula>
                                                                                         </quantifiedformula>
                                                                                         <constantatomformula symbol="=">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{109}"/>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{109}"/>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </constantatomformula>
                                                                                        </formulalist>
                                                                                       </sequent>
                                                                                       <rule type="weakr" symbol="w:r">
                                                                                        <sequent>
                                                                                         <formulalist>
                                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                                         </formulalist>
                                                                                         <formulalist>
                                                                                          <constantatomformula symbol="=">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{109}"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{109}"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </constantatomformula>
                                                                                          <quantifiedformula type="exists">
                                                                                           <variable symbol="p"/>
                                                                                           <quantifiedformula type="exists">
                                                                                            <variable symbol="q"/>
                                                                                            <conjunctiveformula type="and">
                                                                                             <conjunctiveformula type="neg">
                                                                                              <constantatomformula symbol="=">
                                                                                               <variable symbol="p"/>
                                                                                               <variable symbol="q"/>
                                                                                              </constantatomformula>
                                                                                             </conjunctiveformula>
                                                                                             <constantatomformula symbol="=">
                                                                                              <function symbol="f">
                                                                                               <variable symbol="p"/>
                                                                                              </function>
                                                                                              <function symbol="f">
                                                                                               <variable symbol="q"/>
                                                                                              </function>
                                                                                             </constantatomformula>
                                                                                            </conjunctiveformula>
                                                                                           </quantifiedformula>
                                                                                          </quantifiedformula>
                                                                                         </formulalist>
                                                                                        </sequent>
                                                                                        <rule type="perml" symbol="\pi:l" param="">
                                                                                         <sequent>
                                                                                          <formulalist>
                                                                                           <constantatomformula symbol="A" definition="A"/>
                                                                                          </formulalist>
                                                                                          <formulalist>
                                                                                           <constantatomformula symbol="=">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{109}"/>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                            </function>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{109}"/>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                            </function>
                                                                                           </constantatomformula>
                                                                                          </formulalist>
                                                                                         </sequent>
                                                                                         <rule type="weakl" symbol="w:l">
                                                                                          <sequent>
                                                                                           <formulalist>
                                                                                            <constantatomformula symbol="A" definition="A"/>
                                                                                           </formulalist>
                                                                                           <formulalist>
                                                                                            <constantatomformula symbol="=">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{109}"/>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{109}"/>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </constantatomformula>
                                                                                           </formulalist>
                                                                                          </sequent>
                                                                                          <rule type="axiom" symbol="axiom">
                                                                                           <sequent>
                                                                                            <formulalist/>
                                                                                            <formulalist>
                                                                                             <constantatomformula symbol="=">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{109}"/>
                                                                                                <constant symbol="1"/>
                                                                                               </function>
                                                                                              </function>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{109}"/>
                                                                                                <constant symbol="1"/>
                                                                                               </function>
                                                                                              </function>
                                                                                             </constantatomformula>
                                                                                            </formulalist>
                                                                                           </sequent>
                                                                                          </rule>
                                                                                         </rule>
                                                                                        </rule>
                                                                                       </rule>
                                                                                      </rule>
                                                                                     </rule>
                                                                                    </rule>
                                                                                   </rule>
                                                                                  </rule>
                                                                                 </rule>
                                                                                 <rule type="perml" symbol="\pi:l" param="(1 2)">
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                                                                                   <formulalist>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <variable symbol="x_{109}"/>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
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                                                                                       <variable symbol="x_{110}"/>
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                                                                                     </function>
                                                                                     <constant symbol="1"/>
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                                                                                    <constantatomformula symbol="=">
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                                                                                       <constant symbol="1"/>
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                                                                                     <constant symbol="1"/>
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                                                                                   <formulalist>
                                                                                    <quantifiedformula type="exists">
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                                                                                       <constantatomformula symbol="=">
                                                                                        <function symbol="f">
                                                                                         <variable symbol="p"/>
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                                                                                        <function symbol="f">
                                                                                         <variable symbol="q"/>
                                                                                        </function>
                                                                                       </constantatomformula>
                                                                                      </conjunctiveformula>
                                                                                     </quantifiedformula>
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                                                                                   </formulalist>
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                                                                                  <rule type="permr" symbol="\pi:r" param="">
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                                                                                     <constantatomformula symbol="=">
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                                                                                        <variable symbol="x_{109}"/>
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                                                                                      <constant symbol="1"/>
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                                                                                     <constantatomformula symbol="=">
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                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
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                                                                                          <variable symbol="x_{109}"/>
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                                                                                         <constant symbol="1"/>
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                                                                                        <variable symbol="x_{110}"/>
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                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </constantatomformula>
                                                                                     <constantatomformula symbol="A" definition="A"/>
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                                                                                    <formulalist>
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                                                                                        </constantatomformula>
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                                                                                      </quantifiedformula>
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                                                                                    </formulalist>
                                                                                   </sequent>
                                                                                   <rule type="perml" symbol="\pi:l" param="(1 2 3)">
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                                                                                      <constantatomformula symbol="=">
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                                                                                           <variable symbol="x_{109}"/>
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                                                                                         <variable symbol="x_{110}"/>
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                                                                                       </function>
                                                                                       <constant symbol="1"/>
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                                                                                      <constantatomformula symbol="A" definition="A"/>
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                                                                                     <formulalist>
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                                                                                       </quantifiedformula>
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                                                                                    <rule type="contrr" symbol="c:r" param="2">
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                                                                                      <formulalist>
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                                                                                       <constantatomformula symbol="=">
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                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
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                                                                                          <variable symbol="x_{110}"/>
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                                                                                        </quantifiedformula>
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                                                                                      </formulalist>
                                                                                     </sequent>
                                                                                     <rule type="permr" symbol="\pi:r" param="(1 2)">
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                                                                                        <constantatomformula symbol="=">
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                                                                                           <variable symbol="x_{110}"/>
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                                                                                         </quantifiedformula>
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                                                                                       </formulalist>
                                                                                      </sequent>
                                                                                      <rule type="weakr" symbol="w:r">
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                                                                                             <constant symbol="1"/>
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                                                                                            <variable symbol="x_{110}"/>
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                                                                                        <formulalist>
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                                                                                            </constantatomformula>
                                                                                           </conjunctiveformula>
                                                                                          </quantifiedformula>
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                                                                                            </constantatomformula>
                                                                                           </conjunctiveformula>
                                                                                          </quantifiedformula>
                                                                                         </quantifiedformula>
                                                                                        </formulalist>
                                                                                       </sequent>
                                                                                       <rule type="weakl" symbol="w:l">
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                                                                                         <formulalist>
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                                                                                             <variable symbol="x_{110}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
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                                                                                         </formulalist>
                                                                                         <formulalist>
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                                                                                              </function>
                                                                                             </constantatomformula>
                                                                                            </conjunctiveformula>
                                                                                           </quantifiedformula>
                                                                                          </quantifiedformula>
                                                                                         </formulalist>
                                                                                        </sequent>
                                                                                        <rule type="existsr" symbol="\ex:r">
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                                                                                          <formulalist>
                                                                                           <constantatomformula symbol="=">
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                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </constantatomformula>
                                                                                           <constantatomformula symbol="=">
                                                                                            <function symbol="f">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <variable symbol="x_{109}"/>
                                                                                               </function>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                              <variable symbol="x_{110}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </constantatomformula>
                                                                                          </formulalist>
                                                                                          <formulalist>
                                                                                           <quantifiedformula type="exists">
                                                                                            <variable symbol="p"/>
                                                                                            <quantifiedformula type="exists">
                                                                                             <variable symbol="q"/>
                                                                                             <conjunctiveformula type="and">
                                                                                              <conjunctiveformula type="neg">
                                                                                               <constantatomformula symbol="=">
                                                                                                <variable symbol="p"/>
                                                                                                <variable symbol="q"/>
                                                                                               </constantatomformula>
                                                                                              </conjunctiveformula>
                                                                                              <constantatomformula symbol="=">
                                                                                               <function symbol="f">
                                                                                                <variable symbol="p"/>
                                                                                               </function>
                                                                                               <function symbol="f">
                                                                                                <variable symbol="q"/>
                                                                                               </function>
                                                                                              </constantatomformula>
                                                                                             </conjunctiveformula>
                                                                                            </quantifiedformula>
                                                                                           </quantifiedformula>
                                                                                          </formulalist>
                                                                                         </sequent>
                                                                                         <rule type="existsr" symbol="\ex:r">
                                                                                          <sequent>
                                                                                           <formulalist>
                                                                                            <constantatomformula symbol="=">
                                                                                             <function symbol="f">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <variable symbol="x_{109}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </constantatomformula>
                                                                                            <constantatomformula symbol="=">
                                                                                             <function symbol="f">
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                                <constant symbol="1"/>
                                                                                               </function>
                                                                                               <variable symbol="x_{110}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </constantatomformula>
                                                                                           </formulalist>
                                                                                           <formulalist>
                                                                                            <quantifiedformula type="exists">
                                                                                             <variable symbol="q"/>
                                                                                             <conjunctiveformula type="and">
                                                                                              <conjunctiveformula type="neg">
                                                                                               <constantatomformula symbol="=">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                                <variable symbol="q"/>
                                                                                               </constantatomformula>
                                                                                              </conjunctiveformula>
                                                                                              <constantatomformula symbol="=">
                                                                                               <function symbol="f">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                               <function symbol="f">
                                                                                                <variable symbol="q"/>
                                                                                               </function>
                                                                                              </constantatomformula>
                                                                                             </conjunctiveformula>
                                                                                            </quantifiedformula>
                                                                                           </formulalist>
                                                                                          </sequent>
                                                                                          <rule type="andr" symbol="\and:r">
                                                                                           <sequent>
                                                                                            <formulalist>
                                                                                             <constantatomformula symbol="=">
                                                                                              <function symbol="f">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <variable symbol="x_{109}"/>
                                                                                               </function>
                                                                                              </function>
                                                                                              <constant symbol="1"/>
                                                                                             </constantatomformula>
                                                                                             <constantatomformula symbol="=">
                                                                                              <function symbol="f">
                                                                                               <function symbol="+">
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <variable symbol="x_{109}"/>
                                                                                                 </function>
                                                                                                 <constant symbol="1"/>
                                                                                                </function>
                                                                                                <variable symbol="x_{110}"/>
                                                                                               </function>
                                                                                              </function>
                                                                                              <constant symbol="1"/>
                                                                                             </constantatomformula>
                                                                                            </formulalist>
                                                                                            <formulalist>
                                                                                             <conjunctiveformula type="and">
                                                                                              <conjunctiveformula type="neg">
                                                                                               <constantatomformula symbol="=">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{109}"/>
                                                                                                  </function>
                                                                                                  <constant symbol="1"/>
                                                                                                 </function>
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                </function>
                                                                                               </constantatomformula>
                                                                                              </conjunctiveformula>
                                                                                              <constantatomformula symbol="=">
                                                                                               <function symbol="f">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                               <function symbol="f">
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{109}"/>
                                                                                                  </function>
                                                                                                  <constant symbol="1"/>
                                                                                                 </function>
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </constantatomformula>
                                                                                             </conjunctiveformula>
                                                                                            </formulalist>
                                                                                           </sequent>
                                                                                           <rule type="negr" symbol="\neg:r">
                                                                                            <sequent>
                                                                                             <formulalist/>
                                                                                             <formulalist>
                                                                                              <conjunctiveformula type="neg">
                                                                                               <constantatomformula symbol="=">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{109}"/>
                                                                                                  </function>
                                                                                                  <constant symbol="1"/>
                                                                                                 </function>
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                </function>
                                                                                               </constantatomformula>
                                                                                              </conjunctiveformula>
                                                                                             </formulalist>
                                                                                            </sequent>
                                                                                            <rule type="eql1" symbol="=:l1" param="(2)">
                                                                                             <sequent>
                                                                                              <formulalist>
                                                                                               <constantatomformula symbol="=">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{109}"/>
                                                                                                  </function>
                                                                                                  <constant symbol="1"/>
                                                                                                 </function>
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                </function>
                                                                                               </constantatomformula>
                                                                                              </formulalist>
                                                                                              <formulalist/>
                                                                                             </sequent>
                                                                                             <rule type="axiom" symbol="axiom">
                                                                                              <sequent>
                                                                                               <formulalist/>
                                                                                               <formulalist>
                                                                                                <constantatomformula symbol="=">
                                                                                                 <function symbol="+">
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{109}"/>
                                                                                                  </function>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{110}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                 <function symbol="+">
                                                                                                  <function symbol="+">
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <variable symbol="x_{109}"/>
                                                                                                   </function>
                                                                                                   <constant symbol="1"/>
                                                                                                  </function>
                                                                                                  <variable symbol="x_{110}"/>
                                                                                                 </function>
                                                                                                </constantatomformula>
                                                                                               </formulalist>
                                                                                              </sequent>
                                                                                             </rule>
                                                                                             <rule type="axiom" symbol="axiom">
                                                                                              <sequent>
                                                                                               <formulalist>
                                                                                                <constantatomformula symbol="=">
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <variable symbol="x_{109}"/>
                                                                                                 </function>
                                                                                                 <function symbol="+">
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{109}"/>
                                                                                                  </function>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{110}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </constantatomformula>
                                                                                               </formulalist>
                                                                                               <formulalist/>
                                                                                              </sequent>
                                                                                             </rule>
                                                                                            </rule>
                                                                                           </rule>
                                                                                           <rule type="eqr2" symbol="=:r2" param="(1)">
                                                                                            <sequent>
                                                                                             <formulalist>
                                                                                              <constantatomformula symbol="=">
                                                                                               <function symbol="f">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                               <constant symbol="1"/>
                                                                                              </constantatomformula>
                                                                                              <constantatomformula symbol="=">
                                                                                               <function symbol="f">
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{109}"/>
                                                                                                  </function>
                                                                                                  <constant symbol="1"/>
                                                                                                 </function>
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                               <constant symbol="1"/>
                                                                                              </constantatomformula>
                                                                                             </formulalist>
                                                                                             <formulalist>
                                                                                              <constantatomformula symbol="=">
                                                                                               <function symbol="f">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                               <function symbol="f">
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{109}"/>
                                                                                                  </function>
                                                                                                  <constant symbol="1"/>
                                                                                                 </function>
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </constantatomformula>
                                                                                             </formulalist>
                                                                                            </sequent>
                                                                                            <rule type="axiom" symbol="axiom">
                                                                                             <sequent>
                                                                                              <formulalist>
                                                                                               <constantatomformula symbol="=">
                                                                                                <function symbol="f">
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <variable symbol="x_{109}"/>
                                                                                                 </function>
                                                                                                </function>
                                                                                                <constant symbol="1"/>
                                                                                               </constantatomformula>
                                                                                              </formulalist>
                                                                                              <formulalist>
                                                                                               <constantatomformula symbol="=">
                                                                                                <function symbol="f">
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <variable symbol="x_{109}"/>
                                                                                                 </function>
                                                                                                </function>
                                                                                                <constant symbol="1"/>
                                                                                               </constantatomformula>
                                                                                              </formulalist>
                                                                                             </sequent>
                                                                                            </rule>
                                                                                            <rule type="eqr2" symbol="=:r2" param="(2)">
                                                                                             <sequent>
                                                                                              <formulalist>
                                                                                               <constantatomformula symbol="=">
                                                                                                <function symbol="f">
                                                                                                 <function symbol="+">
                                                                                                  <function symbol="+">
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <variable symbol="x_{109}"/>
                                                                                                   </function>
                                                                                                   <constant symbol="1"/>
                                                                                                  </function>
                                                                                                  <variable symbol="x_{110}"/>
                                                                                                 </function>
                                                                                                </function>
                                                                                                <constant symbol="1"/>
                                                                                               </constantatomformula>
                                                                                              </formulalist>
                                                                                              <formulalist>
                                                                                               <constantatomformula symbol="=">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="f">
                                                                                                 <function symbol="+">
                                                                                                  <function symbol="+">
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <variable symbol="x_{109}"/>
                                                                                                   </function>
                                                                                                   <constant symbol="1"/>
                                                                                                  </function>
                                                                                                  <variable symbol="x_{110}"/>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </constantatomformula>
                                                                                              </formulalist>
                                                                                             </sequent>
                                                                                             <rule type="axiom" symbol="axiom">
                                                                                              <sequent>
                                                                                               <formulalist>
                                                                                                <constantatomformula symbol="=">
                                                                                                 <function symbol="f">
                                                                                                  <function symbol="+">
                                                                                                   <function symbol="+">
                                                                                                    <function symbol="+">
                                                                                                     <constant symbol="1"/>
                                                                                                     <variable symbol="x_{109}"/>
                                                                                                    </function>
                                                                                                    <constant symbol="1"/>
                                                                                                   </function>
                                                                                                   <variable symbol="x_{110}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                 <constant symbol="1"/>
                                                                                                </constantatomformula>
                                                                                               </formulalist>
                                                                                               <formulalist>
                                                                                                <constantatomformula symbol="=">
                                                                                                 <function symbol="f">
                                                                                                  <function symbol="+">
                                                                                                   <function symbol="+">
                                                                                                    <function symbol="+">
                                                                                                     <constant symbol="1"/>
                                                                                                     <variable symbol="x_{109}"/>
                                                                                                    </function>
                                                                                                    <constant symbol="1"/>
                                                                                                   </function>
                                                                                                   <variable symbol="x_{110}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                 <constant symbol="1"/>
                                                                                                </constantatomformula>
                                                                                               </formulalist>
                                                                                              </sequent>
                                                                                             </rule>
                                                                                             <rule type="axiom" symbol="axiom">
                                                                                              <sequent>
                                                                                               <formulalist/>
                                                                                               <formulalist>
                                                                                                <constantatomformula symbol="=">
                                                                                                 <constant symbol="1"/>
                                                                                                 <constant symbol="1"/>
                                                                                                </constantatomformula>
                                                                                               </formulalist>
                                                                                              </sequent>
                                                                                             </rule>
                                                                                            </rule>
                                                                                           </rule>
                                                                                          </rule>
                                                                                         </rule>
                                                                                        </rule>
                                                                                       </rule>
                                                                                      </rule>
                                                                                     </rule>
                                                                                    </rule>
                                                                                   </rule>
                                                                                  </rule>
                                                                                 </rule>
                                                                                </rule>
                                                                               </rule>
                                                                              </rule>
                                                                             </rule>
                                                                            </rule>
                                                                           </rule>
                                                                          </rule>
                                                                         </rule>
                                                                        </rule>
                                                                       </rule>
                                                                      </rule>
                                                                     </rule>
                                                                    </rule>
                                                                   </rule>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                       <rule type="perml" symbol="\pi:l" param="(1 2)">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="=">
                                           <function symbol="f">
                                            <function symbol="+">
                                             <variable symbol="x_{109}"/>
                                             <constant symbol="1"/>
                                            </function>
                                           </function>
                                           <constant symbol="0"/>
                                          </constantatomformula>
                                          <constantatomformula symbol="=">
                                           <function symbol="f">
                                            <function symbol="+">
                                             <variable symbol="x_{109}"/>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <variable symbol="x_{110}"/>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                           <constant symbol="0"/>
                                          </constantatomformula>
                                          <constantatomformula symbol="A" definition="A"/>
                                         </formulalist>
                                         <formulalist>
                                          <quantifiedformula type="exists">
                                           <variable symbol="p"/>
                                           <quantifiedformula type="exists">
                                            <variable symbol="q"/>
                                            <conjunctiveformula type="and">
                                             <conjunctiveformula type="neg">
                                              <constantatomformula symbol="=">
                                               <variable symbol="p"/>
                                               <variable symbol="q"/>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <variable symbol="p"/>
                                              </function>
                                              <function symbol="f">
                                               <variable symbol="q"/>
                                              </function>
                                             </constantatomformula>
                                            </conjunctiveformula>
                                           </quantifiedformula>
                                          </quantifiedformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="contrr" symbol="c:r" param="2">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="=">
                                            <function symbol="f">
                                             <function symbol="+">
                                              <variable symbol="x_{109}"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                            <constant symbol="0"/>
                                           </constantatomformula>
                                           <constantatomformula symbol="=">
                                            <function symbol="f">
                                             <function symbol="+">
                                              <variable symbol="x_{109}"/>
                                              <constant symbol="1"/>
                                             </function>
                                            </function>
                                            <constant symbol="0"/>
                                           </constantatomformula>
                                           <constantatomformula symbol="A" definition="A"/>
                                          </formulalist>
                                          <formulalist>
                                           <quantifiedformula type="exists">
                                            <variable symbol="p"/>
                                            <quantifiedformula type="exists">
                                             <variable symbol="q"/>
                                             <conjunctiveformula type="and">
                                              <conjunctiveformula type="neg">
                                               <constantatomformula symbol="=">
                                                <variable symbol="p"/>
                                                <variable symbol="q"/>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <variable symbol="p"/>
                                               </function>
                                               <function symbol="f">
                                                <variable symbol="q"/>
                                               </function>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                            </quantifiedformula>
                                           </quantifiedformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="permr" symbol="\pi:r" param="">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="=">
                                             <function symbol="f">
                                              <function symbol="+">
                                               <variable symbol="x_{109}"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                             <constant symbol="0"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="f">
                                              <function symbol="+">
                                               <variable symbol="x_{109}"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                             <constant symbol="0"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="A" definition="A"/>
                                           </formulalist>
                                           <formulalist>
                                            <quantifiedformula type="exists">
                                             <variable symbol="p"/>
                                             <quantifiedformula type="exists">
                                              <variable symbol="q"/>
                                              <conjunctiveformula type="and">
                                               <conjunctiveformula type="neg">
                                                <constantatomformula symbol="=">
                                                 <variable symbol="p"/>
                                                 <variable symbol="q"/>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <variable symbol="p"/>
                                                </function>
                                                <function symbol="f">
                                                 <variable symbol="q"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                             </quantifiedformula>
                                            </quantifiedformula>
                                            <quantifiedformula type="exists">
                                             <variable symbol="p"/>
                                             <quantifiedformula type="exists">
                                              <variable symbol="q"/>
                                              <conjunctiveformula type="and">
                                               <conjunctiveformula type="neg">
                                                <constantatomformula symbol="=">
                                                 <variable symbol="p"/>
                                                 <variable symbol="q"/>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <variable symbol="p"/>
                                                </function>
                                                <function symbol="f">
                                                 <variable symbol="q"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                             </quantifiedformula>
                                            </quantifiedformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="contrl" symbol="c:l" param="1,1,2">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <function symbol="+">
                                                <variable symbol="x_{109}"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                              <constant symbol="0"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <function symbol="+">
                                                <variable symbol="x_{109}"/>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                              <constant symbol="0"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="A" definition="A"/>
                                            </formulalist>
                                            <formulalist>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="perml" symbol="\pi:l" param="(2 3)">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <variable symbol="x_{109}"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                               <constant symbol="0"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <variable symbol="x_{109}"/>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                               <constant symbol="0"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="A" definition="A"/>
                                              <constantatomformula symbol="A" definition="A"/>
                                             </formulalist>
                                             <formulalist>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="eql1" symbol="=:l1" param="(1 1 2)">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <variable symbol="x_{109}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <constant symbol="0"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="A" definition="A"/>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <variable symbol="x_{109}"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                                <constant symbol="0"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="A" definition="A"/>
                                              </formulalist>
                                              <formulalist>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="contrr" symbol="c:r" param="2,1">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="permr" symbol="\pi:r" param="">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="contrl" symbol="c:l" param="2">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="perml" symbol="\pi:l" param="">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="eqr1" symbol="=:r1" param="(1)">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="weakr" symbol="w:r">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </constantatomformula>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="perml" symbol="\pi:l" param="">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="weakl" symbol="w:l">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="axiom" symbol="axiom">
                                                       <sequent>
                                                        <formulalist/>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                  <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="weakr" symbol="w:r">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </constantatomformula>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="perml" symbol="\pi:l" param="">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="weakl" symbol="w:l">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="axiom" symbol="axiom">
                                                       <sequent>
                                                        <formulalist/>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                             <rule type="perml" symbol="\pi:l" param="(1 2)">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="perml" symbol="\pi:l" param="(1 2)">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="contrr" symbol="c:r" param="2">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="permr" symbol="\pi:r" param="">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="contrl" symbol="c:l" param="1,1,2">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="perml" symbol="\pi:l" param="(2 3)">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="eql1" symbol="=:l1" param="(1 1)">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="contrr" symbol="c:r" param="2,1">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="permr" symbol="\pi:r" param="">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="contrl" symbol="c:l" param="2">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
                                                            <conjunctiveformula type="neg">
                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
                                                            <conjunctiveformula type="neg">
                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="perml" symbol="\pi:l" param="">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
                                                            <conjunctiveformula type="and">
                                                             <conjunctiveformula type="neg">
                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
                                                            <conjunctiveformula type="and">
                                                             <conjunctiveformula type="neg">
                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="eqr1" symbol="=:r1" param="(1)">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="p"/>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="q"/>
                                                             <conjunctiveformula type="and">
                                                              <conjunctiveformula type="neg">
                                                               <constantatomformula symbol="=">
                                                                <variable symbol="p"/>
                                                                <variable symbol="q"/>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="p"/>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="q"/>
                                                             <conjunctiveformula type="and">
                                                              <conjunctiveformula type="neg">
                                                               <constantatomformula symbol="=">
                                                                <variable symbol="p"/>
                                                                <variable symbol="q"/>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="p"/>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="q"/>
                                                              <conjunctiveformula type="and">
                                                               <conjunctiveformula type="neg">
                                                                <constantatomformula symbol="=">
                                                                 <variable symbol="p"/>
                                                                 <variable symbol="q"/>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <variable symbol="p"/>
                                                                </function>
                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </quantifiedformula>
                                                            </quantifiedformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="weakr" symbol="w:r">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="A" definition="A"/>
                                                            </formulalist>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </constantatomformula>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="p"/>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="q"/>
                                                               <conjunctiveformula type="and">
                                                                <conjunctiveformula type="neg">
                                                                 <constantatomformula symbol="=">
                                                                  <variable symbol="p"/>
                                                                  <variable symbol="q"/>
                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <variable symbol="p"/>
                                                                 </function>
                                                                 <function symbol="f">
                                                                  <variable symbol="q"/>
                                                                 </function>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                              </quantifiedformula>
                                                             </quantifiedformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="perml" symbol="\pi:l" param="">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="A" definition="A"/>
                                                             </formulalist>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="weakl" symbol="w:l">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="A" definition="A"/>
                                                              </formulalist>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </constantatomformula>
                                                              </formulalist>
                                                             </sequent>
                                                             <rule type="axiom" symbol="axiom">
                                                              <sequent>
                                                               <formulalist/>
                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{109}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </constantatomformula>
                                                               </formulalist>
                                                              </sequent>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                         <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="p"/>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="q"/>
                                                              <conjunctiveformula type="and">
                                                               <conjunctiveformula type="neg">
                                                                <constantatomformula symbol="=">
                                                                 <variable symbol="p"/>
                                                                 <variable symbol="q"/>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <variable symbol="p"/>
                                                                </function>
                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </quantifiedformula>
                                                            </quantifiedformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="weakr" symbol="w:r">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="A" definition="A"/>
                                                            </formulalist>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </constantatomformula>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="p"/>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="q"/>
                                                               <conjunctiveformula type="and">
                                                                <conjunctiveformula type="neg">
                                                                 <constantatomformula symbol="=">
                                                                  <variable symbol="p"/>
                                                                  <variable symbol="q"/>
                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <variable symbol="p"/>
                                                                 </function>
                                                                 <function symbol="f">
                                                                  <variable symbol="q"/>
                                                                 </function>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                              </quantifiedformula>
                                                             </quantifiedformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="perml" symbol="\pi:l" param="">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="A" definition="A"/>
                                                             </formulalist>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="weakl" symbol="w:l">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="A" definition="A"/>
                                                              </formulalist>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </constantatomformula>
                                                              </formulalist>
                                                             </sequent>
                                                             <rule type="axiom" symbol="axiom">
                                                              <sequent>
                                                               <formulalist/>
                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </constantatomformula>
                                                               </formulalist>
                                                              </sequent>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                    <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="0"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="0"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="0"/>
                                                        </constantatomformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="0"/>
                                                        </constantatomformula>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
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                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="contrr" symbol="c:r" param="2">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
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                                                             <constantatomformula symbol="=">
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                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="permr" symbol="\pi:r" param="">
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                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="0"/>
                                                          </constantatomformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <constant symbol="1"/>
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                                                           </function>
                                                           <constant symbol="0"/>
                                                          </constantatomformula>
                                                          <constantatomformula symbol="A" definition="A"/>
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                                                         <formulalist>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
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                                                              <constantatomformula symbol="=">
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                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
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                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
                                                            <conjunctiveformula type="and">
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                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="contrl" symbol="c:l" param="1,1,2">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="0"/>
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                                                           <constantatomformula symbol="=">
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                                                              <variable symbol="x_{109}"/>
                                                              <constant symbol="1"/>
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                                                            </function>
                                                            <constant symbol="0"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="A" definition="A"/>
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                                                          <formulalist>
                                                           <quantifiedformula type="exists">
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                                                            <quantifiedformula type="exists">
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                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
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                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                           <quantifiedformula type="exists">
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                                                             <variable symbol="q"/>
                                                             <conjunctiveformula type="and">
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                                                               <constantatomformula symbol="=">
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                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                              <constantatomformula symbol="=">
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                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="perml" symbol="\pi:l" param="(2 3)">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
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                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
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                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="0"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="=">
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                                                               <variable symbol="x_{109}"/>
                                                               <constant symbol="1"/>
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                                                             </function>
                                                             <constant symbol="0"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="p"/>
                                                             <quantifiedformula type="exists">
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                                                              <conjunctiveformula type="and">
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                                                                <constantatomformula symbol="=">
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                                                               </conjunctiveformula>
                                                               <constantatomformula symbol="=">
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                                                                </function>
                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </quantifiedformula>
                                                            </quantifiedformula>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="p"/>
                                                             <quantifiedformula type="exists">
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                                                                <constantatomformula symbol="=">
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                                                               </conjunctiveformula>
                                                               <constantatomformula symbol="=">
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                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </quantifiedformula>
                                                            </quantifiedformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="eql1" symbol="=:l1" param="(1 1 1)">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                             <constantatomformula symbol="A" definition="A"/>
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                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                             <constantatomformula symbol="A" definition="A"/>
                                                            </formulalist>
                                                            <formulalist>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="p"/>
                                                              <quantifiedformula type="exists">
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                                                               <conjunctiveformula type="and">
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                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                                <constantatomformula symbol="=">
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                                                                 </function>
                                                                 <function symbol="f">
                                                                  <variable symbol="q"/>
                                                                 </function>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                              </quantifiedformula>
                                                             </quantifiedformula>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="p"/>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="q"/>
                                                               <conjunctiveformula type="and">
                                                                <conjunctiveformula type="neg">
                                                                 <constantatomformula symbol="=">
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                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <variable symbol="p"/>
                                                                 </function>
                                                                 <function symbol="f">
                                                                  <variable symbol="q"/>
                                                                 </function>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                              </quantifiedformula>
                                                             </quantifiedformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="contrr" symbol="c:r" param="2,1">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="A" definition="A"/>
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                                                             <formulalist>
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                                                               <quantifiedformula type="exists">
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                                                               <function symbol="+">
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                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="permr" symbol="\pi:r" param="">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="A" definition="A"/>
                                                              </formulalist>
                                                              <formulalist>
                                                               <quantifiedformula type="exists">
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                                                                <quantifiedformula type="exists">
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                                                                  <constantatomformula symbol="=">
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                                                               <constantatomformula symbol="=">
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                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </constantatomformula>
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                                                             </sequent>
                                                             <rule type="contrl" symbol="c:l" param="2">
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                                                               <formulalist>
                                                                <constantatomformula symbol="A" definition="A"/>
                                                               </formulalist>
                                                               <formulalist>
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                                                                <constantatomformula symbol="=">
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                                                                   <variable symbol="x_{109}"/>
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                                                              </sequent>
                                                              <rule type="perml" symbol="\pi:l" param="">
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                                                                <formulalist>
                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                 <constantatomformula symbol="A" definition="A"/>
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                                                                <formulalist>
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                                                               </sequent>
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                                                                 <formulalist>
                                                                  <constantatomformula symbol="A" definition="A"/>
                                                                  <constantatomformula symbol="A" definition="A"/>
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                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="0"/>
                                                               </constantatomformula>
                                                               <constantatomformula symbol="A" definition="A"/>
                                                              </formulalist>
                                                              <formulalist>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="p"/>
                                                                <quantifiedformula type="exists">
                                                                 <variable symbol="q"/>
                                                                 <conjunctiveformula type="and">
                                                                  <conjunctiveformula type="neg">
                                                                   <constantatomformula symbol="=">
                                                                    <variable symbol="p"/>
                                                                    <variable symbol="q"/>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <variable symbol="p"/>
                                                                   </function>
                                                                   <function symbol="f">
                                                                    <variable symbol="q"/>
                                                                   </function>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                </quantifiedformula>
                                                               </quantifiedformula>
                                                              </formulalist>
                                                             </sequent>
                                                             <rule type="perml" symbol="\pi:l" param="(1 2 3)">
                                                              <sequent>
                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{109}"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="0"/>
                                                                </constantatomformula>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{109}"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="0"/>
                                                                </constantatomformula>
                                                                <constantatomformula symbol="A" definition="A"/>
                                                               </formulalist>
                                                               <formulalist>
                                                                <quantifiedformula type="exists">
                                                                 <variable symbol="p"/>
                                                                 <quantifiedformula type="exists">
                                                                  <variable symbol="q"/>
                                                                  <conjunctiveformula type="and">
                                                                   <conjunctiveformula type="neg">
                                                                    <constantatomformula symbol="=">
                                                                     <variable symbol="p"/>
                                                                     <variable symbol="q"/>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <variable symbol="p"/>
                                                                    </function>
                                                                    <function symbol="f">
                                                                     <variable symbol="q"/>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                 </quantifiedformula>
                                                                </quantifiedformula>
                                                               </formulalist>
                                                              </sequent>
                                                              <rule type="contrr" symbol="c:r" param="2">
                                                               <sequent>
                                                                <formulalist>
                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{109}"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="0"/>
                                                                 </constantatomformula>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{109}"/>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="0"/>
                                                                 </constantatomformula>
                                                                </formulalist>
                                                                <formulalist>
                                                                 <quantifiedformula type="exists">
                                                                  <variable symbol="p"/>
                                                                  <quantifiedformula type="exists">
                                                                   <variable symbol="q"/>
                                                                   <conjunctiveformula type="and">
                                                                    <conjunctiveformula type="neg">
                                                                     <constantatomformula symbol="=">
                                                                      <variable symbol="p"/>
                                                                      <variable symbol="q"/>
                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <variable symbol="p"/>
                                                                     </function>
                                                                     <function symbol="f">
                                                                      <variable symbol="q"/>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                  </quantifiedformula>
                                                                 </quantifiedformula>
                                                                </formulalist>
                                                               </sequent>
                                                               <rule type="weakr" symbol="w:r">
                                                                <sequent>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="A" definition="A"/>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{109}"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="0"/>
                                                                  </constantatomformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="0"/>
                                                                  </constantatomformula>
                                                                 </formulalist>
                                                                 <formulalist>
                                                                  <quantifiedformula type="exists">
                                                                   <variable symbol="p"/>
                                                                   <quantifiedformula type="exists">
                                                                    <variable symbol="q"/>
                                                                    <conjunctiveformula type="and">
                                                                     <conjunctiveformula type="neg">
                                                                      <constantatomformula symbol="=">
                                                                       <variable symbol="p"/>
                                                                       <variable symbol="q"/>
                                                                      </constantatomformula>
                                                                     </conjunctiveformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <variable symbol="p"/>
                                                                      </function>
                                                                      <function symbol="f">
                                                                       <variable symbol="q"/>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                   </quantifiedformula>
                                                                  </quantifiedformula>
                                                                  <quantifiedformula type="exists">
                                                                   <variable symbol="p"/>
                                                                   <quantifiedformula type="exists">
                                                                    <variable symbol="q"/>
                                                                    <conjunctiveformula type="and">
                                                                     <conjunctiveformula type="neg">
                                                                      <constantatomformula symbol="=">
                                                                       <variable symbol="p"/>
                                                                       <variable symbol="q"/>
                                                                      </constantatomformula>
                                                                     </conjunctiveformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <variable symbol="p"/>
                                                                      </function>
                                                                      <function symbol="f">
                                                                       <variable symbol="q"/>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                   </quantifiedformula>
                                                                  </quantifiedformula>
                                                                 </formulalist>
                                                                </sequent>
                                                                <rule type="weakl" symbol="w:l">
                                                                 <sequent>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="A" definition="A"/>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{109}"/>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="0"/>
                                                                   </constantatomformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="0"/>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                  <formulalist>
                                                                   <quantifiedformula type="exists">
                                                                    <variable symbol="p"/>
                                                                    <quantifiedformula type="exists">
                                                                     <variable symbol="q"/>
                                                                     <conjunctiveformula type="and">
                                                                      <conjunctiveformula type="neg">
                                                                       <constantatomformula symbol="=">
                                                                        <variable symbol="p"/>
                                                                        <variable symbol="q"/>
                                                                       </constantatomformula>
                                                                      </conjunctiveformula>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="f">
                                                                        <variable symbol="p"/>
                                                                       </function>
                                                                       <function symbol="f">
                                                                        <variable symbol="q"/>
                                                                       </function>
                                                                      </constantatomformula>
                                                                     </conjunctiveformula>
                                                                    </quantifiedformula>
                                                                   </quantifiedformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                 <rule type="existsr" symbol="\ex:r">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="0"/>
                                                                    </constantatomformula>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{109}"/>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="0"/>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                   <formulalist>
                                                                    <quantifiedformula type="exists">
                                                                     <variable symbol="p"/>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="q"/>
                                                                      <conjunctiveformula type="and">
                                                                       <conjunctiveformula type="neg">
                                                                        <constantatomformula symbol="=">
                                                                         <variable symbol="p"/>
                                                                         <variable symbol="q"/>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <variable symbol="p"/>
                                                                        </function>
                                                                        <function symbol="f">
                                                                         <variable symbol="q"/>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </conjunctiveformula>
                                                                     </quantifiedformula>
                                                                    </quantifiedformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                  <rule type="existsr" symbol="\ex:r">
                                                                   <sequent>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="0"/>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="0"/>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                    <formulalist>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="q"/>
                                                                      <conjunctiveformula type="and">
                                                                       <conjunctiveformula type="neg">
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                         <variable symbol="q"/>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <function symbol="f">
                                                                         <variable symbol="q"/>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </conjunctiveformula>
                                                                     </quantifiedformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                   <rule type="andr" symbol="\and:r">
                                                                    <sequent>
                                                                     <formulalist>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="f">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{109}"/>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="0"/>
                                                                      </constantatomformula>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="f">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="0"/>
                                                                      </constantatomformula>
                                                                     </formulalist>
                                                                     <formulalist>
                                                                      <conjunctiveformula type="and">
                                                                       <conjunctiveformula type="neg">
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <function symbol="f">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </conjunctiveformula>
                                                                     </formulalist>
                                                                    </sequent>
                                                                    <rule type="negr" symbol="\neg:r">
                                                                     <sequent>
                                                                      <formulalist/>
                                                                      <formulalist>
                                                                       <conjunctiveformula type="neg">
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                      </formulalist>
                                                                     </sequent>
                                                                     <rule type="eql1" symbol="=:l1" param="(2)">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                       <formulalist/>
                                                                      </sequent>
                                                                      <rule type="axiom" symbol="axiom">
                                                                       <sequent>
                                                                        <formulalist/>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{109}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                      </rule>
                                                                      <rule type="axiom" symbol="axiom">
                                                                       <sequent>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </formulalist>
                                                                        <formulalist/>
                                                                       </sequent>
                                                                      </rule>
                                                                     </rule>
                                                                    </rule>
                                                                    <rule type="eqr2" symbol="=:r2" param="(1)">
                                                                     <sequent>
                                                                      <formulalist>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="0"/>
                                                                       </constantatomformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="0"/>
                                                                       </constantatomformula>
                                                                      </formulalist>
                                                                      <formulalist>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <function symbol="f">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{109}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </formulalist>
                                                                     </sequent>
                                                                     <rule type="axiom" symbol="axiom">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="0"/>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{109}"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="0"/>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                     </rule>
                                                                     <rule type="eqr2" symbol="=:r2" param="(2)">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{109}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="0"/>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <constant symbol="0"/>
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{109}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                      <rule type="axiom" symbol="axiom">
                                                                       <sequent>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{109}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="0"/>
                                                                         </constantatomformula>
                                                                        </formulalist>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{109}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="0"/>
                                                                         </constantatomformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                      </rule>
                                                                      <rule type="axiom" symbol="axiom">
                                                                       <sequent>
                                                                        <formulalist/>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <constant symbol="0"/>
                                                                          <constant symbol="0"/>
                                                                         </constantatomformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                      </rule>
                                                                     </rule>
                                                                    </rule>
                                                                   </rule>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                           <rule type="perml" symbol="\pi:l" param="(1 2)">
                            <sequent>
                             <formulalist>
                              <constantatomformula symbol="=">
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                                <function symbol="+">
                                 <variable symbol="x_{109}"/>
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <function symbol="+">
                                   <variable symbol="x_{110}"/>
                                   <constant symbol="1"/>
                                  </function>
                                 </function>
                                </function>
                               </function>
                               <constant symbol="0"/>
                              </constantatomformula>
                              <constantatomformula symbol="=">
                               <function symbol="f">
                                <function symbol="+">
                                 <variable symbol="x_{109}"/>
                                 <function symbol="+">
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <function symbol="+">
                                    <variable symbol="x_{110}"/>
                                    <constant symbol="1"/>
                                   </function>
                                  </function>
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <variable symbol="x_{111}"/>
                                  </function>
                                 </function>
                                </function>
                               </function>
                               <constant symbol="0"/>
                              </constantatomformula>
                              <constantatomformula symbol="A" definition="A"/>
                             </formulalist>
                             <formulalist>
                              <quantifiedformula type="exists">
                               <variable symbol="p"/>
                               <quantifiedformula type="exists">
                                <variable symbol="q"/>
                                <conjunctiveformula type="and">
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                                  <constantatomformula symbol="=">
                                   <variable symbol="p"/>
                                   <variable symbol="q"/>
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                                 </conjunctiveformula>
                                 <constantatomformula symbol="=">
                                  <function symbol="f">
                                   <variable symbol="p"/>
                                  </function>
                                  <function symbol="f">
                                   <variable symbol="q"/>
                                  </function>
                                 </constantatomformula>
                                </conjunctiveformula>
                               </quantifiedformula>
                              </quantifiedformula>
                             </formulalist>
                            </sequent>
                            <rule type="contrr" symbol="c:r" param="2">
                             <sequent>
                              <formulalist>
                               <constantatomformula symbol="=">
                                <function symbol="f">
                                 <function symbol="+">
                                  <variable symbol="x_{109}"/>
                                  <function symbol="+">
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <function symbol="+">
                                     <variable symbol="x_{110}"/>
                                     <constant symbol="1"/>
                                    </function>
                                   </function>
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <variable symbol="x_{111}"/>
                                   </function>
                                  </function>
                                 </function>
                                </function>
                                <constant symbol="0"/>
                               </constantatomformula>
                               <constantatomformula symbol="=">
                                <function symbol="f">
                                 <function symbol="+">
                                  <variable symbol="x_{109}"/>
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <function symbol="+">
                                    <variable symbol="x_{110}"/>
                                    <constant symbol="1"/>
                                   </function>
                                  </function>
                                 </function>
                                </function>
                                <constant symbol="0"/>
                               </constantatomformula>
                               <constantatomformula symbol="A" definition="A"/>
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                              <formulalist>
                               <quantifiedformula type="exists">
                                <variable symbol="p"/>
                                <quantifiedformula type="exists">
                                 <variable symbol="q"/>
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                                   <constantatomformula symbol="=">
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                                    <variable symbol="q"/>
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                                  <constantatomformula symbol="=">
                                   <function symbol="f">
                                    <variable symbol="p"/>
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                                   <function symbol="f">
                                    <variable symbol="q"/>
                                   </function>
                                  </constantatomformula>
                                 </conjunctiveformula>
                                </quantifiedformula>
                               </quantifiedformula>
                              </formulalist>
                             </sequent>
                             <rule type="permr" symbol="\pi:r" param="">
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                               <formulalist>
                                <constantatomformula symbol="=">
                                 <function symbol="f">
                                  <function symbol="+">
                                   <variable symbol="x_{109}"/>
                                   <function symbol="+">
                                    <function symbol="+">
                                     <constant symbol="1"/>
                                     <function symbol="+">
                                      <variable symbol="x_{110}"/>
                                      <constant symbol="1"/>
                                     </function>
                                    </function>
                                    <function symbol="+">
                                     <constant symbol="1"/>
                                     <variable symbol="x_{111}"/>
                                    </function>
                                   </function>
                                  </function>
                                 </function>
                                 <constant symbol="0"/>
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                                <constantatomformula symbol="=">
                                 <function symbol="f">
                                  <function symbol="+">
                                   <variable symbol="x_{109}"/>
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <function symbol="+">
                                     <variable symbol="x_{110}"/>
                                     <constant symbol="1"/>
                                    </function>
                                   </function>
                                  </function>
                                 </function>
                                 <constant symbol="0"/>
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                                <constantatomformula symbol="A" definition="A"/>
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                               <formulalist>
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                                 <quantifiedformula type="exists">
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                                     <variable symbol="q"/>
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                                   </conjunctiveformula>
                                   <constantatomformula symbol="=">
                                    <function symbol="f">
                                     <variable symbol="p"/>
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                                    <function symbol="f">
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                                <quantifiedformula type="exists">
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                                   <constantatomformula symbol="=">
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                                    <function symbol="f">
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                                     <function symbol="+">
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                                        <constantatomformula symbol="A" definition="A"/>
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                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </function>
                                                  <variable symbol="x_{111}"/>
                                                 </function>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="permr" symbol="\pi:r" param="(1 2)">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="+">
                                                  <variable symbol="x_{109}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <variable symbol="x_{111}"/>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <variable symbol="x_{111}"/>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="weakr" symbol="w:r">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{111}"/>
                                                  </function>
                                                 </constantatomformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="perml" symbol="\pi:l" param="">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="weakl" symbol="w:l">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="axiom" symbol="axiom">
                                                  <sequent>
                                                   <formulalist/>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                             <rule type="permr" symbol="\pi:r" param="(1 2)">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="+">
                                                  <variable symbol="x_{109}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <variable symbol="x_{111}"/>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <variable symbol="x_{109}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <variable symbol="x_{111}"/>
                                                  </function>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="weakr" symbol="w:r">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="perml" symbol="\pi:l" param="">
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                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="weakl" symbol="w:l">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="axiom" symbol="axiom">
                                                  <sequent>
                                                   <formulalist/>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                        <rule type="perml" symbol="\pi:l" param="(1 2)">
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                                          <formulalist>
                                           <constantatomformula symbol="=">
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                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                                <constant symbol="1"/>
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                                              </function>
                                              <variable symbol="x_{111}"/>
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                                           <constantatomformula symbol="=">
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                                               <function symbol="+">
                                                <variable symbol="x_{110}"/>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                            <constant symbol="0"/>
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                                          <formulalist>
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                                              <constantatomformula symbol="=">
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                                              </constantatomformula>
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                                          </formulalist>
                                         </sequent>
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                                           <formulalist>
                                            <constantatomformula symbol="=">
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                                              <function symbol="+">
                                               <variable symbol="x_{109}"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <variable symbol="x_{110}"/>
                                                 <constant symbol="1"/>
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                                               </function>
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                                             <constant symbol="0"/>
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                                            <constantatomformula symbol="=">
                                             <function symbol="f">
                                              <function symbol="+">
                                               <function symbol="+">
                                                <variable symbol="x_{109}"/>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="1"/>
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                                               </function>
                                               <variable symbol="x_{111}"/>
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                                             </function>
                                             <constant symbol="0"/>
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                                            <constantatomformula symbol="A" definition="A"/>
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                                           <formulalist>
                                            <quantifiedformula type="exists">
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                                               <constantatomformula symbol="=">
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                                                <function symbol="f">
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                                          <rule type="contrr" symbol="c:r" param="2">
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                                             <constantatomformula symbol="=">
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                                               <function symbol="+">
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                                                 <variable symbol="x_{109}"/>
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                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <constant symbol="1"/>
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                                                  </function>
                                                  <constant symbol="1"/>
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                                             <constantatomformula symbol="=">
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                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                              <constant symbol="0"/>
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                                            <formulalist>
                                             <quantifiedformula type="exists">
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                                                </constantatomformula>
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                                              </quantifiedformula>
                                             </quantifiedformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="permr" symbol="\pi:r" param="">
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                                             <formulalist>
                                              <constantatomformula symbol="=">
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                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <variable symbol="x_{109}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                 <variable symbol="x_{111}"/>
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                                               </function>
                                               <constant symbol="0"/>
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                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <variable symbol="x_{109}"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                               <constant symbol="0"/>
                                              </constantatomformula>
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                                             <formulalist>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="contrl" symbol="c:l" param="1,1,2">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <variable symbol="x_{111}"/>
                                                 </function>
                                                </function>
                                                <constant symbol="0"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <variable symbol="x_{109}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <constant symbol="0"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="A" definition="A"/>
                                              </formulalist>
                                              <formulalist>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="perml" symbol="\pi:l" param="(2 3)">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{111}"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="A" definition="A"/>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="eql1" symbol="=:l1" param="(1 1 1)">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="contrr" symbol="c:r" param="2,1">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="permr" symbol="\pi:r" param="">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="contrl" symbol="c:l" param="2">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="perml" symbol="\pi:l" param="">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="eqr1" symbol="=:r1" param="(1)">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="weakr" symbol="w:r">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </constantatomformula>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="perml" symbol="\pi:l" param="">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="weakl" symbol="w:l">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="axiom" symbol="axiom">
                                                         <sequent>
                                                          <formulalist/>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                    <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="weakr" symbol="w:r">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </constantatomformula>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="perml" symbol="\pi:l" param="">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="weakl" symbol="w:l">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="axiom" symbol="axiom">
                                                         <sequent>
                                                          <formulalist/>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                               <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="permr" symbol="\pi:r" param="">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="perml" symbol="\pi:l" param="(1 2 3)">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="contrr" symbol="c:r" param="2">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="weakr" symbol="w:r">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="weakl" symbol="w:l">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="0"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="0"/>
                                                       </constantatomformula>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="existsr" symbol="\ex:r">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="0"/>
                                                        </constantatomformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="0"/>
                                                        </constantatomformula>
                                                       </formulalist>
                                                       <formulalist>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="existsr" symbol="\ex:r">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                        </formulalist>
                                                        <formulalist>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="andr" symbol="\and:r">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="0"/>
                                                          </constantatomformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="0"/>
                                                          </constantatomformula>
                                                         </formulalist>
                                                         <formulalist>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="negr" symbol="\neg:r">
                                                         <sequent>
                                                          <formulalist/>
                                                          <formulalist>
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="eql1" symbol="=:l1" param="(2)">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </formulalist>
                                                           <formulalist/>
                                                          </sequent>
                                                          <rule type="axiom" symbol="axiom">
                                                           <sequent>
                                                            <formulalist/>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                          </rule>
                                                          <rule type="axiom" symbol="axiom">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </constantatomformula>
                                                            </formulalist>
                                                            <formulalist/>
                                                           </sequent>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                        <rule type="eqr2" symbol="=:r2" param="(1)">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="0"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="0"/>
                                                           </constantatomformula>
                                                          </formulalist>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="axiom" symbol="axiom">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="0"/>
                                                            </constantatomformula>
                                                           </formulalist>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="0"/>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                         </rule>
                                                         <rule type="eqr2" symbol="=:r2" param="(2)">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="0"/>
                                                            </constantatomformula>
                                                           </formulalist>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <constant symbol="0"/>
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="axiom" symbol="axiom">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                            </formulalist>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                          </rule>
                                                          <rule type="axiom" symbol="axiom">
                                                           <sequent>
                                                            <formulalist/>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <constant symbol="0"/>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                          </rule>
                         </rule>
                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
       <rule type="contrr" symbol="c:r" param="2">
        <sequent>
         <formulalist>
          <constantatomformula symbol="=">
           <function symbol="f">
            <function symbol="+">
             <variable symbol="x_{109}"/>
             <function symbol="+">
              <constant symbol="1"/>
              <function symbol="+">
               <variable symbol="x_{110}"/>
               <function symbol="+">
                <constant symbol="1"/>
                <function symbol="+">
                 <constant symbol="1"/>
                 <variable symbol="x_{111}"/>
                </function>
               </function>
              </function>
             </function>
            </function>
           </function>
           <constant symbol="1"/>
          </constantatomformula>
          <constantatomformula symbol="A" definition="A"/>
         </formulalist>
         <formulalist>
          <quantifiedformula type="exists">
           <variable symbol="p"/>
           <quantifiedformula type="exists">
            <variable symbol="q"/>
            <conjunctiveformula type="and">
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                          <variable symbol="q"/>
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                       <quantifiedformula type="exists">
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                          <constantatomformula symbol="=">
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                           <variable symbol="q"/>
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                         <constantatomformula symbol="=">
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                          <function symbol="f">
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                         <function symbol="+">
                          <variable symbol="x_{112}"/>
                          <function symbol="+">
                           <function symbol="+">
                            <constant symbol="1"/>
                            <function symbol="+">
                             <variable symbol="x_{110}"/>
                             <function symbol="+">
                              <constant symbol="1"/>
                              <function symbol="+">
                               <constant symbol="1"/>
                               <variable symbol="x_{111}"/>
                              </function>
                             </function>
                            </function>
                           </function>
                           <variable symbol="x_{109}"/>
                          </function>
                         </function>
                        </function>
                       </function>
                       <constant symbol="1"/>
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                     </formulalist>
                    </sequent>
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                      <formulalist>
                       <constantatomformula symbol="A" definition="A"/>
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                        <quantifiedformula type="exists">
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                            <variable symbol="q"/>
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                           <function symbol="f">
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                            <variable symbol="q"/>
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                          <constantatomformula symbol="=">
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                           <function symbol="f">
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                          <function symbol="+">
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                           <function symbol="+">
                            <function symbol="+">
                             <constant symbol="1"/>
                             <function symbol="+">
                              <variable symbol="x_{110}"/>
                              <function symbol="+">
                               <constant symbol="1"/>
                               <function symbol="+">
                                <constant symbol="1"/>
                                <variable symbol="x_{111}"/>
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                              </function>
                             </function>
                            </function>
                            <variable symbol="x_{109}"/>
                           </function>
                          </function>
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                        </function>
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                        <constantatomformula symbol="A" definition="A"/>
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                        <quantifiedformula type="exists">
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                           <constantatomformula symbol="=">
                            <function symbol="f">
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                            <function symbol="f">
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                        <quantifiedformula type="exists">
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                         <quantifiedformula type="exists">
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                           <conjunctiveformula type="neg">
                            <constantatomformula symbol="=">
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                             <variable symbol="q"/>
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                           <constantatomformula symbol="=">
                            <function symbol="f">
                             <variable symbol="p"/>
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                            <function symbol="f">
                             <variable symbol="q"/>
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                        <constantatomformula symbol="=">
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                          <function symbol="+">
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                           <function symbol="+">
                            <variable symbol="x_{112}"/>
                            <function symbol="+">
                             <function symbol="+">
                              <constant symbol="1"/>
                              <function symbol="+">
                               <variable symbol="x_{110}"/>
                               <function symbol="+">
                                <constant symbol="1"/>
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <variable symbol="x_{111}"/>
                                </function>
                               </function>
                              </function>
                             </function>
                             <variable symbol="x_{109}"/>
                            </function>
                           </function>
                          </function>
                         </function>
                         <constant symbol="1"/>
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                      <rule type="perml" symbol="\pi:l" param="">
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                        <formulalist>
                         <constantatomformula symbol="A" definition="A"/>
                         <constantatomformula symbol="A" definition="A"/>
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                        <formulalist>
                         <quantifiedformula type="exists">
                          <variable symbol="p"/>
                          <quantifiedformula type="exists">
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                            <conjunctiveformula type="neg">
                             <constantatomformula symbol="=">
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                              <variable symbol="q"/>
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                            <constantatomformula symbol="=">
                             <function symbol="f">
                              <variable symbol="p"/>
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                             <function symbol="f">
                              <variable symbol="q"/>
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                            </constantatomformula>
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                         <quantifiedformula type="exists">
                          <variable symbol="p"/>
                          <quantifiedformula type="exists">
                           <variable symbol="q"/>
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                            <conjunctiveformula type="neg">
                             <constantatomformula symbol="=">
                              <variable symbol="p"/>
                              <variable symbol="q"/>
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                            <constantatomformula symbol="=">
                             <function symbol="f">
                              <variable symbol="p"/>
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                             <function symbol="f">
                              <variable symbol="q"/>
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                           </conjunctiveformula>
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                         <constantatomformula symbol="=">
                          <function symbol="f">
                           <function symbol="+">
                            <constant symbol="1"/>
                            <function symbol="+">
                             <variable symbol="x_{112}"/>
                             <function symbol="+">
                              <function symbol="+">
                               <constant symbol="1"/>
                               <function symbol="+">
                                <variable symbol="x_{110}"/>
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <variable symbol="x_{111}"/>
                                 </function>
                                </function>
                               </function>
                              </function>
                              <variable symbol="x_{109}"/>
                             </function>
                            </function>
                           </function>
                          </function>
                          <constant symbol="1"/>
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                        </formulalist>
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                       <rule type="cut" symbol="cut">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="A" definition="A"/>
                          <constantatomformula symbol="A" definition="A"/>
                         </formulalist>
                         <formulalist>
                          <quantifiedformula type="exists">
                           <variable symbol="p"/>
                           <quantifiedformula type="exists">
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                             <constantatomformula symbol="=">
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                               <variable symbol="p"/>
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                              <function symbol="f">
                               <variable symbol="q"/>
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                          <quantifiedformula type="exists">
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                           <quantifiedformula type="exists">
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                              <constantatomformula symbol="=">
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                               <variable symbol="q"/>
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                             <constantatomformula symbol="=">
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                               <variable symbol="p"/>
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                              <function symbol="f">
                               <variable symbol="q"/>
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                          </quantifiedformula>
                          <constantatomformula symbol="=">
                           <function symbol="f">
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                             <function symbol="+">
                              <variable symbol="x_{112}"/>
                              <function symbol="+">
                               <function symbol="+">
                                <constant symbol="1"/>
                                <function symbol="+">
                                 <variable symbol="x_{110}"/>
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <variable symbol="x_{111}"/>
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                                 </function>
                                </function>
                               </function>
                               <variable symbol="x_{109}"/>
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                             </function>
                            </function>
                           </function>
                           <constant symbol="1"/>
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                        <rule type="contrr" symbol="c:r" param="1,2">
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                          <formulalist>
                           <constantatomformula symbol="A" definition="A"/>
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                              <constantatomformula symbol="=">
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                           <constantatomformula symbol="=">
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                               <function symbol="+">
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <function symbol="+">
                                  <variable symbol="x_{110}"/>
                                  <function symbol="+">
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                                   <function symbol="+">
                                    <constant symbol="1"/>
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                                 </function>
                                </function>
                                <variable symbol="x_{109}"/>
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                              <function symbol="+">
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                               <function symbol="+">
                                <constant symbol="1"/>
                                <constant symbol="1"/>
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                               <constantatomformula symbol="=">
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                            <constantatomformula symbol="=">
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                                 </function>
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                               <function symbol="+">
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                                <function symbol="+">
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                                 <constant symbol="1"/>
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                            <constantatomformula symbol="=">
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                                   <function symbol="+">
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                                    <function symbol="+">
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                                <constantatomformula symbol="=">
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                             <quantifiedformula type="exists">
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                                <constantatomformula symbol="=">
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                                   <function symbol="+">
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                                     <function symbol="+">
                                      <variable symbol="x_{110}"/>
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                                        <variable symbol="x_{111}"/>
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                                    </function>
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                                           <constant symbol="1"/>
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                                        </function>
                                       </function>
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                                      <formulalist>
                                       <constantatomformula symbol="A" definition="A"/>
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                                      <formulalist>
                                       <quantifiedformula type="exists">
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                                          <constantatomformula symbol="=">
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                                       <constantatomformula symbol="=">
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                                          <constant symbol="1"/>
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                                          <function symbol="+">
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                                           <constant symbol="1"/>
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                                         </function>
                                        </function>
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                                            <constant symbol="1"/>
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                                         </function>
                                        </function>
                                       </constantatomformula>
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                                       <formulalist>
                                        <constantatomformula symbol="A" definition="A"/>
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                                       <formulalist>
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                                        <constantatomformula symbol="=">
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                                           <constant symbol="1"/>
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                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <constant symbol="1"/>
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                                          </function>
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                                         <function symbol="+">
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                                          <function symbol="+">
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                                             <constant symbol="1"/>
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                                        <formulalist>
                                         <constantatomformula symbol="A" definition="A"/>
                                         <constantatomformula symbol="A" definition="A"/>
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                                        <formulalist>
                                         <quantifiedformula type="exists">
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                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <constant symbol="1"/>
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                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
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                                              <constant symbol="1"/>
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                                            </function>
                                           </function>
                                          </function>
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                                        </formulalist>
                                       </sequent>
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                                         <formulalist>
                                          <constantatomformula symbol="A" definition="A"/>
                                          <constantatomformula symbol="A" definition="A"/>
                                         </formulalist>
                                         <formulalist>
                                          <quantifiedformula type="exists">
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                                          <quantifiedformula type="exists">
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                                              <constant symbol="1"/>
                                              <constant symbol="1"/>
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                                               <constant symbol="1"/>
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                                          <formulalist>
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                                                <constant symbol="1"/>
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                                              <constant symbol="1"/>
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                                                <constant symbol="1"/>
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                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
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                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="+">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <variable symbol="x_{112}"/>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <variable symbol="x_{109}"/>
                                                 </function>
                                                </function>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                              </function>
                                              <function symbol="+">
                                               <function symbol="+">
                                                <variable symbol="x_{112}"/>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <variable symbol="x_{109}"/>
                                                </function>
                                               </function>
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <constant symbol="1"/>
                                                </function>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="perml" symbol="\pi:l" param="">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="A" definition="A"/>
                                              <constantatomformula symbol="A" definition="A"/>
                                             </formulalist>
                                             <formulalist>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <variable symbol="x_{112}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{109}"/>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                               </function>
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <variable symbol="x_{112}"/>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <variable symbol="x_{109}"/>
                                                 </function>
                                                </function>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="eqr1" symbol="=:r1" param="(1)">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="A" definition="A"/>
                                               <constantatomformula symbol="A" definition="A"/>
                                              </formulalist>
                                              <formulalist>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{112}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                </function>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <variable symbol="x_{112}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{109}"/>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="permr" symbol="\pi:r" param="(1 2)">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{112}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{112}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="weakr" symbol="w:r">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{112}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="perml" symbol="\pi:l" param="">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="weakl" symbol="w:l">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="axiom" symbol="axiom">
                                                  <sequent>
                                                   <formulalist/>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                             <rule type="permr" symbol="\pi:r" param="(1 2)">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{112}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{112}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="weakr" symbol="w:r">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{112}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{112}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="perml" symbol="\pi:l" param="">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="weakl" symbol="w:l">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="axiom" symbol="axiom">
                                                  <sequent>
                                                   <formulalist/>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                        <rule type="permr" symbol="\pi:r" param="(2 3)">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="A" definition="A"/>
                                          </formulalist>
                                          <formulalist>
                                           <quantifiedformula type="exists">
                                            <variable symbol="p"/>
                                            <quantifiedformula type="exists">
                                             <variable symbol="q"/>
                                             <conjunctiveformula type="and">
                                              <conjunctiveformula type="neg">
                                               <constantatomformula symbol="=">
                                                <variable symbol="p"/>
                                                <variable symbol="q"/>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <variable symbol="p"/>
                                               </function>
                                               <function symbol="f">
                                                <variable symbol="q"/>
                                               </function>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                            </quantifiedformula>
                                           </quantifiedformula>
                                           <constantatomformula symbol="=">
                                            <function symbol="f">
                                             <function symbol="+">
                                              <function symbol="+">
                                               <variable symbol="x_{112}"/>
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <variable symbol="x_{109}"/>
                                               </function>
                                              </function>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                            <constant symbol="0"/>
                                           </constantatomformula>
                                           <constantatomformula symbol="=">
                                            <function symbol="f">
                                             <function symbol="+">
                                              <function symbol="+">
                                               <function symbol="+">
                                                <variable symbol="x_{112}"/>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <variable symbol="x_{109}"/>
                                                </function>
                                               </function>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                            <constant symbol="0"/>
                                           </constantatomformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="contrr" symbol="c:r" param="2,1,1">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="A" definition="A"/>
                                           </formulalist>
                                           <formulalist>
                                            <quantifiedformula type="exists">
                                             <variable symbol="p"/>
                                             <quantifiedformula type="exists">
                                              <variable symbol="q"/>
                                              <conjunctiveformula type="and">
                                               <conjunctiveformula type="neg">
                                                <constantatomformula symbol="=">
                                                 <variable symbol="p"/>
                                                 <variable symbol="q"/>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <variable symbol="p"/>
                                                </function>
                                                <function symbol="f">
                                                 <variable symbol="q"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                             </quantifiedformula>
                                            </quantifiedformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="f">
                                              <function symbol="+">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <variable symbol="x_{112}"/>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <variable symbol="x_{109}"/>
                                                 </function>
                                                </function>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                             <constant symbol="0"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="f">
                                              <function symbol="+">
                                               <function symbol="+">
                                                <variable symbol="x_{112}"/>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <variable symbol="x_{109}"/>
                                                </function>
                                               </function>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                             <constant symbol="0"/>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="permr" symbol="\pi:r" param="(2 3)">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="A" definition="A"/>
                                            </formulalist>
                                            <formulalist>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <variable symbol="x_{112}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{109}"/>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                              <constant symbol="0"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <variable symbol="x_{112}"/>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <variable symbol="x_{109}"/>
                                                 </function>
                                                </function>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                              <constant symbol="0"/>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="contrl" symbol="c:l" param="2">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="A" definition="A"/>
                                             </formulalist>
                                             <formulalist>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{112}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                               <constant symbol="0"/>
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                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
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                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
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                                              </quantifiedformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <variable symbol="x_{112}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{109}"/>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                               <constant symbol="0"/>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="perml" symbol="\pi:l" param="">
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                                              <formulalist>
                                               <constantatomformula symbol="A" definition="A"/>
                                               <constantatomformula symbol="A" definition="A"/>
                                              </formulalist>
                                              <formulalist>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
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                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
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                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
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                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{112}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <constant symbol="0"/>
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                                                <quantifiedformula type="exists">
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                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
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                                                  </constantatomformula>
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                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{112}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <constant symbol="0"/>
                                               </constantatomformula>
                                              </formulalist>
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                                             <rule type="cut" symbol="cut">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                                <constantatomformula symbol="A" definition="A"/>
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                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
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                                                 <constant symbol="0"/>
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                                                 <quantifiedformula type="exists">
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                                                   <constantatomformula symbol="=">
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                                                    <function symbol="f">
                                                     <variable symbol="q"/>
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                                                   </constantatomformula>
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                                                <constantatomformula symbol="=">
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                                                        <function symbol="+">
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                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
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                                                   </function>
                                                   <function symbol="+">
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                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
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                                              <rule type="permr" symbol="\pi:r" param="(1 3 2)">
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                                                 <constantatomformula symbol="A" definition="A"/>
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                                                 <quantifiedformula type="exists">
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                                                    <constantatomformula symbol="=">
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                                                     <function symbol="f">
                                                      <variable symbol="q"/>
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                                                 <constantatomformula symbol="=">
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                                                    <function symbol="+">
                                                     <function symbol="+">
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                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
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                                                          <function symbol="+">
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                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
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                                                     </function>
                                                     <function symbol="+">
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                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
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                                                      <constant symbol="1"/>
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                                                    </function>
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                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
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                                                       <function symbol="+">
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                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
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                                                           <variable symbol="x_{111}"/>
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                                                         </function>
                                                        </function>
                                                       </function>
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                                                     </function>
                                                     <function symbol="+">
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                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="1"/>
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                                               </sequent>
                                               <rule type="perml" symbol="\pi:l" param="">
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                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
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                                                  <constantatomformula symbol="=">
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                                                     <function symbol="+">
                                                      <function symbol="+">
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                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
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                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
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                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
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                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
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                                                     </function>
                                                    </function>
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                                                   <constant symbol="0"/>
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                                                  <constantatomformula symbol="=">
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                                                       <constant symbol="1"/>
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                                                     </function>
                                                     <function symbol="+">
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                                                       <function symbol="+">
                                                        <function symbol="+">
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                                                         <function symbol="+">
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                                                           <function symbol="+">
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                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
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                                                      </function>
                                                      <function symbol="+">
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                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <constant symbol="1"/>
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                                                  <quantifiedformula type="exists">
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                                                   <quantifiedformula type="exists">
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                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
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                                                      <function symbol="f">
                                                       <variable symbol="q"/>
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                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                 </formulalist>
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                                                <rule type="contrr" symbol="c:r" param="1,1,2">
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                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
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                                                  <formulalist>
                                                   <constantatomformula symbol="=">
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                                                         </function>
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                                                      </function>
                                                      <function symbol="+">
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                                                        <constant symbol="1"/>
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                                                     </function>
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                                                      </function>
                                                      <function symbol="+">
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                                                           </function>
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                                                         </function>
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                                                       </function>
                                                       <function symbol="+">
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                                                        <constant symbol="1"/>
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                                                      </function>
                                                     </function>
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                                                   <quantifiedformula type="exists">
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                                                      <constantatomformula symbol="=">
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                                                      </constantatomformula>
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                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="weakr" symbol="w:r">
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                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
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                                                   <formulalist>
                                                    <constantatomformula symbol="=">
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                                                       <function symbol="+">
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                                                              <variable symbol="x_{111}"/>
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                                                            </function>
                                                           </function>
                                                          </function>
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                                                        </function>
                                                        <function symbol="+">
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                                                         <constant symbol="1"/>
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                                                       </function>
                                                       <function symbol="+">
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                                                         <constant symbol="1"/>
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                                                       </function>
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                                                    <constantatomformula symbol="=">
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                                                         <constant symbol="1"/>
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                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
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                                                         <function symbol="+">
                                                          <function symbol="+">
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                                                           <function symbol="+">
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                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
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                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
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                                                    <quantifiedformula type="exists">
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                                                     <quantifiedformula type="exists">
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                                                       <constantatomformula symbol="=">
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                                                        <function symbol="f">
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                                                       </constantatomformula>
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                                                    </quantifiedformula>
                                                    <quantifiedformula type="exists">
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                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="weakr" symbol="w:r">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </constantatomformula>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="defl" symbol="d:l" param="();[()];()">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="foralll" symbol="\all:l">
                                                     <sequent>
                                                      <formulalist>
                                                       <quantifiedformula type="all">
                                                        <variable symbol="x"/>
                                                        <conjunctiveformula type="or">
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="x"/>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="x"/>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </formulalist>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="0"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="orl" symbol="\or:l">
                                                      <sequent>
                                                       <formulalist>
                                                        <conjunctiveformula type="or">
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </formulalist>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="0"/>
                                                        </constantatomformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="axiom" symbol="axiom">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                        </formulalist>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                      </rule>
                                                      <rule type="eql2" symbol="=:l2" param="(1 1)">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </constantatomformula>
                                                        </formulalist>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="axiom" symbol="axiom">
                                                        <sequent>
                                                         <formulalist/>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                       </rule>
                                                       <rule type="axiom" symbol="axiom">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </constantatomformula>
                                                         </formulalist>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                              <rule type="contrr" symbol="c:r" param="2,1">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="1"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="permr" symbol="\pi:r" param="(2 3)">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="contrl" symbol="c:l" param="1,2">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </constantatomformula>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="cut" symbol="cut">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="permr" symbol="\pi:r" param="(1 3 2)">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="perml" symbol="\pi:l" param="">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="0"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </constantatomformula>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="contrr" symbol="c:r" param="1,1,2">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="0"/>
                                                        </constantatomformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </constantatomformula>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="weakr" symbol="w:r">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </constantatomformula>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
                                                            <conjunctiveformula type="neg">
                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
                                                            <conjunctiveformula type="neg">
                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="weakr" symbol="w:r">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="0"/>
                                                          </constantatomformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </constantatomformula>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
                                                            <conjunctiveformula type="and">
                                                             <conjunctiveformula type="neg">
                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="defl" symbol="d:l" param="();[()];()">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="0"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="foralll" symbol="\all:l">
                                                          <sequent>
                                                           <formulalist>
                                                            <quantifiedformula type="all">
                                                             <variable symbol="x"/>
                                                             <conjunctiveformula type="or">
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="x"/>
                                                               </function>
                                                               <constant symbol="0"/>
                                                              </constantatomformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="x"/>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </formulalist>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="0"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="orl" symbol="\or:l">
                                                           <sequent>
                                                            <formulalist>
                                                             <conjunctiveformula type="or">
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="0"/>
                                                              </constantatomformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </formulalist>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{112}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <variable symbol="x_{109}"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{112}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <variable symbol="x_{109}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="axiom" symbol="axiom">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="0"/>
                                                              </constantatomformula>
                                                             </formulalist>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="0"/>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                           </rule>
                                                           <rule type="eql2" symbol="=:l2" param="(1 1)">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </constantatomformula>
                                                             </formulalist>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="axiom" symbol="axiom">
                                                             <sequent>
                                                              <formulalist/>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </constantatomformula>
                                                              </formulalist>
                                                             </sequent>
                                                            </rule>
                                                            <rule type="axiom" symbol="axiom">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </constantatomformula>
                                                              </formulalist>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </constantatomformula>
                                                              </formulalist>
                                                             </sequent>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                   <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
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                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
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                                                         <constantatomformula symbol="=">
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                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
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                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="contrr" symbol="c:r" param="2">
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                                                      <formulalist>
                                                       <constantatomformula symbol="=">
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                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
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                                                       <constantatomformula symbol="=">
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                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
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                                                       <constantatomformula symbol="A" definition="A"/>
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                                                      <formulalist>
                                                       <quantifiedformula type="exists">
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                                                          <constantatomformula symbol="=">
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                                                           <function symbol="f">
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                                                          </constantatomformula>
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                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="permr" symbol="\pi:r" param="">
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                                                       <formulalist>
                                                        <constantatomformula symbol="=">
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                                                             <constant symbol="1"/>
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                                                           </function>
                                                           <function symbol="+">
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                                                                <variable symbol="x_{110}"/>
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                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
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                                                        <constantatomformula symbol="=">
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                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
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                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
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                                                        <constantatomformula symbol="A" definition="A"/>
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                                                       <formulalist>
                                                        <quantifiedformula type="exists">
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                                                           <constantatomformula symbol="=">
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                                                        <quantifiedformula type="exists">
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                                                           <constantatomformula symbol="=">
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                                                           </constantatomformula>
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                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="contrl" symbol="c:l" param="1,1,2">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
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                                                           <function symbol="+">
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                                                              <constant symbol="1"/>
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                                                            </function>
                                                            <function symbol="+">
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                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
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                                                             </function>
                                                             <function symbol="+">
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                                                              <constant symbol="1"/>
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                                                            </function>
                                                           </function>
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                                                         <constantatomformula symbol="=">
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                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
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                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
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                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </constantatomformula>
                                                         <constantatomformula symbol="A" definition="A"/>
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                                                        <formulalist>
                                                         <quantifiedformula type="exists">
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                                                            <constantatomformula symbol="=">
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                                                         <quantifiedformula type="exists">
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                                                            <constantatomformula symbol="=">
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                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="perml" symbol="\pi:l" param="(2 3)">
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                                                         <formulalist>
                                                          <constantatomformula symbol="=">
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                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
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                                                                  <variable symbol="x_{110}"/>
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                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
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                                                              </function>
                                                              <function symbol="+">
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                                                               <constant symbol="1"/>
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                                                             </function>
                                                            </function>
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                                                             </function>
                                                             <function symbol="+">
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                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
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                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
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                                                          <constantatomformula symbol="A" definition="A"/>
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                                                         </formulalist>
                                                         <formulalist>
                                                          <quantifiedformula type="exists">
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                                                           <quantifiedformula type="exists">
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                                                               <variable symbol="q"/>
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                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
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                                                              <function symbol="f">
                                                               <variable symbol="q"/>
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                                                             </constantatomformula>
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                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                          <quantifiedformula type="exists">
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                                                           <quantifiedformula type="exists">
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                                                               <variable symbol="q"/>
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                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
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                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="eql1" symbol="=:l1" param="(1 1 2)">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="p"/>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="q"/>
                                                             <conjunctiveformula type="and">
                                                              <conjunctiveformula type="neg">
                                                               <constantatomformula symbol="=">
                                                                <variable symbol="p"/>
                                                                <variable symbol="q"/>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                           <quantifiedformula type="exists">
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                                                            <quantifiedformula type="exists">
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                                                               <constantatomformula symbol="=">
                                                                <variable symbol="p"/>
                                                                <variable symbol="q"/>
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                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="contrr" symbol="c:r" param="2,1">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <quantifiedformula type="exists">
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                                                             <quantifiedformula type="exists">
                                                              <variable symbol="q"/>
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                                                                <constantatomformula symbol="=">
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                                                                 <variable symbol="q"/>
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                                                               </conjunctiveformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <variable symbol="p"/>
                                                                </function>
                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </quantifiedformula>
                                                            </quantifiedformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="permr" symbol="\pi:r" param="">
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                                                            <formulalist>
                                                             <constantatomformula symbol="A" definition="A"/>
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                                                            <formulalist>
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                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <variable symbol="p"/>
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                                                                 <function symbol="f">
                                                                  <variable symbol="q"/>
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                                                                </constantatomformula>
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                                                             <quantifiedformula type="exists">
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                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
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                                                                  <variable symbol="q"/>
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                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                              </quantifiedformula>
                                                             </quantifiedformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{112}"/>
                                                                 <function symbol="+">
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                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
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                                                                      <variable symbol="x_{111}"/>
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                                                                    </function>
                                                                   </function>
                                                                  </function>
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                                                               <constant symbol="1"/>
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                                                                <function symbol="+">
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                                                                   </function>
                                                                  </function>
                                                                 </function>
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                                                               </function>
                                                               <function symbol="+">
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                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="contrl" symbol="c:l" param="2">
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                                                                 </constantatomformula>
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                                                              <quantifiedformula type="exists">
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                                                                     </function>
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                                                                   </function>
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                                                                 </function>
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                                                                <constant symbol="1"/>
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                                                               <function symbol="+">
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                                                                    </function>
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                                                                </function>
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                                                                 <constant symbol="1"/>
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                                                               </function>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="perml" symbol="\pi:l" param="">
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                                                              <formulalist>
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                                                                  <constantatomformula symbol="=">
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                                                               <quantifiedformula type="exists">
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                                                                </quantifiedformula>
                                                               </quantifiedformula>
                                                               <constantatomformula symbol="=">
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                                                                   <function symbol="+">
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                                                                      </function>
                                                                     </function>
                                                                    </function>
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                                                                  </function>
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                                                                 <constant symbol="1"/>
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                                                                     </function>
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                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
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                                                                 </function>
                                                                 <function symbol="+">
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                                                                  <constant symbol="1"/>
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                                                                </function>
                                                               </constantatomformula>
                                                              </formulalist>
                                                             </sequent>
                                                             <rule type="eqr1" symbol="=:r1" param="(1)">
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                                                               <formulalist>
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                                                                <constantatomformula symbol="A" definition="A"/>
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                                                               <formulalist>
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                                                                   <constantatomformula symbol="=">
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                                                                   </constantatomformula>
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                                                                   </constantatomformula>
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                                                                <constantatomformula symbol="=">
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                                                                    <function symbol="+">
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                                                                       </function>
                                                                      </function>
                                                                     </function>
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                                                                   <function symbol="+">
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                                                                      </function>
                                                                     </function>
                                                                    </function>
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                                                                  </function>
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                                                                  </function>
                                                                 </function>
                                                                </constantatomformula>
                                                               </formulalist>
                                                              </sequent>
                                                              <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                               <sequent>
                                                                <formulalist>
                                                                 <constantatomformula symbol="A" definition="A"/>
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                                                                <formulalist>
                                                                 <quantifiedformula type="exists">
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                                                                    <constantatomformula symbol="=">
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                                                                     <function symbol="f">
                                                                      <variable symbol="q"/>
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                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                  </quantifiedformula>
                                                                 </quantifiedformula>
                                                                 <constantatomformula symbol="=">
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                                                                    <function symbol="+">
                                                                     <function symbol="+">
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                                                                       <variable symbol="x_{110}"/>
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                                                                        <function symbol="+">
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                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </constantatomformula>
                                                                </formulalist>
                                                               </sequent>
                                                               <rule type="weakr" symbol="w:r">
                                                                <sequent>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="A" definition="A"/>
                                                                 </formulalist>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{112}"/>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="x_{109}"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </constantatomformula>
                                                                  <quantifiedformula type="exists">
                                                                   <variable symbol="p"/>
                                                                   <quantifiedformula type="exists">
                                                                    <variable symbol="q"/>
                                                                    <conjunctiveformula type="and">
                                                                     <conjunctiveformula type="neg">
                                                                      <constantatomformula symbol="=">
                                                                       <variable symbol="p"/>
                                                                       <variable symbol="q"/>
                                                                      </constantatomformula>
                                                                     </conjunctiveformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <variable symbol="p"/>
                                                                      </function>
                                                                      <function symbol="f">
                                                                       <variable symbol="q"/>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                   </quantifiedformula>
                                                                  </quantifiedformula>
                                                                 </formulalist>
                                                                </sequent>
                                                                <rule type="perml" symbol="\pi:l" param="">
                                                                 <sequent>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="A" definition="A"/>
                                                                  </formulalist>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{112}"/>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="x_{109}"/>
                                                                      </function>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                 <rule type="weakl" symbol="w:l">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="A" definition="A"/>
                                                                   </formulalist>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                  <rule type="axiom" symbol="axiom">
                                                                   <sequent>
                                                                    <formulalist/>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                              <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                               <sequent>
                                                                <formulalist>
                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                </formulalist>
                                                                <formulalist>
                                                                 <quantifiedformula type="exists">
                                                                  <variable symbol="p"/>
                                                                  <quantifiedformula type="exists">
                                                                   <variable symbol="q"/>
                                                                   <conjunctiveformula type="and">
                                                                    <conjunctiveformula type="neg">
                                                                     <constantatomformula symbol="=">
                                                                      <variable symbol="p"/>
                                                                      <variable symbol="q"/>
                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <variable symbol="p"/>
                                                                     </function>
                                                                     <function symbol="f">
                                                                      <variable symbol="q"/>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                  </quantifiedformula>
                                                                 </quantifiedformula>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                 </constantatomformula>
                                                                </formulalist>
                                                               </sequent>
                                                               <rule type="weakr" symbol="w:r">
                                                                <sequent>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="A" definition="A"/>
                                                                 </formulalist>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                  </constantatomformula>
                                                                  <quantifiedformula type="exists">
                                                                   <variable symbol="p"/>
                                                                   <quantifiedformula type="exists">
                                                                    <variable symbol="q"/>
                                                                    <conjunctiveformula type="and">
                                                                     <conjunctiveformula type="neg">
                                                                      <constantatomformula symbol="=">
                                                                       <variable symbol="p"/>
                                                                       <variable symbol="q"/>
                                                                      </constantatomformula>
                                                                     </conjunctiveformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <variable symbol="p"/>
                                                                      </function>
                                                                      <function symbol="f">
                                                                       <variable symbol="q"/>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                   </quantifiedformula>
                                                                  </quantifiedformula>
                                                                 </formulalist>
                                                                </sequent>
                                                                <rule type="perml" symbol="\pi:l" param="">
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                                                                  <formulalist>
                                                                   <constantatomformula symbol="A" definition="A"/>
                                                                  </formulalist>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{112}"/>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="x_{109}"/>
                                                                      </function>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{112}"/>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="x_{109}"/>
                                                                      </function>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                 <rule type="weakl" symbol="w:l">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="A" definition="A"/>
                                                                   </formulalist>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                  <rule type="axiom" symbol="axiom">
                                                                   <sequent>
                                                                    <formulalist/>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                         <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="p"/>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="q"/>
                                                              <conjunctiveformula type="and">
                                                               <conjunctiveformula type="neg">
                                                                <constantatomformula symbol="=">
                                                                 <variable symbol="p"/>
                                                                 <variable symbol="q"/>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <variable symbol="p"/>
                                                                </function>
                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </quantifiedformula>
                                                            </quantifiedformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{112}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <variable symbol="x_{109}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </constantatomformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
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                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </constantatomformula>
                                                             <constantatomformula symbol="A" definition="A"/>
                                                            </formulalist>
                                                            <formulalist>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="p"/>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="q"/>
                                                               <conjunctiveformula type="and">
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                                                                 <constantatomformula symbol="=">
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                                                                  <variable symbol="q"/>
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                                                                <constantatomformula symbol="=">
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                                                                  <variable symbol="q"/>
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                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                              </quantifiedformula>
                                                             </quantifiedformula>
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                                                           <rule type="contrr" symbol="c:r" param="2">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </constantatomformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </constantatomformula>
                                                              <constantatomformula symbol="A" definition="A"/>
                                                             </formulalist>
                                                             <formulalist>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="p"/>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="q"/>
                                                                <conjunctiveformula type="and">
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                                                                  <constantatomformula symbol="=">
                                                                   <variable symbol="p"/>
                                                                   <variable symbol="q"/>
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                                                                 <constantatomformula symbol="=">
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                                                                  <function symbol="f">
                                                                   <variable symbol="q"/>
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                                                                 </constantatomformula>
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                                                               </quantifiedformula>
                                                              </quantifiedformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="permr" symbol="\pi:r" param="">
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                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
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                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
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                                                               <constantatomformula symbol="A" definition="A"/>
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                                                              <formulalist>
                                                               <quantifiedformula type="exists">
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                                                                <quantifiedformula type="exists">
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                                                                  <constantatomformula symbol="=">
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                                                             </sequent>
                                                             <rule type="contrl" symbol="c:l" param="1,1,2">
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                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
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                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <constant symbol="1"/>
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                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
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                                                                        <function symbol="+">
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                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="x_{109}"/>
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                                                                    <constant symbol="1"/>
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                                                                     <constant symbol="1"/>
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                                                                     <function symbol="+">
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                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
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                                                               <rule type="eql1" symbol="=:l1" param="(1 1)">
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                                                                  <constantatomformula symbol="=">
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                                                                          </function>
                                                                         </function>
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                                                                         </function>
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                                                                       </function>
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                                                                     </function>
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                                                                   <constant symbol="1"/>
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                                                                  <quantifiedformula type="exists">
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                                                                     <constantatomformula symbol="=">
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                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                   </quantifiedformula>
                                                                  </quantifiedformula>
                                                                 </formulalist>
                                                                </sequent>
                                                                <rule type="contrr" symbol="c:r" param="2,1">
                                                                 <sequent>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="A" definition="A"/>
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                                                                   <quantifiedformula type="exists">
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                                                                      </constantatomformula>
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                                                                   <constantatomformula symbol="=">
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                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
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                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
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                                                                     </function>
                                                                     <constant symbol="1"/>
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                                                                    <function symbol="+">
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                                                                      <function symbol="+">
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                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                 <rule type="permr" symbol="\pi:r" param="">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="A" definition="A"/>
                                                                   </formulalist>
                                                                   <formulalist>
                                                                    <quantifiedformula type="exists">
                                                                     <variable symbol="p"/>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="q"/>
                                                                      <conjunctiveformula type="and">
                                                                       <conjunctiveformula type="neg">
                                                                        <constantatomformula symbol="=">
                                                                         <variable symbol="p"/>
                                                                         <variable symbol="q"/>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <variable symbol="p"/>
                                                                        </function>
                                                                        <function symbol="f">
                                                                         <variable symbol="q"/>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </conjunctiveformula>
                                                                     </quantifiedformula>
                                                                    </quantifiedformula>
                                                                    <quantifiedformula type="exists">
                                                                     <variable symbol="p"/>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="q"/>
                                                                      <conjunctiveformula type="and">
                                                                       <conjunctiveformula type="neg">
                                                                        <constantatomformula symbol="=">
                                                                         <variable symbol="p"/>
                                                                         <variable symbol="q"/>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <variable symbol="p"/>
                                                                        </function>
                                                                        <function symbol="f">
                                                                         <variable symbol="q"/>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </conjunctiveformula>
                                                                     </quantifiedformula>
                                                                    </quantifiedformula>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                  <rule type="contrl" symbol="c:l" param="2">
                                                                   <sequent>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="A" definition="A"/>
                                                                    </formulalist>
                                                                    <formulalist>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="p"/>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="q"/>
                                                                       <conjunctiveformula type="and">
                                                                        <conjunctiveformula type="neg">
                                                                         <constantatomformula symbol="=">
                                                                          <variable symbol="p"/>
                                                                          <variable symbol="q"/>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <variable symbol="p"/>
                                                                         </function>
                                                                         <function symbol="f">
                                                                          <variable symbol="q"/>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                      </quantifiedformula>
                                                                     </quantifiedformula>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="p"/>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="q"/>
                                                                       <conjunctiveformula type="and">
                                                                        <conjunctiveformula type="neg">
                                                                         <constantatomformula symbol="=">
                                                                          <variable symbol="p"/>
                                                                          <variable symbol="q"/>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <variable symbol="p"/>
                                                                         </function>
                                                                         <function symbol="f">
                                                                          <variable symbol="q"/>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                      </quantifiedformula>
                                                                     </quantifiedformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{112}"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{111}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{112}"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{111}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                   <rule type="perml" symbol="\pi:l" param="">
                                                                    <sequent>
                                                                     <formulalist>
                                                                      <constantatomformula symbol="A" definition="A"/>
                                                                      <constantatomformula symbol="A" definition="A"/>
                                                                     </formulalist>
                                                                     <formulalist>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="p"/>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="q"/>
                                                                        <conjunctiveformula type="and">
                                                                         <conjunctiveformula type="neg">
                                                                          <constantatomformula symbol="=">
                                                                           <variable symbol="p"/>
                                                                           <variable symbol="q"/>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <variable symbol="p"/>
                                                                          </function>
                                                                          <function symbol="f">
                                                                           <variable symbol="q"/>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                       </quantifiedformula>
                                                                      </quantifiedformula>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="p"/>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="q"/>
                                                                        <conjunctiveformula type="and">
                                                                         <conjunctiveformula type="neg">
                                                                          <constantatomformula symbol="=">
                                                                           <variable symbol="p"/>
                                                                           <variable symbol="q"/>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <variable symbol="p"/>
                                                                          </function>
                                                                          <function symbol="f">
                                                                           <variable symbol="q"/>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                       </quantifiedformula>
                                                                      </quantifiedformula>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{112}"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <variable symbol="x_{111}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{112}"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <variable symbol="x_{111}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                      </constantatomformula>
                                                                     </formulalist>
                                                                    </sequent>
                                                                    <rule type="eqr1" symbol="=:r1" param="(1)">
                                                                     <sequent>
                                                                      <formulalist>
                                                                       <constantatomformula symbol="A" definition="A"/>
                                                                       <constantatomformula symbol="A" definition="A"/>
                                                                      </formulalist>
                                                                      <formulalist>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="p"/>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="q"/>
                                                                         <conjunctiveformula type="and">
                                                                          <conjunctiveformula type="neg">
                                                                           <constantatomformula symbol="=">
                                                                            <variable symbol="p"/>
                                                                            <variable symbol="q"/>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <variable symbol="p"/>
                                                                           </function>
                                                                           <function symbol="f">
                                                                            <variable symbol="q"/>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                        </quantifiedformula>
                                                                       </quantifiedformula>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="p"/>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="q"/>
                                                                         <conjunctiveformula type="and">
                                                                          <conjunctiveformula type="neg">
                                                                           <constantatomformula symbol="=">
                                                                            <variable symbol="p"/>
                                                                            <variable symbol="q"/>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <variable symbol="p"/>
                                                                           </function>
                                                                           <function symbol="f">
                                                                            <variable symbol="q"/>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                        </quantifiedformula>
                                                                       </quantifiedformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{112}"/>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{110}"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <variable symbol="x_{111}"/>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                             <variable symbol="x_{109}"/>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{112}"/>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{110}"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <variable symbol="x_{111}"/>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                             <variable symbol="x_{109}"/>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </formulalist>
                                                                     </sequent>
                                                                     <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="A" definition="A"/>
                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="p"/>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="q"/>
                                                                          <conjunctiveformula type="and">
                                                                           <conjunctiveformula type="neg">
                                                                            <constantatomformula symbol="=">
                                                                             <variable symbol="p"/>
                                                                             <variable symbol="q"/>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <variable symbol="p"/>
                                                                            </function>
                                                                            <function symbol="f">
                                                                             <variable symbol="q"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                         </quantifiedformula>
                                                                        </quantifiedformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{112}"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{111}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <variable symbol="x_{109}"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{112}"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{111}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <variable symbol="x_{109}"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                      <rule type="weakr" symbol="w:r">
                                                                       <sequent>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="A" definition="A"/>
                                                                        </formulalist>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </constantatomformula>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="p"/>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="q"/>
                                                                           <conjunctiveformula type="and">
                                                                            <conjunctiveformula type="neg">
                                                                             <constantatomformula symbol="=">
                                                                              <variable symbol="p"/>
                                                                              <variable symbol="q"/>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <variable symbol="p"/>
                                                                             </function>
                                                                             <function symbol="f">
                                                                              <variable symbol="q"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                          </quantifiedformula>
                                                                         </quantifiedformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                       <rule type="perml" symbol="\pi:l" param="">
                                                                        <sequent>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                         </formulalist>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{112}"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <variable symbol="x_{111}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <variable symbol="x_{109}"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{112}"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <variable symbol="x_{111}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <variable symbol="x_{109}"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </formulalist>
                                                                        </sequent>
                                                                        <rule type="weakl" symbol="w:l">
                                                                         <sequent>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="A" definition="A"/>
                                                                          </formulalist>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{112}"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <variable symbol="x_{111}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <variable symbol="x_{109}"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{112}"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <variable symbol="x_{111}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <variable symbol="x_{109}"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </formulalist>
                                                                         </sequent>
                                                                         <rule type="axiom" symbol="axiom">
                                                                          <sequent>
                                                                           <formulalist/>
                                                                           <formulalist>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{112}"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <variable symbol="x_{111}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{109}"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{112}"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <variable symbol="x_{111}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{109}"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </formulalist>
                                                                          </sequent>
                                                                         </rule>
                                                                        </rule>
                                                                       </rule>
                                                                      </rule>
                                                                     </rule>
                                                                     <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="A" definition="A"/>
                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="p"/>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="q"/>
                                                                          <conjunctiveformula type="and">
                                                                           <conjunctiveformula type="neg">
                                                                            <constantatomformula symbol="=">
                                                                             <variable symbol="p"/>
                                                                             <variable symbol="q"/>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <variable symbol="p"/>
                                                                            </function>
                                                                            <function symbol="f">
                                                                             <variable symbol="q"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                         </quantifiedformula>
                                                                        </quantifiedformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{112}"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{111}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <variable symbol="x_{109}"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{112}"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{111}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <variable symbol="x_{109}"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                      <rule type="weakr" symbol="w:r">
                                                                       <sequent>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="A" definition="A"/>
                                                                        </formulalist>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                         </constantatomformula>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="p"/>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="q"/>
                                                                           <conjunctiveformula type="and">
                                                                            <conjunctiveformula type="neg">
                                                                             <constantatomformula symbol="=">
                                                                              <variable symbol="p"/>
                                                                              <variable symbol="q"/>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <variable symbol="p"/>
                                                                             </function>
                                                                             <function symbol="f">
                                                                              <variable symbol="q"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                          </quantifiedformula>
                                                                         </quantifiedformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                       <rule type="perml" symbol="\pi:l" param="">
                                                                        <sequent>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                         </formulalist>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{112}"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <variable symbol="x_{111}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <variable symbol="x_{109}"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{112}"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <variable symbol="x_{111}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <variable symbol="x_{109}"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </formulalist>
                                                                        </sequent>
                                                                        <rule type="weakl" symbol="w:l">
                                                                         <sequent>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="A" definition="A"/>
                                                                          </formulalist>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{112}"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <variable symbol="x_{111}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <variable symbol="x_{109}"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{112}"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <variable symbol="x_{111}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <variable symbol="x_{109}"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </formulalist>
                                                                         </sequent>
                                                                         <rule type="axiom" symbol="axiom">
                                                                          <sequent>
                                                                           <formulalist/>
                                                                           <formulalist>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{112}"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <variable symbol="x_{111}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{109}"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{112}"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <variable symbol="x_{111}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{109}"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </formulalist>
                                                                          </sequent>
                                                                         </rule>
                                                                        </rule>
                                                                       </rule>
                                                                      </rule>
                                                                     </rule>
                                                                    </rule>
                                                                   </rule>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                                <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                                 <sequent>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </constantatomformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </constantatomformula>
                                                                   <constantatomformula symbol="A" definition="A"/>
                                                                  </formulalist>
                                                                  <formulalist>
                                                                   <quantifiedformula type="exists">
                                                                    <variable symbol="p"/>
                                                                    <quantifiedformula type="exists">
                                                                     <variable symbol="q"/>
                                                                     <conjunctiveformula type="and">
                                                                      <conjunctiveformula type="neg">
                                                                       <constantatomformula symbol="=">
                                                                        <variable symbol="p"/>
                                                                        <variable symbol="q"/>
                                                                       </constantatomformula>
                                                                      </conjunctiveformula>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="f">
                                                                        <variable symbol="p"/>
                                                                       </function>
                                                                       <function symbol="f">
                                                                        <variable symbol="q"/>
                                                                       </function>
                                                                      </constantatomformula>
                                                                     </conjunctiveformula>
                                                                    </quantifiedformula>
                                                                   </quantifiedformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                 <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </constantatomformula>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{112}"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{111}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </constantatomformula>
                                                                    <constantatomformula symbol="A" definition="A"/>
                                                                   </formulalist>
                                                                   <formulalist>
                                                                    <quantifiedformula type="exists">
                                                                     <variable symbol="p"/>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="q"/>
                                                                      <conjunctiveformula type="and">
                                                                       <conjunctiveformula type="neg">
                                                                        <constantatomformula symbol="=">
                                                                         <variable symbol="p"/>
                                                                         <variable symbol="q"/>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <variable symbol="p"/>
                                                                        </function>
                                                                        <function symbol="f">
                                                                         <variable symbol="q"/>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </conjunctiveformula>
                                                                     </quantifiedformula>
                                                                    </quantifiedformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                  <rule type="contrr" symbol="c:r" param="2">
                                                                   <sequent>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{112}"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <variable symbol="x_{111}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="A" definition="A"/>
                                                                    </formulalist>
                                                                    <formulalist>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="p"/>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="q"/>
                                                                       <conjunctiveformula type="and">
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                                                                         <constantatomformula symbol="=">
                                                                          <variable symbol="p"/>
                                                                          <variable symbol="q"/>
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                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <variable symbol="p"/>
                                                                         </function>
                                                                         <function symbol="f">
                                                                          <variable symbol="q"/>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                      </quantifiedformula>
                                                                     </quantifiedformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                   <rule type="permr" symbol="\pi:r" param="">
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                                                                     <formulalist>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="f">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{112}"/>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{110}"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <variable symbol="x_{111}"/>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                             <variable symbol="x_{109}"/>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
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                                                                       </function>
                                                                       <constant symbol="1"/>
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                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="f">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{112}"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{111}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </constantatomformula>
                                                                      <constantatomformula symbol="A" definition="A"/>
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                                                                     <formulalist>
                                                                      <quantifiedformula type="exists">
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                                                                          <constantatomformula symbol="=">
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                                                                         <constantatomformula symbol="=">
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                                                                          <function symbol="f">
                                                                           <variable symbol="q"/>
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                                                                         </constantatomformula>
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                                                                         <constantatomformula symbol="=">
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                                                                           <variable symbol="p"/>
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                                                                          <function symbol="f">
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                                                                          </function>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                       </quantifiedformula>
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                                                                     </formulalist>
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                                                                    <rule type="contrl" symbol="c:l" param="1,1,2">
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                                                                      <formulalist>
                                                                       <constantatomformula symbol="=">
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                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{112}"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
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                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{111}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <variable symbol="x_{109}"/>
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                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
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                                                                        </function>
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                                                                       <constantatomformula symbol="=">
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                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{112}"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
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                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
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                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <variable symbol="x_{109}"/>
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                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
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                                                                       <constantatomformula symbol="A" definition="A"/>
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                                                                      <formulalist>
                                                                       <quantifiedformula type="exists">
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                                                                        <quantifiedformula type="exists">
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                                                                          <constantatomformula symbol="=">
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                                                                           <function symbol="f">
                                                                            <variable symbol="q"/>
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                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                        </quantifiedformula>
                                                                       </quantifiedformula>
                                                                       <quantifiedformula type="exists">
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                                                                        <quantifiedformula type="exists">
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                                                                           <constantatomformula symbol="=">
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                                                                          <constantatomformula symbol="=">
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                                                                           <function symbol="f">
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                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                        </quantifiedformula>
                                                                       </quantifiedformula>
                                                                      </formulalist>
                                                                     </sequent>
                                                                     <rule type="perml" symbol="\pi:l" param="(2 3)">
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                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
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                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
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                                                                                </function>
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                                                                             <constant symbol="1"/>
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                                                                           </function>
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                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                             <variable symbol="x_{109}"/>
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                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
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                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <quantifiedformula type="exists">
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                                                                         <quantifiedformula type="exists">
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                                                                           </constantatomformula>
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                                                                        <quantifiedformula type="exists">
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                                                                            <constantatomformula symbol="=">
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                                                                           <constantatomformula symbol="=">
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                                                                             <variable symbol="p"/>
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                                                                            <function symbol="f">
                                                                             <variable symbol="q"/>
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                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                         </quantifiedformula>
                                                                        </quantifiedformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                      <rule type="eql1" symbol="=:l1" param="(1 1 1)">
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                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
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                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{112}"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <function symbol="+">
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                                                                                    <constant symbol="1"/>
                                                                                    <variable symbol="x_{111}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <variable symbol="x_{109}"/>
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                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </constantatomformula>
                                                                         <constantatomformula symbol="A" definition="A"/>
                                                                         <constantatomformula symbol="=">
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                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{112}"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{111}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <variable symbol="x_{109}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </constantatomformula>
                                                                         <constantatomformula symbol="A" definition="A"/>
                                                                        </formulalist>
                                                                        <formulalist>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="p"/>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="q"/>
                                                                           <conjunctiveformula type="and">
                                                                            <conjunctiveformula type="neg">
                                                                             <constantatomformula symbol="=">
                                                                              <variable symbol="p"/>
                                                                              <variable symbol="q"/>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <variable symbol="p"/>
                                                                             </function>
                                                                             <function symbol="f">
                                                                              <variable symbol="q"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                          </quantifiedformula>
                                                                         </quantifiedformula>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="p"/>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="q"/>
                                                                           <conjunctiveformula type="and">
                                                                            <conjunctiveformula type="neg">
                                                                             <constantatomformula symbol="=">
                                                                              <variable symbol="p"/>
                                                                              <variable symbol="q"/>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <variable symbol="p"/>
                                                                             </function>
                                                                             <function symbol="f">
                                                                              <variable symbol="q"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                          </quantifiedformula>
                                                                         </quantifiedformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                       <rule type="contrr" symbol="c:r" param="2,1">
                                                                        <sequent>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                         </formulalist>
                                                                         <formulalist>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="p"/>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="q"/>
                                                                            <conjunctiveformula type="and">
                                                                             <conjunctiveformula type="neg">
                                                                              <constantatomformula symbol="=">
                                                                               <variable symbol="p"/>
                                                                               <variable symbol="q"/>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <variable symbol="p"/>
                                                                              </function>
                                                                              <function symbol="f">
                                                                               <variable symbol="q"/>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                           </quantifiedformula>
                                                                          </quantifiedformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </formulalist>
                                                                        </sequent>
                                                                        <rule type="permr" symbol="\pi:r" param="">
                                                                         <sequent>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="A" definition="A"/>
                                                                          </formulalist>
                                                                          <formulalist>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="p"/>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="q"/>
                                                                             <conjunctiveformula type="and">
                                                                              <conjunctiveformula type="neg">
                                                                               <constantatomformula symbol="=">
                                                                                <variable symbol="p"/>
                                                                                <variable symbol="q"/>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <variable symbol="p"/>
                                                                               </function>
                                                                               <function symbol="f">
                                                                                <variable symbol="q"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                            </quantifiedformula>
                                                                           </quantifiedformula>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="p"/>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="q"/>
                                                                             <conjunctiveformula type="and">
                                                                              <conjunctiveformula type="neg">
                                                                               <constantatomformula symbol="=">
                                                                                <variable symbol="p"/>
                                                                                <variable symbol="q"/>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <variable symbol="p"/>
                                                                               </function>
                                                                               <function symbol="f">
                                                                                <variable symbol="q"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                            </quantifiedformula>
                                                                           </quantifiedformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{112}"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <variable symbol="x_{111}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <variable symbol="x_{109}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{112}"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <variable symbol="x_{111}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <variable symbol="x_{109}"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </formulalist>
                                                                         </sequent>
                                                                         <rule type="contrl" symbol="c:l" param="2">
                                                                          <sequent>
                                                                           <formulalist>
                                                                            <constantatomformula symbol="A" definition="A"/>
                                                                           </formulalist>
                                                                           <formulalist>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="p"/>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="q"/>
                                                                              <conjunctiveformula type="and">
                                                                               <conjunctiveformula type="neg">
                                                                                <constantatomformula symbol="=">
                                                                                 <variable symbol="p"/>
                                                                                 <variable symbol="q"/>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <variable symbol="p"/>
                                                                                </function>
                                                                                <function symbol="f">
                                                                                 <variable symbol="q"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                             </quantifiedformula>
                                                                            </quantifiedformula>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="p"/>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="q"/>
                                                                              <conjunctiveformula type="and">
                                                                               <conjunctiveformula type="neg">
                                                                                <constantatomformula symbol="=">
                                                                                 <variable symbol="p"/>
                                                                                 <variable symbol="q"/>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <variable symbol="p"/>
                                                                                </function>
                                                                                <function symbol="f">
                                                                                 <variable symbol="q"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                             </quantifiedformula>
                                                                            </quantifiedformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{112}"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <variable symbol="x_{111}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <variable symbol="x_{109}"/>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{112}"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <variable symbol="x_{111}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <variable symbol="x_{109}"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </formulalist>
                                                                          </sequent>
                                                                          <rule type="perml" symbol="\pi:l" param="">
                                                                           <sequent>
                                                                            <formulalist>
                                                                             <constantatomformula symbol="A" definition="A"/>
                                                                             <constantatomformula symbol="A" definition="A"/>
                                                                            </formulalist>
                                                                            <formulalist>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="p"/>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="q"/>
                                                                               <conjunctiveformula type="and">
                                                                                <conjunctiveformula type="neg">
                                                                                 <constantatomformula symbol="=">
                                                                                  <variable symbol="p"/>
                                                                                  <variable symbol="q"/>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <variable symbol="p"/>
                                                                                 </function>
                                                                                 <function symbol="f">
                                                                                  <variable symbol="q"/>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                              </quantifiedformula>
                                                                             </quantifiedformula>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="p"/>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="q"/>
                                                                               <conjunctiveformula type="and">
                                                                                <conjunctiveformula type="neg">
                                                                                 <constantatomformula symbol="=">
                                                                                  <variable symbol="p"/>
                                                                                  <variable symbol="q"/>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <variable symbol="p"/>
                                                                                 </function>
                                                                                 <function symbol="f">
                                                                                  <variable symbol="q"/>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                              </quantifiedformula>
                                                                             </quantifiedformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{112}"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <variable symbol="x_{111}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{109}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{112}"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <variable symbol="x_{111}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{109}"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </formulalist>
                                                                           </sequent>
                                                                           <rule type="eqr1" symbol="=:r1" param="(1)">
                                                                            <sequent>
                                                                             <formulalist>
                                                                              <constantatomformula symbol="A" definition="A"/>
                                                                              <constantatomformula symbol="A" definition="A"/>
                                                                             </formulalist>
                                                                             <formulalist>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="p"/>
                                                                               <quantifiedformula type="exists">
                                                                                <variable symbol="q"/>
                                                                                <conjunctiveformula type="and">
                                                                                 <conjunctiveformula type="neg">
                                                                                  <constantatomformula symbol="=">
                                                                                   <variable symbol="p"/>
                                                                                   <variable symbol="q"/>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="f">
                                                                                   <variable symbol="p"/>
                                                                                  </function>
                                                                                  <function symbol="f">
                                                                                   <variable symbol="q"/>
                                                                                  </function>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                               </quantifiedformula>
                                                                              </quantifiedformula>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="p"/>
                                                                               <quantifiedformula type="exists">
                                                                                <variable symbol="q"/>
                                                                                <conjunctiveformula type="and">
                                                                                 <conjunctiveformula type="neg">
                                                                                  <constantatomformula symbol="=">
                                                                                   <variable symbol="p"/>
                                                                                   <variable symbol="q"/>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="f">
                                                                                   <variable symbol="p"/>
                                                                                  </function>
                                                                                  <function symbol="f">
                                                                                   <variable symbol="q"/>
                                                                                  </function>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                               </quantifiedformula>
                                                                              </quantifiedformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{112}"/>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{110}"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <variable symbol="x_{111}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                   <variable symbol="x_{109}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{112}"/>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{110}"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <variable symbol="x_{111}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                   <variable symbol="x_{109}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </formulalist>
                                                                            </sequent>
                                                                            <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                             <sequent>
                                                                              <formulalist>
                                                                               <constantatomformula symbol="A" definition="A"/>
                                                                              </formulalist>
                                                                              <formulalist>
                                                                               <quantifiedformula type="exists">
                                                                                <variable symbol="p"/>
                                                                                <quantifiedformula type="exists">
                                                                                 <variable symbol="q"/>
                                                                                 <conjunctiveformula type="and">
                                                                                  <conjunctiveformula type="neg">
                                                                                   <constantatomformula symbol="=">
                                                                                    <variable symbol="p"/>
                                                                                    <variable symbol="q"/>
                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="f">
                                                                                    <variable symbol="p"/>
                                                                                   </function>
                                                                                   <function symbol="f">
                                                                                    <variable symbol="q"/>
                                                                                   </function>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                </quantifiedformula>
                                                                               </quantifiedformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{112}"/>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{110}"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <variable symbol="x_{111}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                    <variable symbol="x_{109}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{112}"/>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{110}"/>
                                                                                      <function symbol="+">
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                                                                                       <function symbol="+">
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                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                    <variable symbol="x_{109}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </formulalist>
                                                                             </sequent>
                                                                             <rule type="weakr" symbol="w:r">
                                                                              <sequent>
                                                                               <formulalist>
                                                                                <constantatomformula symbol="A" definition="A"/>
                                                                               </formulalist>
                                                                               <formulalist>
                                                                                <constantatomformula symbol="=">
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                                                                                   <constant symbol="1"/>
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                                                                                    <constant symbol="1"/>
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                                                                                  </function>
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                                                                                         <constant symbol="1"/>
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                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
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                                                                                   </function>
                                                                                   <constant symbol="1"/>
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                                                                                 </function>
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                                                                                  <function symbol="+">
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                                                                                     <constant symbol="1"/>
                                                                                     <constant symbol="1"/>
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                                                                                   </function>
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                                                                                    <function symbol="+">
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                                                                                </constantatomformula>
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                                                                                   <constantatomformula symbol="=">
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                                                                                   </constantatomformula>
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                                                                                 </quantifiedformula>
                                                                                </quantifiedformula>
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                                                                              <rule type="perml" symbol="\pi:l" param="">
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                                                                                <formulalist>
                                                                                 <constantatomformula symbol="=">
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                                                                                          <variable symbol="x_{111}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="x_{109}"/>
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                                                                                    </function>
                                                                                    <constant symbol="1"/>
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                                                                                  </function>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
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                                                                                     <variable symbol="x_{112}"/>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
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                                                                                         <constant symbol="1"/>
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                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="x_{109}"/>
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                                                                                    </function>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
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                                                                                 </constantatomformula>
                                                                                </formulalist>
                                                                               </sequent>
                                                                               <rule type="weakl" symbol="w:l">
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                                                                                 <formulalist>
                                                                                  <constantatomformula symbol="A" definition="A"/>
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                                                                                 <formulalist>
                                                                                  <constantatomformula symbol="=">
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                                                                                      <constant symbol="1"/>
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                                                                                       <constant symbol="1"/>
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                                                                                     </function>
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                                                                                <rule type="axiom" symbol="axiom">
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                                                                                  <formulalist/>
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                                                                                   <constantatomformula symbol="=">
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                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
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                                                                                       <constant symbol="1"/>
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                                                                                     <function symbol="+">
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                                                                               </rule>
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                                                                            <rule type="permr" symbol="\pi:r" param="(1 2)">
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                                                                              <formulalist>
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                                                                             <rule type="weakr" symbol="w:r">
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                                                                                </sequent>
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                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{112}"/>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{110}"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <variable symbol="x_{111}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <variable symbol="x_{109}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </constantatomformula>
                                                                                  </formulalist>
                                                                                 </sequent>
                                                                                </rule>
                                                                               </rule>
                                                                              </rule>
                                                                             </rule>
                                                                            </rule>
                                                                           </rule>
                                                                          </rule>
                                                                         </rule>
                                                                        </rule>
                                                                       </rule>
                                                                       <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                                        <sequent>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{112}"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <variable symbol="x_{111}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <variable symbol="x_{109}"/>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </constantatomformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </constantatomformula>
                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                         </formulalist>
                                                                         <formulalist>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="p"/>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="q"/>
                                                                            <conjunctiveformula type="and">
                                                                             <conjunctiveformula type="neg">
                                                                              <constantatomformula symbol="=">
                                                                               <variable symbol="p"/>
                                                                               <variable symbol="q"/>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <variable symbol="p"/>
                                                                              </function>
                                                                              <function symbol="f">
                                                                               <variable symbol="q"/>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                           </quantifiedformula>
                                                                          </quantifiedformula>
                                                                         </formulalist>
                                                                        </sequent>
                                                                        <rule type="permr" symbol="\pi:r" param="">
                                                                         <sequent>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{112}"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <variable symbol="x_{111}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <variable symbol="x_{109}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </constantatomformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{112}"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <variable symbol="x_{111}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{109}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </constantatomformula>
                                                                           <constantatomformula symbol="A" definition="A"/>
                                                                          </formulalist>
                                                                          <formulalist>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="p"/>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="q"/>
                                                                             <conjunctiveformula type="and">
                                                                              <conjunctiveformula type="neg">
                                                                               <constantatomformula symbol="=">
                                                                                <variable symbol="p"/>
                                                                                <variable symbol="q"/>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <variable symbol="p"/>
                                                                               </function>
                                                                               <function symbol="f">
                                                                                <variable symbol="q"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                            </quantifiedformula>
                                                                           </quantifiedformula>
                                                                          </formulalist>
                                                                         </sequent>
                                                                         <rule type="perml" symbol="\pi:l" param="(1 2 3)">
                                                                          <sequent>
                                                                           <formulalist>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{112}"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <variable symbol="x_{111}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <variable symbol="x_{109}"/>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </constantatomformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{112}"/>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{110}"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <variable symbol="x_{111}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                   <variable symbol="x_{109}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </constantatomformula>
                                                                            <constantatomformula symbol="A" definition="A"/>
                                                                           </formulalist>
                                                                           <formulalist>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="p"/>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="q"/>
                                                                              <conjunctiveformula type="and">
                                                                               <conjunctiveformula type="neg">
                                                                                <constantatomformula symbol="=">
                                                                                 <variable symbol="p"/>
                                                                                 <variable symbol="q"/>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <variable symbol="p"/>
                                                                                </function>
                                                                                <function symbol="f">
                                                                                 <variable symbol="q"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                             </quantifiedformula>
                                                                            </quantifiedformula>
                                                                           </formulalist>
                                                                          </sequent>
                                                                          <rule type="contrr" symbol="c:r" param="2">
                                                                           <sequent>
                                                                            <formulalist>
                                                                             <constantatomformula symbol="A" definition="A"/>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{112}"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <variable symbol="x_{111}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{109}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </constantatomformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{112}"/>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{110}"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <variable symbol="x_{111}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                    <variable symbol="x_{109}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </constantatomformula>
                                                                            </formulalist>
                                                                            <formulalist>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="p"/>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="q"/>
                                                                               <conjunctiveformula type="and">
                                                                                <conjunctiveformula type="neg">
                                                                                 <constantatomformula symbol="=">
                                                                                  <variable symbol="p"/>
                                                                                  <variable symbol="q"/>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <variable symbol="p"/>
                                                                                 </function>
                                                                                 <function symbol="f">
                                                                                  <variable symbol="q"/>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                              </quantifiedformula>
                                                                             </quantifiedformula>
                                                                            </formulalist>
                                                                           </sequent>
                                                                           <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                            <sequent>
                                                                             <formulalist>
                                                                              <constantatomformula symbol="A" definition="A"/>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{112}"/>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{110}"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <variable symbol="x_{111}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                   <variable symbol="x_{109}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </constantatomformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{112}"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{110}"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <variable symbol="x_{111}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <variable symbol="x_{109}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </constantatomformula>
                                                                             </formulalist>
                                                                             <formulalist>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="p"/>
                                                                               <quantifiedformula type="exists">
                                                                                <variable symbol="q"/>
                                                                                <conjunctiveformula type="and">
                                                                                 <conjunctiveformula type="neg">
                                                                                  <constantatomformula symbol="=">
                                                                                   <variable symbol="p"/>
                                                                                   <variable symbol="q"/>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="f">
                                                                                   <variable symbol="p"/>
                                                                                  </function>
                                                                                  <function symbol="f">
                                                                                   <variable symbol="q"/>
                                                                                  </function>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                               </quantifiedformula>
                                                                              </quantifiedformula>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="p"/>
                                                                               <quantifiedformula type="exists">
                                                                                <variable symbol="q"/>
                                                                                <conjunctiveformula type="and">
                                                                                 <conjunctiveformula type="neg">
                                                                                  <constantatomformula symbol="=">
                                                                                   <variable symbol="p"/>
                                                                                   <variable symbol="q"/>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="f">
                                                                                   <variable symbol="p"/>
                                                                                  </function>
                                                                                  <function symbol="f">
                                                                                   <variable symbol="q"/>
                                                                                  </function>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                               </quantifiedformula>
                                                                              </quantifiedformula>
                                                                             </formulalist>
                                                                            </sequent>
                                                                            <rule type="weakr" symbol="w:r">
                                                                             <sequent>
                                                                              <formulalist>
                                                                               <constantatomformula symbol="A" definition="A"/>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{112}"/>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{110}"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <variable symbol="x_{111}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                    <variable symbol="x_{109}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </constantatomformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{112}"/>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{110}"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <variable symbol="x_{111}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="x_{109}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </constantatomformula>
                                                                              </formulalist>
                                                                              <formulalist>
                                                                               <quantifiedformula type="exists">
                                                                                <variable symbol="p"/>
                                                                                <quantifiedformula type="exists">
                                                                                 <variable symbol="q"/>
                                                                                 <conjunctiveformula type="and">
                                                                                  <conjunctiveformula type="neg">
                                                                                   <constantatomformula symbol="=">
                                                                                    <variable symbol="p"/>
                                                                                    <variable symbol="q"/>
                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="f">
                                                                                    <variable symbol="p"/>
                                                                                   </function>
                                                                                   <function symbol="f">
                                                                                    <variable symbol="q"/>
                                                                                   </function>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                </quantifiedformula>
                                                                               </quantifiedformula>
                                                                               <quantifiedformula type="exists">
                                                                                <variable symbol="p"/>
                                                                                <quantifiedformula type="exists">
                                                                                 <variable symbol="q"/>
                                                                                 <conjunctiveformula type="and">
                                                                                  <conjunctiveformula type="neg">
                                                                                   <constantatomformula symbol="=">
                                                                                    <variable symbol="p"/>
                                                                                    <variable symbol="q"/>
                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="f">
                                                                                    <variable symbol="p"/>
                                                                                   </function>
                                                                                   <function symbol="f">
                                                                                    <variable symbol="q"/>
                                                                                   </function>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                </quantifiedformula>
                                                                               </quantifiedformula>
                                                                              </formulalist>
                                                                             </sequent>
                                                                             <rule type="weakl" symbol="w:l">
                                                                              <sequent>
                                                                               <formulalist>
                                                                                <constantatomformula symbol="A" definition="A"/>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{112}"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{110}"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <variable symbol="x_{111}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <variable symbol="x_{109}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </constantatomformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{112}"/>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{110}"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <variable symbol="x_{111}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                       <variable symbol="x_{109}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </constantatomformula>
                                                                               </formulalist>
                                                                               <formulalist>
                                                                                <quantifiedformula type="exists">
                                                                                 <variable symbol="p"/>
                                                                                 <quantifiedformula type="exists">
                                                                                  <variable symbol="q"/>
                                                                                  <conjunctiveformula type="and">
                                                                                   <conjunctiveformula type="neg">
                                                                                    <constantatomformula symbol="=">
                                                                                     <variable symbol="p"/>
                                                                                     <variable symbol="q"/>
                                                                                    </constantatomformula>
                                                                                   </conjunctiveformula>
                                                                                   <constantatomformula symbol="=">
                                                                                    <function symbol="f">
                                                                                     <variable symbol="p"/>
                                                                                    </function>
                                                                                    <function symbol="f">
                                                                                     <variable symbol="q"/>
                                                                                    </function>
                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                 </quantifiedformula>
                                                                                </quantifiedformula>
                                                                               </formulalist>
                                                                              </sequent>
                                                                              <rule type="existsr" symbol="\ex:r">
                                                                               <sequent>
                                                                                <formulalist>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="f">
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{112}"/>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{110}"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <variable symbol="x_{111}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="x_{109}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </constantatomformula>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="f">
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{112}"/>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{110}"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <variable symbol="x_{111}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <variable symbol="x_{109}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </constantatomformula>
                                                                                </formulalist>
                                                                                <formulalist>
                                                                                 <quantifiedformula type="exists">
                                                                                  <variable symbol="p"/>
                                                                                  <quantifiedformula type="exists">
                                                                                   <variable symbol="q"/>
                                                                                   <conjunctiveformula type="and">
                                                                                    <conjunctiveformula type="neg">
                                                                                     <constantatomformula symbol="=">
                                                                                      <variable symbol="p"/>
                                                                                      <variable symbol="q"/>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
                                                                                      <variable symbol="p"/>
                                                                                     </function>
                                                                                     <function symbol="f">
                                                                                      <variable symbol="q"/>
                                                                                     </function>
                                                                                    </constantatomformula>
                                                                                   </conjunctiveformula>
                                                                                  </quantifiedformula>
                                                                                 </quantifiedformula>
                                                                                </formulalist>
                                                                               </sequent>
                                                                               <rule type="existsr" symbol="\ex:r">
                                                                                <sequent>
                                                                                 <formulalist>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="f">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{112}"/>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{110}"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <variable symbol="x_{111}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                       <variable symbol="x_{109}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </constantatomformula>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="f">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{112}"/>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{110}"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <variable symbol="x_{111}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <variable symbol="x_{109}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </constantatomformula>
                                                                                 </formulalist>
                                                                                 <formulalist>
                                                                                  <quantifiedformula type="exists">
                                                                                   <variable symbol="q"/>
                                                                                   <conjunctiveformula type="and">
                                                                                    <conjunctiveformula type="neg">
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{112}"/>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{110}"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <variable symbol="x_{111}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <variable symbol="x_{109}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="q"/>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{112}"/>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{110}"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <variable symbol="x_{111}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <variable symbol="x_{109}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <function symbol="f">
                                                                                      <variable symbol="q"/>
                                                                                     </function>
                                                                                    </constantatomformula>
                                                                                   </conjunctiveformula>
                                                                                  </quantifiedformula>
                                                                                 </formulalist>
                                                                                </sequent>
                                                                                <rule type="andr" symbol="\and:r">
                                                                                 <sequent>
                                                                                  <formulalist>
                                                                                   <constantatomformula symbol="=">
                                                                                    <function symbol="f">
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{112}"/>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{110}"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <variable symbol="x_{111}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <variable symbol="x_{109}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </constantatomformula>
                                                                                   <constantatomformula symbol="=">
                                                                                    <function symbol="f">
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                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{112}"/>
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{110}"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <variable symbol="x_{111}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                          <variable symbol="x_{109}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </constantatomformula>
                                                                                  </formulalist>
                                                                                  <formulalist>
                                                                                   <conjunctiveformula type="and">
                                                                                    <conjunctiveformula type="neg">
                                                                                     <constantatomformula symbol="=">
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                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{112}"/>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{110}"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <variable symbol="x_{111}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <variable symbol="x_{109}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{112}"/>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <variable symbol="x_{111}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{112}"/>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{110}"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <variable symbol="x_{111}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <variable symbol="x_{109}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <function symbol="f">
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                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
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                                                                                           <constant symbol="1"/>
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                                                                                         </function>
                                                                                         <function symbol="+">
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                                                                                            <constant symbol="1"/>
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                                                                                             <variable symbol="x_{110}"/>
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                                                                                              <constant symbol="1"/>
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                                                                                               <constant symbol="1"/>
                                                                                               <variable symbol="x_{111}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </constantatomformula>
                                                                                   </conjunctiveformula>
                                                                                  </formulalist>
                                                                                 </sequent>
                                                                                 <rule type="negr" symbol="\neg:r">
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                                                                                   <formulalist/>
                                                                                   <formulalist>
                                                                                    <conjunctiveformula type="neg">
                                                                                     <constantatomformula symbol="=">
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                                                                                        <constant symbol="1"/>
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                                                                                         <constant symbol="1"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
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                                                                                        <function symbol="+">
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                                                                                             <variable symbol="x_{111}"/>
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                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <variable symbol="x_{109}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
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                                                                                           <constant symbol="1"/>
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                                                                                         </function>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{112}"/>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
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                                                                                             <variable symbol="x_{110}"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
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                                                                                               <constant symbol="1"/>
                                                                                               <variable symbol="x_{111}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                   </formulalist>
                                                                                  </sequent>
                                                                                  <rule type="eql1" symbol="=:l1" param="(2)">
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                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{112}"/>
                                                                                        <function symbol="+">
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                                                                                          <constant symbol="1"/>
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                                                                                           <variable symbol="x_{110}"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
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                                                                                             <constant symbol="1"/>
                                                                                             <variable symbol="x_{111}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <variable symbol="x_{109}"/>
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                                                                                       </function>
                                                                                      </function>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
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                                                                                           <constant symbol="1"/>
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                                                                                         </function>
                                                                                         <function symbol="+">
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                                                                                          <function symbol="+">
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                                                                                             <variable symbol="x_{110}"/>
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                                                                                               <constant symbol="1"/>
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                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </formulalist>
                                                                                    <formulalist/>
                                                                                   </sequent>
                                                                                   <rule type="axiom" symbol="axiom">
                                                                                    <sequent>
                                                                                     <formulalist/>
                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{112}"/>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <variable symbol="x_{111}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{112}"/>
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{110}"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <variable symbol="x_{111}"/>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <variable symbol="x_{109}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </constantatomformula>
                                                                                     </formulalist>
                                                                                    </sequent>
                                                                                   </rule>
                                                                                   <rule type="axiom" symbol="axiom">
                                                                                    <sequent>
                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{112}"/>
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{110}"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <variable symbol="x_{111}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                          <variable symbol="x_{109}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{112}"/>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <variable symbol="x_{111}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </constantatomformula>
                                                                                     </formulalist>
                                                                                     <formulalist/>
                                                                                    </sequent>
                                                                                   </rule>
                                                                                  </rule>
                                                                                 </rule>
                                                                                 <rule type="eqr2" symbol="=:r2" param="(1)">
                                                                                  <sequent>
                                                                                   <formulalist>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{112}"/>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{110}"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <variable symbol="x_{111}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <variable symbol="x_{109}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </constantatomformula>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{112}"/>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <variable symbol="x_{111}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </constantatomformula>
                                                                                   </formulalist>
                                                                                   <formulalist>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{112}"/>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{110}"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <variable symbol="x_{111}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <variable symbol="x_{109}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <function symbol="f">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{112}"/>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <variable symbol="x_{111}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </constantatomformula>
                                                                                   </formulalist>
                                                                                  </sequent>
                                                                                  <rule type="axiom" symbol="axiom">
                                                                                   <sequent>
                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="f">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{112}"/>
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{110}"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <variable symbol="x_{111}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                          <variable symbol="x_{109}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </constantatomformula>
                                                                                    </formulalist>
                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="f">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{112}"/>
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{110}"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <variable symbol="x_{111}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                          <variable symbol="x_{109}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </constantatomformula>
                                                                                    </formulalist>
                                                                                   </sequent>
                                                                                  </rule>
                                                                                  <rule type="eqr2" symbol="=:r2" param="(2)">
                                                                                   <sequent>
                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="f">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{112}"/>
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{110}"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <variable symbol="x_{111}"/>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <variable symbol="x_{109}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </constantatomformula>
                                                                                    </formulalist>
                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="=">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="f">
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{112}"/>
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{110}"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <variable symbol="x_{111}"/>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <variable symbol="x_{109}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </formulalist>
                                                                                   </sequent>
                                                                                   <rule type="axiom" symbol="axiom">
                                                                                    <sequent>
                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="f">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{112}"/>
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{111}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </constantatomformula>
                                                                                     </formulalist>
                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="f">
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{112}"/>
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{111}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </constantatomformula>
                                                                                     </formulalist>
                                                                                    </sequent>
                                                                                   </rule>
                                                                                   <rule type="axiom" symbol="axiom">
                                                                                    <sequent>
                                                                                     <formulalist/>
                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="=">
                                                                                       <constant symbol="1"/>
                                                                                       <constant symbol="1"/>
                                                                                      </constantatomformula>
                                                                                     </formulalist>
                                                                                    </sequent>
                                                                                   </rule>
                                                                                  </rule>
                                                                                 </rule>
                                                                                </rule>
                                                                               </rule>
                                                                              </rule>
                                                                             </rule>
                                                                            </rule>
                                                                           </rule>
                                                                          </rule>
                                                                         </rule>
                                                                        </rule>
                                                                       </rule>
                                                                      </rule>
                                                                     </rule>
                                                                    </rule>
                                                                   </rule>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                              <rule type="contrr" symbol="c:r" param="2,1">
                               <sequent>
                                <formulalist>
                                 <constantatomformula symbol="=">
                                  <function symbol="f">
                                   <function symbol="+">
                                    <function symbol="+">
                                     <variable symbol="x_{112}"/>
                                     <function symbol="+">
                                      <function symbol="+">
                                       <constant symbol="1"/>
                                       <function symbol="+">
                                        <variable symbol="x_{110}"/>
                                        <function symbol="+">
                                         <constant symbol="1"/>
                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <variable symbol="x_{111}"/>
                                         </function>
                                        </function>
                                       </function>
                                      </function>
                                      <variable symbol="x_{109}"/>
                                     </function>
                                    </function>
                                    <function symbol="+">
                                     <constant symbol="1"/>
                                     <function symbol="+">
                                      <constant symbol="1"/>
                                      <function symbol="+">
                                       <constant symbol="1"/>
                                       <function symbol="+">
                                        <constant symbol="1"/>
                                        <constant symbol="1"/>
                                       </function>
                                      </function>
                                     </function>
                                    </function>
                                   </function>
                                  </function>
                                  <constant symbol="0"/>
                                 </constantatomformula>
                                 <constantatomformula symbol="A" definition="A"/>
                                </formulalist>
                                <formulalist>
                                 <quantifiedformula type="exists">
                                  <variable symbol="p"/>
                                  <quantifiedformula type="exists">
                                   <variable symbol="q"/>
                                   <conjunctiveformula type="and">
                                    <conjunctiveformula type="neg">
                                     <constantatomformula symbol="=">
                                      <variable symbol="p"/>
                                      <variable symbol="q"/>
                                     </constantatomformula>
                                    </conjunctiveformula>
                                    <constantatomformula symbol="=">
                                     <function symbol="f">
                                      <variable symbol="p"/>
                                     </function>
                                     <function symbol="f">
                                      <variable symbol="q"/>
                                     </function>
                                    </constantatomformula>
                                   </conjunctiveformula>
                                  </quantifiedformula>
                                 </quantifiedformula>
                                 <constantatomformula symbol="=">
                                  <function symbol="f">
                                   <function symbol="+">
                                    <function symbol="+">
                                     <variable symbol="x_{112}"/>
                                     <function symbol="+">
                                      <function symbol="+">
                                       <constant symbol="1"/>
                                       <function symbol="+">
                                        <variable symbol="x_{110}"/>
                                        <function symbol="+">
                                         <constant symbol="1"/>
                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <variable symbol="x_{111}"/>
                                         </function>
                                        </function>
                                       </function>
                                      </function>
                                      <variable symbol="x_{109}"/>
                                     </function>
                                    </function>
                                    <function symbol="+">
                                     <constant symbol="1"/>
                                     <function symbol="+">
                                      <constant symbol="1"/>
                                      <constant symbol="1"/>
                                     </function>
                                    </function>
                                   </function>
                                  </function>
                                  <constant symbol="0"/>
                                 </constantatomformula>
                                </formulalist>
                               </sequent>
                               <rule type="permr" symbol="\pi:r" param="(2 3)">
                                <sequent>
                                 <formulalist>
                                  <constantatomformula symbol="=">
                                   <function symbol="f">
                                    <function symbol="+">
                                     <function symbol="+">
                                      <variable symbol="x_{112}"/>
                                      <function symbol="+">
                                       <function symbol="+">
                                        <constant symbol="1"/>
                                        <function symbol="+">
                                         <variable symbol="x_{110}"/>
                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <variable symbol="x_{111}"/>
                                          </function>
                                         </function>
                                        </function>
                                       </function>
                                       <variable symbol="x_{109}"/>
                                      </function>
                                     </function>
                                     <function symbol="+">
                                      <constant symbol="1"/>
                                      <function symbol="+">
                                       <constant symbol="1"/>
                                       <function symbol="+">
                                        <constant symbol="1"/>
                                        <function symbol="+">
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                                               </function>
                                              </function>
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                                            <function symbol="+">
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                                               <function symbol="+">
                                                <variable symbol="x_{110}"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <variable symbol="x_{111}"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
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                                            </function>
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                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <constant symbol="1"/>
                                             </function>
                                            </function>
                                           </function>
                                          </function>
                                          <constant symbol="0"/>
                                         </constantatomformula>
                                        </formulalist>
                                       </sequent>
                                       <rule type="perml" symbol="\pi:l" param="">
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                                         <formulalist>
                                          <constantatomformula symbol="A" definition="A"/>
                                          <constantatomformula symbol="A" definition="A"/>
                                         </formulalist>
                                         <formulalist>
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                                              <constantatomformula symbol="=">
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                                               <variable symbol="q"/>
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                                             <constantatomformula symbol="=">
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                                             <constantatomformula symbol="=">
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                                             <function symbol="+">
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                                              <function symbol="+">
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                                                 <variable symbol="x_{110}"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{111}"/>
                                                  </function>
                                                 </function>
                                                </function>
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                                                <function symbol="+">
                                                 <variable symbol="x_{110}"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{111}"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                               <variable symbol="x_{109}"/>
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                                             </function>
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                                               <constant symbol="1"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
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                                          <formulalist>
                                           <constantatomformula symbol="A" definition="A"/>
                                           <constantatomformula symbol="A" definition="A"/>
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                                          <formulalist>
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                                              <constantatomformula symbol="=">
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                                           <constantatomformula symbol="=">
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                                              <function symbol="+">
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                                               <function symbol="+">
                                                <function symbol="+">
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                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <variable symbol="x_{109}"/>
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                                              </function>
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                                           <constantatomformula symbol="=">
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                                               <function symbol="+">
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                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <variable symbol="x_{109}"/>
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                                              <function symbol="+">
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                                               <function symbol="+">
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                                                <constant symbol="1"/>
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                                              </function>
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                                               <constant symbol="1"/>
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                                                  <constantatomformula symbol="=">
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                                                  <formulalist>
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                                                   <constantatomformula symbol="=">
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                                                 <rule type="weakl" symbol="w:l">
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                                                    <constantatomformula symbol="=">
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                                                   </formulalist>
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                                                  <rule type="axiom" symbol="axiom">
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                                                    <formulalist/>
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                                                 </quantifiedformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="weakr" symbol="w:r">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="perml" symbol="\pi:l" param="">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="weakl" symbol="w:l">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="axiom" symbol="axiom">
                                                   <sequent>
                                                    <formulalist/>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                         <rule type="contrr" symbol="c:r" param="2,1,1">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="A" definition="A"/>
                                           </formulalist>
                                           <formulalist>
                                            <quantifiedformula type="exists">
                                             <variable symbol="p"/>
                                             <quantifiedformula type="exists">
                                              <variable symbol="q"/>
                                              <conjunctiveformula type="and">
                                               <conjunctiveformula type="neg">
                                                <constantatomformula symbol="=">
                                                 <variable symbol="p"/>
                                                 <variable symbol="q"/>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <variable symbol="p"/>
                                                </function>
                                                <function symbol="f">
                                                 <variable symbol="q"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                             </quantifiedformula>
                                            </quantifiedformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="f">
                                              <function symbol="+">
                                               <function symbol="+">
                                                <variable symbol="x_{112}"/>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <variable symbol="x_{109}"/>
                                                </function>
                                               </function>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                             <constant symbol="0"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="f">
                                              <function symbol="+">
                                               <function symbol="+">
                                                <variable symbol="x_{112}"/>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <variable symbol="x_{109}"/>
                                                </function>
                                               </function>
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <constant symbol="1"/>
                                                </function>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                             </function>
                                             <constant symbol="0"/>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="permr" symbol="\pi:r" param="">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="A" definition="A"/>
                                            </formulalist>
                                            <formulalist>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <variable symbol="x_{112}"/>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <variable symbol="x_{109}"/>
                                                 </function>
                                                </function>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                              <constant symbol="0"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <variable symbol="x_{112}"/>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <variable symbol="x_{109}"/>
                                                 </function>
                                                </function>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                              </function>
                                              <constant symbol="0"/>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="contrl" symbol="c:l" param="2">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="A" definition="A"/>
                                             </formulalist>
                                             <formulalist>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <variable symbol="x_{112}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{109}"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="1"/>
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                                               </function>
                                               <constant symbol="0"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <variable symbol="x_{112}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{109}"/>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                               </function>
                                               <constant symbol="0"/>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="perml" symbol="\pi:l" param="">
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                                              <formulalist>
                                               <constantatomformula symbol="A" definition="A"/>
                                               <constantatomformula symbol="A" definition="A"/>
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                                              <formulalist>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
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                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
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                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
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                                                 <conjunctiveformula type="and">
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                                                   <constantatomformula symbol="=">
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                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{112}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                                <constant symbol="0"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{112}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                </function>
                                                <constant symbol="0"/>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="eqr1" symbol="=:r1" param="(1 1)">
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                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
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                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
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                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
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                                                    <constantatomformula symbol="=">
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                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{112}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
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                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{112}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="contrr" symbol="c:r" param="2,1">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
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                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{112}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="permr" symbol="\pi:r" param="">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="contrl" symbol="c:l" param="2">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="perml" symbol="\pi:l" param="">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="eqr1" symbol="=:r1" param="(1)">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="weakr" symbol="w:r">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </constantatomformula>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="perml" symbol="\pi:l" param="">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="weakl" symbol="w:l">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="axiom" symbol="axiom">
                                                        <sequent>
                                                         <formulalist/>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                   <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="weakr" symbol="w:r">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </constantatomformula>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="perml" symbol="\pi:l" param="">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="weakl" symbol="w:l">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="axiom" symbol="axiom">
                                                        <sequent>
                                                         <formulalist/>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                              <rule type="permr" symbol="\pi:r" param="(2 3)">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="contrr" symbol="c:r" param="2,1,1">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="permr" symbol="\pi:r" param="(2 3)">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="contrl" symbol="c:l" param="2">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="perml" symbol="\pi:l" param="">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="cut" symbol="cut">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
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                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
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                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="permr" symbol="\pi:r" param="(1 3 2)">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
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                                                           <constantatomformula symbol="=">
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                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
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                                                           <function symbol="f">
                                                            <variable symbol="q"/>
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                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
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                                                        </function>
                                                        <constant symbol="0"/>
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                                                       <constantatomformula symbol="=">
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                                                            <function symbol="+">
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                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
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                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
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                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="perml" symbol="\pi:l" param="">
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                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
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                                                         </function>
                                                         <constant symbol="0"/>
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                                                        <constantatomformula symbol="=">
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                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
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                                                        <quantifiedformula type="exists">
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                                                         <quantifiedformula type="exists">
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                                                           <constantatomformula symbol="=">
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                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="contrr" symbol="c:r" param="1,1,2">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
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                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
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                                                          </function>
                                                          <constant symbol="0"/>
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                                                         <constantatomformula symbol="=">
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                                                              <function symbol="+">
                                                               <function symbol="+">
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                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
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                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </constantatomformula>
                                                         <quantifiedformula type="exists">
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                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
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                                                             <constantatomformula symbol="=">
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                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
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                                                             <function symbol="f">
                                                              <variable symbol="q"/>
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                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="weakr" symbol="w:r">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
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                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
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                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
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                                                           </function>
                                                           <constant symbol="0"/>
                                                          </constantatomformula>
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                                                               <function symbol="+">
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                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
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                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
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                                                          <quantifiedformula type="exists">
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                                                           <quantifiedformula type="exists">
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                                                            <conjunctiveformula type="and">
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                                                              <constantatomformula symbol="=">
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                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
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                                                              <function symbol="f">
                                                               <variable symbol="q"/>
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                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
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                                                             <conjunctiveformula type="neg">
                                                              <constantatomformula symbol="=">
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                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
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                                                              <function symbol="f">
                                                               <variable symbol="q"/>
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                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="weakr" symbol="w:r">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
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                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="0"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </constantatomformula>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="p"/>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="q"/>
                                                             <conjunctiveformula type="and">
                                                              <conjunctiveformula type="neg">
                                                               <constantatomformula symbol="=">
                                                                <variable symbol="p"/>
                                                                <variable symbol="q"/>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="defl" symbol="d:l" param="();[()];()">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{112}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <variable symbol="x_{109}"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="0"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{112}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <variable symbol="x_{109}"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="foralll" symbol="\all:l">
                                                           <sequent>
                                                            <formulalist>
                                                             <quantifiedformula type="all">
                                                              <variable symbol="x"/>
                                                              <conjunctiveformula type="or">
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <variable symbol="x"/>
                                                                </function>
                                                                <constant symbol="0"/>
                                                               </constantatomformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <variable symbol="x"/>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </quantifiedformula>
                                                            </formulalist>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="orl" symbol="\or:l">
                                                            <sequent>
                                                             <formulalist>
                                                              <conjunctiveformula type="or">
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="0"/>
                                                               </constantatomformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </formulalist>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <constant symbol="0"/>
                                                              </constantatomformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="axiom" symbol="axiom">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="0"/>
                                                               </constantatomformula>
                                                              </formulalist>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="0"/>
                                                               </constantatomformula>
                                                              </formulalist>
                                                             </sequent>
                                                            </rule>
                                                            <rule type="eql2" symbol="=:l2" param="(1 1)">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </constantatomformula>
                                                              </formulalist>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </constantatomformula>
                                                              </formulalist>
                                                             </sequent>
                                                             <rule type="axiom" symbol="axiom">
                                                              <sequent>
                                                               <formulalist/>
                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </constantatomformula>
                                                               </formulalist>
                                                              </sequent>
                                                             </rule>
                                                             <rule type="axiom" symbol="axiom">
                                                              <sequent>
                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </constantatomformula>
                                                               </formulalist>
                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </constantatomformula>
                                                               </formulalist>
                                                              </sequent>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                    <rule type="contrr" symbol="c:r" param="2,1">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="0"/>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="permr" symbol="\pi:r" param="(2 3)">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
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                                                         <quantifiedformula type="exists">
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                                                           <constantatomformula symbol="=">
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                                                           <function symbol="+">
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                                                            <function symbol="+">
                                                             <function symbol="+">
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                                                         </function>
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                                                                 <function symbol="+">
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                                                             <function symbol="+">
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                                                              <constant symbol="1"/>
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                                                            </function>
                                                           </function>
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                                                                </function>
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                                                              </function>
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                                                              <function symbol="+">
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                                                               <constant symbol="1"/>
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                                                                 <variable symbol="q"/>
                                                                 <conjunctiveformula type="and">
                                                                  <conjunctiveformula type="neg">
                                                                   <constantatomformula symbol="=">
                                                                    <variable symbol="p"/>
                                                                    <variable symbol="q"/>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <variable symbol="p"/>
                                                                   </function>
                                                                   <function symbol="f">
                                                                    <variable symbol="q"/>
                                                                   </function>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                </quantifiedformula>
                                                               </quantifiedformula>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="p"/>
                                                                <quantifiedformula type="exists">
                                                                 <variable symbol="q"/>
                                                                 <conjunctiveformula type="and">
                                                                  <conjunctiveformula type="neg">
                                                                   <constantatomformula symbol="=">
                                                                    <variable symbol="p"/>
                                                                    <variable symbol="q"/>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <variable symbol="p"/>
                                                                   </function>
                                                                   <function symbol="f">
                                                                    <variable symbol="q"/>
                                                                   </function>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                </quantifiedformula>
                                                               </quantifiedformula>
                                                              </formulalist>
                                                             </sequent>
                                                             <rule type="weakr" symbol="w:r">
                                                              <sequent>
                                                               <formulalist>
                                                                <constantatomformula symbol="A" definition="A"/>
                                                               </formulalist>
                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="0"/>
                                                                </constantatomformula>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </constantatomformula>
                                                                <quantifiedformula type="exists">
                                                                 <variable symbol="p"/>
                                                                 <quantifiedformula type="exists">
                                                                  <variable symbol="q"/>
                                                                  <conjunctiveformula type="and">
                                                                   <conjunctiveformula type="neg">
                                                                    <constantatomformula symbol="=">
                                                                     <variable symbol="p"/>
                                                                     <variable symbol="q"/>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <variable symbol="p"/>
                                                                    </function>
                                                                    <function symbol="f">
                                                                     <variable symbol="q"/>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                 </quantifiedformula>
                                                                </quantifiedformula>
                                                               </formulalist>
                                                              </sequent>
                                                              <rule type="defl" symbol="d:l" param="();[()];()">
                                                               <sequent>
                                                                <formulalist>
                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                </formulalist>
                                                                <formulalist>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="0"/>
                                                                 </constantatomformula>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </constantatomformula>
                                                                </formulalist>
                                                               </sequent>
                                                               <rule type="foralll" symbol="\all:l">
                                                                <sequent>
                                                                 <formulalist>
                                                                  <quantifiedformula type="all">
                                                                   <variable symbol="x"/>
                                                                   <conjunctiveformula type="or">
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <variable symbol="x"/>
                                                                     </function>
                                                                     <constant symbol="0"/>
                                                                    </constantatomformula>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <variable symbol="x"/>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                  </quantifiedformula>
                                                                 </formulalist>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{112}"/>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="x_{109}"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="0"/>
                                                                  </constantatomformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{112}"/>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="x_{109}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </constantatomformula>
                                                                 </formulalist>
                                                                </sequent>
                                                                <rule type="orl" symbol="\or:l">
                                                                 <sequent>
                                                                  <formulalist>
                                                                   <conjunctiveformula type="or">
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="0"/>
                                                                    </constantatomformula>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                  </formulalist>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="0"/>
                                                                   </constantatomformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                 <rule type="axiom" symbol="axiom">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="0"/>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="0"/>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                 </rule>
                                                                 <rule type="eql2" symbol="=:l2" param="(1 1)">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                  <rule type="axiom" symbol="axiom">
                                                                   <sequent>
                                                                    <formulalist/>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                  </rule>
                                                                  <rule type="axiom" symbol="axiom">
                                                                   <sequent>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                         <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{112}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <variable symbol="x_{109}"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="p"/>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="q"/>
                                                              <conjunctiveformula type="and">
                                                               <conjunctiveformula type="neg">
                                                                <constantatomformula symbol="=">
                                                                 <variable symbol="p"/>
                                                                 <variable symbol="q"/>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <variable symbol="p"/>
                                                                </function>
                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </quantifiedformula>
                                                            </quantifiedformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="contrr" symbol="c:r" param="2">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
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                                                                  <constant symbol="1"/>
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                                                              </function>
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                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </constantatomformula>
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                                                            <formulalist>
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                                                                   <constant symbol="1"/>
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                                                               <constant symbol="1"/>
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                                                                    <constant symbol="1"/>
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                                                                  </function>
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                                                                <constant symbol="1"/>
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                                                                         <variable symbol="p"/>
                                                                        </function>
                                                                        <function symbol="f">
                                                                         <variable symbol="q"/>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </conjunctiveformula>
                                                                     </quantifiedformula>
                                                                    </quantifiedformula>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                  <rule type="perml" symbol="\pi:l" param="">
                                                                   <sequent>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="A" definition="A"/>
                                                                     <constantatomformula symbol="A" definition="A"/>
                                                                    </formulalist>
                                                                    <formulalist>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="p"/>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="q"/>
                                                                       <conjunctiveformula type="and">
                                                                        <conjunctiveformula type="neg">
                                                                         <constantatomformula symbol="=">
                                                                          <variable symbol="p"/>
                                                                          <variable symbol="q"/>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <variable symbol="p"/>
                                                                         </function>
                                                                         <function symbol="f">
                                                                          <variable symbol="q"/>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                      </quantifiedformula>
                                                                     </quantifiedformula>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="p"/>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="q"/>
                                                                       <conjunctiveformula type="and">
                                                                        <conjunctiveformula type="neg">
                                                                         <constantatomformula symbol="=">
                                                                          <variable symbol="p"/>
                                                                          <variable symbol="q"/>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <variable symbol="p"/>
                                                                         </function>
                                                                         <function symbol="f">
                                                                          <variable symbol="q"/>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                      </quantifiedformula>
                                                                     </quantifiedformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                   <rule type="eqr1" symbol="=:r1" param="(1)">
                                                                    <sequent>
                                                                     <formulalist>
                                                                      <constantatomformula symbol="A" definition="A"/>
                                                                      <constantatomformula symbol="A" definition="A"/>
                                                                     </formulalist>
                                                                     <formulalist>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="p"/>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="q"/>
                                                                        <conjunctiveformula type="and">
                                                                         <conjunctiveformula type="neg">
                                                                          <constantatomformula symbol="=">
                                                                           <variable symbol="p"/>
                                                                           <variable symbol="q"/>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <variable symbol="p"/>
                                                                          </function>
                                                                          <function symbol="f">
                                                                           <variable symbol="q"/>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                       </quantifiedformula>
                                                                      </quantifiedformula>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="p"/>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="q"/>
                                                                        <conjunctiveformula type="and">
                                                                         <conjunctiveformula type="neg">
                                                                          <constantatomformula symbol="=">
                                                                           <variable symbol="p"/>
                                                                           <variable symbol="q"/>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <variable symbol="p"/>
                                                                          </function>
                                                                          <function symbol="f">
                                                                           <variable symbol="q"/>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                       </quantifiedformula>
                                                                      </quantifiedformula>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{112}"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{111}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                      </constantatomformula>
                                                                     </formulalist>
                                                                    </sequent>
                                                                    <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                     <sequent>
                                                                      <formulalist>
                                                                       <constantatomformula symbol="A" definition="A"/>
                                                                      </formulalist>
                                                                      <formulalist>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="p"/>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="q"/>
                                                                         <conjunctiveformula type="and">
                                                                          <conjunctiveformula type="neg">
                                                                           <constantatomformula symbol="=">
                                                                            <variable symbol="p"/>
                                                                            <variable symbol="q"/>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <variable symbol="p"/>
                                                                           </function>
                                                                           <function symbol="f">
                                                                            <variable symbol="q"/>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                        </quantifiedformula>
                                                                       </quantifiedformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{112}"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{111}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{112}"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <variable symbol="x_{111}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </formulalist>
                                                                     </sequent>
                                                                     <rule type="weakr" symbol="w:r">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="A" definition="A"/>
                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{112}"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <variable symbol="x_{111}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{112}"/>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{110}"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <variable symbol="x_{111}"/>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                             <variable symbol="x_{109}"/>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </constantatomformula>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="p"/>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="q"/>
                                                                          <conjunctiveformula type="and">
                                                                           <conjunctiveformula type="neg">
                                                                            <constantatomformula symbol="=">
                                                                             <variable symbol="p"/>
                                                                             <variable symbol="q"/>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <variable symbol="p"/>
                                                                            </function>
                                                                            <function symbol="f">
                                                                             <variable symbol="q"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                         </quantifiedformula>
                                                                        </quantifiedformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                      <rule type="perml" symbol="\pi:l" param="">
                                                                       <sequent>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="A" definition="A"/>
                                                                        </formulalist>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{112}"/>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{110}"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <variable symbol="x_{111}"/>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                             <variable symbol="x_{109}"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{112}"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{111}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <variable symbol="x_{109}"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                       <rule type="weakl" symbol="w:l">
                                                                        <sequent>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                         </formulalist>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{112}"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{111}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <variable symbol="x_{109}"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </formulalist>
                                                                        </sequent>
                                                                        <rule type="axiom" symbol="axiom">
                                                                         <sequent>
                                                                          <formulalist/>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{112}"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <variable symbol="x_{111}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <variable symbol="x_{109}"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </formulalist>
                                                                         </sequent>
                                                                        </rule>
                                                                       </rule>
                                                                      </rule>
                                                                     </rule>
                                                                    </rule>
                                                                    <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                     <sequent>
                                                                      <formulalist>
                                                                       <constantatomformula symbol="A" definition="A"/>
                                                                      </formulalist>
                                                                      <formulalist>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="p"/>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="q"/>
                                                                         <conjunctiveformula type="and">
                                                                          <conjunctiveformula type="neg">
                                                                           <constantatomformula symbol="=">
                                                                            <variable symbol="p"/>
                                                                            <variable symbol="q"/>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <variable symbol="p"/>
                                                                           </function>
                                                                           <function symbol="f">
                                                                            <variable symbol="q"/>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                        </quantifiedformula>
                                                                       </quantifiedformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{112}"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{111}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{112}"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{111}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </formulalist>
                                                                     </sequent>
                                                                     <rule type="weakr" symbol="w:r">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="A" definition="A"/>
                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{112}"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <variable symbol="x_{111}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{112}"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <variable symbol="x_{111}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                        </constantatomformula>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="p"/>
                                                                         <quantifiedformula type="exists">
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                                                                          <conjunctiveformula type="and">
                                                                           <conjunctiveformula type="neg">
                                                                            <constantatomformula symbol="=">
                                                                             <variable symbol="p"/>
                                                                             <variable symbol="q"/>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <variable symbol="p"/>
                                                                            </function>
                                                                            <function symbol="f">
                                                                             <variable symbol="q"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                         </quantifiedformula>
                                                                        </quantifiedformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                      <rule type="perml" symbol="\pi:l" param="">
                                                                       <sequent>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="A" definition="A"/>
                                                                        </formulalist>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{112}"/>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{110}"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <variable symbol="x_{111}"/>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                             <variable symbol="x_{109}"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{112}"/>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{110}"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <variable symbol="x_{111}"/>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                             <variable symbol="x_{109}"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                       <rule type="weakl" symbol="w:l">
                                                                        <sequent>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                         </formulalist>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{112}"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{111}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <variable symbol="x_{109}"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{112}"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{111}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <variable symbol="x_{109}"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </formulalist>
                                                                        </sequent>
                                                                        <rule type="axiom" symbol="axiom">
                                                                         <sequent>
                                                                          <formulalist/>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </formulalist>
                                                                         </sequent>
                                                                        </rule>
                                                                       </rule>
                                                                      </rule>
                                                                     </rule>
                                                                    </rule>
                                                                   </rule>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                               </rule>
                                                               <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                                <sequent>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </constantatomformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{112}"/>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="x_{109}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </constantatomformula>
                                                                  <constantatomformula symbol="A" definition="A"/>
                                                                 </formulalist>
                                                                 <formulalist>
                                                                  <quantifiedformula type="exists">
                                                                   <variable symbol="p"/>
                                                                   <quantifiedformula type="exists">
                                                                    <variable symbol="q"/>
                                                                    <conjunctiveformula type="and">
                                                                     <conjunctiveformula type="neg">
                                                                      <constantatomformula symbol="=">
                                                                       <variable symbol="p"/>
                                                                       <variable symbol="q"/>
                                                                      </constantatomformula>
                                                                     </conjunctiveformula>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <variable symbol="p"/>
                                                                      </function>
                                                                      <function symbol="f">
                                                                       <variable symbol="q"/>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </conjunctiveformula>
                                                                   </quantifiedformula>
                                                                  </quantifiedformula>
                                                                 </formulalist>
                                                                </sequent>
                                                                <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                                 <sequent>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </constantatomformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </constantatomformula>
                                                                   <constantatomformula symbol="A" definition="A"/>
                                                                  </formulalist>
                                                                  <formulalist>
                                                                   <quantifiedformula type="exists">
                                                                    <variable symbol="p"/>
                                                                    <quantifiedformula type="exists">
                                                                     <variable symbol="q"/>
                                                                     <conjunctiveformula type="and">
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                                                                       <constantatomformula symbol="=">
                                                                        <variable symbol="p"/>
                                                                        <variable symbol="q"/>
                                                                       </constantatomformula>
                                                                      </conjunctiveformula>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="f">
                                                                        <variable symbol="p"/>
                                                                       </function>
                                                                       <function symbol="f">
                                                                        <variable symbol="q"/>
                                                                       </function>
                                                                      </constantatomformula>
                                                                     </conjunctiveformula>
                                                                    </quantifiedformula>
                                                                   </quantifiedformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                 <rule type="contrr" symbol="c:r" param="2">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{112}"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{111}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </constantatomformula>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </constantatomformula>
                                                                    <constantatomformula symbol="A" definition="A"/>
                                                                   </formulalist>
                                                                   <formulalist>
                                                                    <quantifiedformula type="exists">
                                                                     <variable symbol="p"/>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="q"/>
                                                                      <conjunctiveformula type="and">
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                                                                        <constantatomformula symbol="=">
                                                                         <variable symbol="p"/>
                                                                         <variable symbol="q"/>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <variable symbol="p"/>
                                                                        </function>
                                                                        <function symbol="f">
                                                                         <variable symbol="q"/>
                                                                        </function>
                                                                       </constantatomformula>
                                                                      </conjunctiveformula>
                                                                     </quantifiedformula>
                                                                    </quantifiedformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                  <rule type="permr" symbol="\pi:r" param="">
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                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
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                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{112}"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <variable symbol="x_{111}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="=">
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                                                                       <function symbol="+">
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                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </constantatomformula>
                                                                     <constantatomformula symbol="A" definition="A"/>
                                                                    </formulalist>
                                                                    <formulalist>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="p"/>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="q"/>
                                                                       <conjunctiveformula type="and">
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                                                                         <constantatomformula symbol="=">
                                                                          <variable symbol="p"/>
                                                                          <variable symbol="q"/>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <variable symbol="p"/>
                                                                         </function>
                                                                         <function symbol="f">
                                                                          <variable symbol="q"/>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                      </quantifiedformula>
                                                                     </quantifiedformula>
                                                                     <quantifiedformula type="exists">
                                                                      <variable symbol="p"/>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="q"/>
                                                                       <conjunctiveformula type="and">
                                                                        <conjunctiveformula type="neg">
                                                                         <constantatomformula symbol="=">
                                                                          <variable symbol="p"/>
                                                                          <variable symbol="q"/>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <variable symbol="p"/>
                                                                         </function>
                                                                         <function symbol="f">
                                                                          <variable symbol="q"/>
                                                                         </function>
                                                                        </constantatomformula>
                                                                       </conjunctiveformula>
                                                                      </quantifiedformula>
                                                                     </quantifiedformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                   <rule type="contrl" symbol="c:l" param="1,1,2">
                                                                    <sequent>
                                                                     <formulalist>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="f">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{112}"/>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{110}"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <variable symbol="x_{111}"/>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                             <variable symbol="x_{109}"/>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </constantatomformula>
                                                                      <constantatomformula symbol="=">
                                                                       <function symbol="f">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{112}"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{111}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </constantatomformula>
                                                                      <constantatomformula symbol="A" definition="A"/>
                                                                     </formulalist>
                                                                     <formulalist>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="p"/>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="q"/>
                                                                        <conjunctiveformula type="and">
                                                                         <conjunctiveformula type="neg">
                                                                          <constantatomformula symbol="=">
                                                                           <variable symbol="p"/>
                                                                           <variable symbol="q"/>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <variable symbol="p"/>
                                                                          </function>
                                                                          <function symbol="f">
                                                                           <variable symbol="q"/>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                       </quantifiedformula>
                                                                      </quantifiedformula>
                                                                      <quantifiedformula type="exists">
                                                                       <variable symbol="p"/>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="q"/>
                                                                        <conjunctiveformula type="and">
                                                                         <conjunctiveformula type="neg">
                                                                          <constantatomformula symbol="=">
                                                                           <variable symbol="p"/>
                                                                           <variable symbol="q"/>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <variable symbol="p"/>
                                                                          </function>
                                                                          <function symbol="f">
                                                                           <variable symbol="q"/>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </conjunctiveformula>
                                                                       </quantifiedformula>
                                                                      </quantifiedformula>
                                                                     </formulalist>
                                                                    </sequent>
                                                                    <rule type="perml" symbol="\pi:l" param="(2 3)">
                                                                     <sequent>
                                                                      <formulalist>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{112}"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{111}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <variable symbol="x_{109}"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </constantatomformula>
                                                                       <constantatomformula symbol="=">
                                                                        <function symbol="f">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{112}"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <variable symbol="x_{111}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </constantatomformula>
                                                                       <constantatomformula symbol="A" definition="A"/>
                                                                       <constantatomformula symbol="A" definition="A"/>
                                                                      </formulalist>
                                                                      <formulalist>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="p"/>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="q"/>
                                                                         <conjunctiveformula type="and">
                                                                          <conjunctiveformula type="neg">
                                                                           <constantatomformula symbol="=">
                                                                            <variable symbol="p"/>
                                                                            <variable symbol="q"/>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <variable symbol="p"/>
                                                                           </function>
                                                                           <function symbol="f">
                                                                            <variable symbol="q"/>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                        </quantifiedformula>
                                                                       </quantifiedformula>
                                                                       <quantifiedformula type="exists">
                                                                        <variable symbol="p"/>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="q"/>
                                                                         <conjunctiveformula type="and">
                                                                          <conjunctiveformula type="neg">
                                                                           <constantatomformula symbol="=">
                                                                            <variable symbol="p"/>
                                                                            <variable symbol="q"/>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <variable symbol="p"/>
                                                                           </function>
                                                                           <function symbol="f">
                                                                            <variable symbol="q"/>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </conjunctiveformula>
                                                                        </quantifiedformula>
                                                                       </quantifiedformula>
                                                                      </formulalist>
                                                                     </sequent>
                                                                     <rule type="eql1" symbol="=:l1" param="(1 1)">
                                                                      <sequent>
                                                                       <formulalist>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </constantatomformula>
                                                                        <constantatomformula symbol="A" definition="A"/>
                                                                        <constantatomformula symbol="=">
                                                                         <function symbol="f">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{112}"/>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{110}"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <variable symbol="x_{111}"/>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                             <variable symbol="x_{109}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </constantatomformula>
                                                                        <constantatomformula symbol="A" definition="A"/>
                                                                       </formulalist>
                                                                       <formulalist>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="p"/>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="q"/>
                                                                          <conjunctiveformula type="and">
                                                                           <conjunctiveformula type="neg">
                                                                            <constantatomformula symbol="=">
                                                                             <variable symbol="p"/>
                                                                             <variable symbol="q"/>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <variable symbol="p"/>
                                                                            </function>
                                                                            <function symbol="f">
                                                                             <variable symbol="q"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                         </quantifiedformula>
                                                                        </quantifiedformula>
                                                                        <quantifiedformula type="exists">
                                                                         <variable symbol="p"/>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="q"/>
                                                                          <conjunctiveformula type="and">
                                                                           <conjunctiveformula type="neg">
                                                                            <constantatomformula symbol="=">
                                                                             <variable symbol="p"/>
                                                                             <variable symbol="q"/>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <variable symbol="p"/>
                                                                            </function>
                                                                            <function symbol="f">
                                                                             <variable symbol="q"/>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </conjunctiveformula>
                                                                         </quantifiedformula>
                                                                        </quantifiedformula>
                                                                       </formulalist>
                                                                      </sequent>
                                                                      <rule type="contrr" symbol="c:r" param="2,1">
                                                                       <sequent>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="A" definition="A"/>
                                                                        </formulalist>
                                                                        <formulalist>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="p"/>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="q"/>
                                                                           <conjunctiveformula type="and">
                                                                            <conjunctiveformula type="neg">
                                                                             <constantatomformula symbol="=">
                                                                              <variable symbol="p"/>
                                                                              <variable symbol="q"/>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <variable symbol="p"/>
                                                                             </function>
                                                                             <function symbol="f">
                                                                              <variable symbol="q"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                          </quantifiedformula>
                                                                         </quantifiedformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </function>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                         </constantatomformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                       <rule type="permr" symbol="\pi:r" param="">
                                                                        <sequent>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                         </formulalist>
                                                                         <formulalist>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="p"/>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="q"/>
                                                                            <conjunctiveformula type="and">
                                                                             <conjunctiveformula type="neg">
                                                                              <constantatomformula symbol="=">
                                                                               <variable symbol="p"/>
                                                                               <variable symbol="q"/>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <variable symbol="p"/>
                                                                              </function>
                                                                              <function symbol="f">
                                                                               <variable symbol="q"/>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                           </quantifiedformula>
                                                                          </quantifiedformula>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="p"/>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="q"/>
                                                                            <conjunctiveformula type="and">
                                                                             <conjunctiveformula type="neg">
                                                                              <constantatomformula symbol="=">
                                                                               <variable symbol="p"/>
                                                                               <variable symbol="q"/>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <variable symbol="p"/>
                                                                              </function>
                                                                              <function symbol="f">
                                                                               <variable symbol="q"/>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                           </quantifiedformula>
                                                                          </quantifiedformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{112}"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <variable symbol="x_{111}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <variable symbol="x_{109}"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{112}"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <variable symbol="x_{111}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <variable symbol="x_{109}"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                          </constantatomformula>
                                                                         </formulalist>
                                                                        </sequent>
                                                                        <rule type="contrl" symbol="c:l" param="2">
                                                                         <sequent>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="A" definition="A"/>
                                                                          </formulalist>
                                                                          <formulalist>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="p"/>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="q"/>
                                                                             <conjunctiveformula type="and">
                                                                              <conjunctiveformula type="neg">
                                                                               <constantatomformula symbol="=">
                                                                                <variable symbol="p"/>
                                                                                <variable symbol="q"/>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <variable symbol="p"/>
                                                                               </function>
                                                                               <function symbol="f">
                                                                                <variable symbol="q"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                            </quantifiedformula>
                                                                           </quantifiedformula>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="p"/>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="q"/>
                                                                             <conjunctiveformula type="and">
                                                                              <conjunctiveformula type="neg">
                                                                               <constantatomformula symbol="=">
                                                                                <variable symbol="p"/>
                                                                                <variable symbol="q"/>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <variable symbol="p"/>
                                                                               </function>
                                                                               <function symbol="f">
                                                                                <variable symbol="q"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                            </quantifiedformula>
                                                                           </quantifiedformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{112}"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <variable symbol="x_{111}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <variable symbol="x_{109}"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{112}"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <variable symbol="x_{111}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <variable symbol="x_{109}"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                           </constantatomformula>
                                                                          </formulalist>
                                                                         </sequent>
                                                                         <rule type="perml" symbol="\pi:l" param="">
                                                                          <sequent>
                                                                           <formulalist>
                                                                            <constantatomformula symbol="A" definition="A"/>
                                                                            <constantatomformula symbol="A" definition="A"/>
                                                                           </formulalist>
                                                                           <formulalist>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="p"/>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="q"/>
                                                                              <conjunctiveformula type="and">
                                                                               <conjunctiveformula type="neg">
                                                                                <constantatomformula symbol="=">
                                                                                 <variable symbol="p"/>
                                                                                 <variable symbol="q"/>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <variable symbol="p"/>
                                                                                </function>
                                                                                <function symbol="f">
                                                                                 <variable symbol="q"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                             </quantifiedformula>
                                                                            </quantifiedformula>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="p"/>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="q"/>
                                                                              <conjunctiveformula type="and">
                                                                               <conjunctiveformula type="neg">
                                                                                <constantatomformula symbol="=">
                                                                                 <variable symbol="p"/>
                                                                                 <variable symbol="q"/>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <variable symbol="p"/>
                                                                                </function>
                                                                                <function symbol="f">
                                                                                 <variable symbol="q"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                             </quantifiedformula>
                                                                            </quantifiedformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{112}"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <variable symbol="x_{111}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{109}"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{112}"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <variable symbol="x_{111}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{109}"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </formulalist>
                                                                          </sequent>
                                                                          <rule type="eqr1" symbol="=:r1" param="(1)">
                                                                           <sequent>
                                                                            <formulalist>
                                                                             <constantatomformula symbol="A" definition="A"/>
                                                                             <constantatomformula symbol="A" definition="A"/>
                                                                            </formulalist>
                                                                            <formulalist>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="p"/>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="q"/>
                                                                               <conjunctiveformula type="and">
                                                                                <conjunctiveformula type="neg">
                                                                                 <constantatomformula symbol="=">
                                                                                  <variable symbol="p"/>
                                                                                  <variable symbol="q"/>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <variable symbol="p"/>
                                                                                 </function>
                                                                                 <function symbol="f">
                                                                                  <variable symbol="q"/>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                              </quantifiedformula>
                                                                             </quantifiedformula>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="p"/>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="q"/>
                                                                               <conjunctiveformula type="and">
                                                                                <conjunctiveformula type="neg">
                                                                                 <constantatomformula symbol="=">
                                                                                  <variable symbol="p"/>
                                                                                  <variable symbol="q"/>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <variable symbol="p"/>
                                                                                 </function>
                                                                                 <function symbol="f">
                                                                                  <variable symbol="q"/>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                              </quantifiedformula>
                                                                             </quantifiedformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{112}"/>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{110}"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <variable symbol="x_{111}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                   <variable symbol="x_{109}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{112}"/>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{110}"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <variable symbol="x_{111}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                   <variable symbol="x_{109}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </formulalist>
                                                                           </sequent>
                                                                           <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                            <sequent>
                                                                             <formulalist>
                                                                              <constantatomformula symbol="A" definition="A"/>
                                                                             </formulalist>
                                                                             <formulalist>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="p"/>
                                                                               <quantifiedformula type="exists">
                                                                                <variable symbol="q"/>
                                                                                <conjunctiveformula type="and">
                                                                                 <conjunctiveformula type="neg">
                                                                                  <constantatomformula symbol="=">
                                                                                   <variable symbol="p"/>
                                                                                   <variable symbol="q"/>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="f">
                                                                                   <variable symbol="p"/>
                                                                                  </function>
                                                                                  <function symbol="f">
                                                                                   <variable symbol="q"/>
                                                                                  </function>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                               </quantifiedformula>
                                                                              </quantifiedformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{112}"/>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{110}"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <variable symbol="x_{111}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                    <variable symbol="x_{109}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{112}"/>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{110}"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <variable symbol="x_{111}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                    <variable symbol="x_{109}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </formulalist>
                                                                            </sequent>
                                                                            <rule type="weakr" symbol="w:r">
                                                                             <sequent>
                                                                              <formulalist>
                                                                               <constantatomformula symbol="A" definition="A"/>
                                                                              </formulalist>
                                                                              <formulalist>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{112}"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{110}"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <variable symbol="x_{111}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <variable symbol="x_{109}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{112}"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{110}"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <variable symbol="x_{111}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <variable symbol="x_{109}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                               <quantifiedformula type="exists">
                                                                                <variable symbol="p"/>
                                                                                <quantifiedformula type="exists">
                                                                                 <variable symbol="q"/>
                                                                                 <conjunctiveformula type="and">
                                                                                  <conjunctiveformula type="neg">
                                                                                   <constantatomformula symbol="=">
                                                                                    <variable symbol="p"/>
                                                                                    <variable symbol="q"/>
                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="f">
                                                                                    <variable symbol="p"/>
                                                                                   </function>
                                                                                   <function symbol="f">
                                                                                    <variable symbol="q"/>
                                                                                   </function>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                </quantifiedformula>
                                                                               </quantifiedformula>
                                                                              </formulalist>
                                                                             </sequent>
                                                                             <rule type="perml" symbol="\pi:l" param="">
                                                                              <sequent>
                                                                               <formulalist>
                                                                                <constantatomformula symbol="A" definition="A"/>
                                                                               </formulalist>
                                                                               <formulalist>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{112}"/>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{110}"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <variable symbol="x_{111}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="x_{109}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{112}"/>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{110}"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <variable symbol="x_{111}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="x_{109}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </formulalist>
                                                                              </sequent>
                                                                              <rule type="weakl" symbol="w:l">
                                                                               <sequent>
                                                                                <formulalist>
                                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                                </formulalist>
                                                                                <formulalist>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{112}"/>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{110}"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <variable symbol="x_{111}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                       <variable symbol="x_{109}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{112}"/>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{110}"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <variable symbol="x_{111}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                       <variable symbol="x_{109}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </constantatomformula>
                                                                                </formulalist>
                                                                               </sequent>
                                                                               <rule type="axiom" symbol="axiom">
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                                                                                 <formulalist/>
                                                                                 <formulalist>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{112}"/>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{110}"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <variable symbol="x_{111}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <variable symbol="x_{109}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{112}"/>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{110}"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <variable symbol="x_{111}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <variable symbol="x_{109}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </constantatomformula>
                                                                                 </formulalist>
                                                                                </sequent>
                                                                               </rule>
                                                                              </rule>
                                                                             </rule>
                                                                            </rule>
                                                                           </rule>
                                                                           <rule type="permr" symbol="\pi:r" param="(1 2)">
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                                                                             <formulalist>
                                                                              <constantatomformula symbol="A" definition="A"/>
                                                                             </formulalist>
                                                                             <formulalist>
                                                                              <quantifiedformula type="exists">
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                                                                               <quantifiedformula type="exists">
                                                                                <variable symbol="q"/>
                                                                                <conjunctiveformula type="and">
                                                                                 <conjunctiveformula type="neg">
                                                                                  <constantatomformula symbol="=">
                                                                                   <variable symbol="p"/>
                                                                                   <variable symbol="q"/>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="f">
                                                                                   <variable symbol="p"/>
                                                                                  </function>
                                                                                  <function symbol="f">
                                                                                   <variable symbol="q"/>
                                                                                  </function>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                               </quantifiedformula>
                                                                              </quantifiedformula>
                                                                              <constantatomformula symbol="=">
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                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{112}"/>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{110}"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <variable symbol="x_{111}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                    <variable symbol="x_{109}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{112}"/>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{110}"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <variable symbol="x_{111}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                    <variable symbol="x_{109}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </formulalist>
                                                                            </sequent>
                                                                            <rule type="weakr" symbol="w:r">
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                                                                              <formulalist>
                                                                               <constantatomformula symbol="A" definition="A"/>
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                                                                              <formulalist>
                                                                               <constantatomformula symbol="=">
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                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{112}"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{110}"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
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                                                                                         <constant symbol="1"/>
                                                                                         <variable symbol="x_{111}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <variable symbol="x_{109}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <function symbol="+">
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                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{112}"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
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                                                                                       <variable symbol="x_{110}"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <variable symbol="x_{111}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <variable symbol="x_{109}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                               </constantatomformula>
                                                                               <quantifiedformula type="exists">
                                                                                <variable symbol="p"/>
                                                                                <quantifiedformula type="exists">
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                                                                                 <conjunctiveformula type="and">
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                                                                                   <constantatomformula symbol="=">
                                                                                    <variable symbol="p"/>
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                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="f">
                                                                                    <variable symbol="p"/>
                                                                                   </function>
                                                                                   <function symbol="f">
                                                                                    <variable symbol="q"/>
                                                                                   </function>
                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                </quantifiedformula>
                                                                               </quantifiedformula>
                                                                              </formulalist>
                                                                             </sequent>
                                                                             <rule type="perml" symbol="\pi:l" param="">
                                                                              <sequent>
                                                                               <formulalist>
                                                                                <constantatomformula symbol="A" definition="A"/>
                                                                               </formulalist>
                                                                               <formulalist>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{112}"/>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{110}"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <variable symbol="x_{111}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="x_{109}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{112}"/>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{110}"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <variable symbol="x_{111}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="x_{109}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </formulalist>
                                                                              </sequent>
                                                                              <rule type="weakl" symbol="w:l">
                                                                               <sequent>
                                                                                <formulalist>
                                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                                </formulalist>
                                                                                <formulalist>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{112}"/>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{110}"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <variable symbol="x_{111}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                       <variable symbol="x_{109}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{112}"/>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{110}"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <variable symbol="x_{111}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                       <variable symbol="x_{109}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </constantatomformula>
                                                                                </formulalist>
                                                                               </sequent>
                                                                               <rule type="axiom" symbol="axiom">
                                                                                <sequent>
                                                                                 <formulalist/>
                                                                                 <formulalist>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{112}"/>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{110}"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <variable symbol="x_{111}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <variable symbol="x_{109}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{112}"/>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{110}"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <variable symbol="x_{111}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <variable symbol="x_{109}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </constantatomformula>
                                                                                 </formulalist>
                                                                                </sequent>
                                                                               </rule>
                                                                              </rule>
                                                                             </rule>
                                                                            </rule>
                                                                           </rule>
                                                                          </rule>
                                                                         </rule>
                                                                        </rule>
                                                                       </rule>
                                                                      </rule>
                                                                      <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                                       <sequent>
                                                                        <formulalist>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{112}"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <variable symbol="x_{111}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <variable symbol="x_{109}"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </constantatomformula>
                                                                         <constantatomformula symbol="=">
                                                                          <function symbol="f">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{112}"/>
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{110}"/>
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <variable symbol="x_{111}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <variable symbol="x_{109}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </constantatomformula>
                                                                         <constantatomformula symbol="A" definition="A"/>
                                                                        </formulalist>
                                                                        <formulalist>
                                                                         <quantifiedformula type="exists">
                                                                          <variable symbol="p"/>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="q"/>
                                                                           <conjunctiveformula type="and">
                                                                            <conjunctiveformula type="neg">
                                                                             <constantatomformula symbol="=">
                                                                              <variable symbol="p"/>
                                                                              <variable symbol="q"/>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <variable symbol="p"/>
                                                                             </function>
                                                                             <function symbol="f">
                                                                              <variable symbol="q"/>
                                                                             </function>
                                                                            </constantatomformula>
                                                                           </conjunctiveformula>
                                                                          </quantifiedformula>
                                                                         </quantifiedformula>
                                                                        </formulalist>
                                                                       </sequent>
                                                                       <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                                        <sequent>
                                                                         <formulalist>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{112}"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{110}"/>
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <variable symbol="x_{111}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <variable symbol="x_{109}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </constantatomformula>
                                                                          <constantatomformula symbol="=">
                                                                           <function symbol="f">
                                                                            <function symbol="+">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{112}"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <variable symbol="x_{111}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <variable symbol="x_{109}"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </function>
                                                                           </function>
                                                                           <constant symbol="1"/>
                                                                          </constantatomformula>
                                                                          <constantatomformula symbol="A" definition="A"/>
                                                                         </formulalist>
                                                                         <formulalist>
                                                                          <quantifiedformula type="exists">
                                                                           <variable symbol="p"/>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="q"/>
                                                                            <conjunctiveformula type="and">
                                                                             <conjunctiveformula type="neg">
                                                                              <constantatomformula symbol="=">
                                                                               <variable symbol="p"/>
                                                                               <variable symbol="q"/>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <variable symbol="p"/>
                                                                              </function>
                                                                              <function symbol="f">
                                                                               <variable symbol="q"/>
                                                                              </function>
                                                                             </constantatomformula>
                                                                            </conjunctiveformula>
                                                                           </quantifiedformula>
                                                                          </quantifiedformula>
                                                                         </formulalist>
                                                                        </sequent>
                                                                        <rule type="contrr" symbol="c:r" param="2">
                                                                         <sequent>
                                                                          <formulalist>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <function symbol="+">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{112}"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <variable symbol="x_{111}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{109}"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </constantatomformula>
                                                                           <constantatomformula symbol="=">
                                                                            <function symbol="f">
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <variable symbol="x_{112}"/>
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{110}"/>
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <variable symbol="x_{111}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                                <variable symbol="x_{109}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <constant symbol="1"/>
                                                                           </constantatomformula>
                                                                           <constantatomformula symbol="A" definition="A"/>
                                                                          </formulalist>
                                                                          <formulalist>
                                                                           <quantifiedformula type="exists">
                                                                            <variable symbol="p"/>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="q"/>
                                                                             <conjunctiveformula type="and">
                                                                              <conjunctiveformula type="neg">
                                                                               <constantatomformula symbol="=">
                                                                                <variable symbol="p"/>
                                                                                <variable symbol="q"/>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <variable symbol="p"/>
                                                                               </function>
                                                                               <function symbol="f">
                                                                                <variable symbol="q"/>
                                                                               </function>
                                                                              </constantatomformula>
                                                                             </conjunctiveformula>
                                                                            </quantifiedformula>
                                                                           </quantifiedformula>
                                                                          </formulalist>
                                                                         </sequent>
                                                                         <rule type="permr" symbol="\pi:r" param="">
                                                                          <sequent>
                                                                           <formulalist>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <function symbol="+">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <variable symbol="x_{112}"/>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{110}"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <variable symbol="x_{111}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                   <variable symbol="x_{109}"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </constantatomformula>
                                                                            <constantatomformula symbol="=">
                                                                             <function symbol="f">
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <variable symbol="x_{112}"/>
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{110}"/>
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <variable symbol="x_{111}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <variable symbol="x_{109}"/>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                             <constant symbol="1"/>
                                                                            </constantatomformula>
                                                                            <constantatomformula symbol="A" definition="A"/>
                                                                           </formulalist>
                                                                           <formulalist>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="p"/>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="q"/>
                                                                              <conjunctiveformula type="and">
                                                                               <conjunctiveformula type="neg">
                                                                                <constantatomformula symbol="=">
                                                                                 <variable symbol="p"/>
                                                                                 <variable symbol="q"/>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <variable symbol="p"/>
                                                                                </function>
                                                                                <function symbol="f">
                                                                                 <variable symbol="q"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                             </quantifiedformula>
                                                                            </quantifiedformula>
                                                                            <quantifiedformula type="exists">
                                                                             <variable symbol="p"/>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="q"/>
                                                                              <conjunctiveformula type="and">
                                                                               <conjunctiveformula type="neg">
                                                                                <constantatomformula symbol="=">
                                                                                 <variable symbol="p"/>
                                                                                 <variable symbol="q"/>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
                                                                                 <variable symbol="p"/>
                                                                                </function>
                                                                                <function symbol="f">
                                                                                 <variable symbol="q"/>
                                                                                </function>
                                                                               </constantatomformula>
                                                                              </conjunctiveformula>
                                                                             </quantifiedformula>
                                                                            </quantifiedformula>
                                                                           </formulalist>
                                                                          </sequent>
                                                                          <rule type="contrl" symbol="c:l" param="1,1,2">
                                                                           <sequent>
                                                                            <formulalist>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <function symbol="+">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <variable symbol="x_{112}"/>
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{110}"/>
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <variable symbol="x_{111}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                    <variable symbol="x_{109}"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </function>
                                                                                </function>
                                                                                <constant symbol="1"/>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </constantatomformula>
                                                                             <constantatomformula symbol="=">
                                                                              <function symbol="f">
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <function symbol="+">
                                                                                 <variable symbol="x_{112}"/>
                                                                                 <function symbol="+">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{110}"/>
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <variable symbol="x_{111}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <variable symbol="x_{109}"/>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                              </function>
                                                                              <constant symbol="1"/>
                                                                             </constantatomformula>
                                                                             <constantatomformula symbol="A" definition="A"/>
                                                                            </formulalist>
                                                                            <formulalist>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="p"/>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="q"/>
                                                                               <conjunctiveformula type="and">
                                                                                <conjunctiveformula type="neg">
                                                                                 <constantatomformula symbol="=">
                                                                                  <variable symbol="p"/>
                                                                                  <variable symbol="q"/>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <variable symbol="p"/>
                                                                                 </function>
                                                                                 <function symbol="f">
                                                                                  <variable symbol="q"/>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                              </quantifiedformula>
                                                                             </quantifiedformula>
                                                                             <quantifiedformula type="exists">
                                                                              <variable symbol="p"/>
                                                                              <quantifiedformula type="exists">
                                                                               <variable symbol="q"/>
                                                                               <conjunctiveformula type="and">
                                                                                <conjunctiveformula type="neg">
                                                                                 <constantatomformula symbol="=">
                                                                                  <variable symbol="p"/>
                                                                                  <variable symbol="q"/>
                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <variable symbol="p"/>
                                                                                 </function>
                                                                                 <function symbol="f">
                                                                                  <variable symbol="q"/>
                                                                                 </function>
                                                                                </constantatomformula>
                                                                               </conjunctiveformula>
                                                                              </quantifiedformula>
                                                                             </quantifiedformula>
                                                                            </formulalist>
                                                                           </sequent>
                                                                           <rule type="perml" symbol="\pi:l" param="(2 3)">
                                                                            <sequent>
                                                                             <formulalist>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <function symbol="+">
                                                                                 <function symbol="+">
                                                                                  <constant symbol="1"/>
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{112}"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{110}"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <variable symbol="x_{111}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <variable symbol="x_{109}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
                                                                              </constantatomformula>
                                                                              <constantatomformula symbol="=">
                                                                               <function symbol="f">
                                                                                <function symbol="+">
                                                                                 <constant symbol="1"/>
                                                                                 <function symbol="+">
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                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
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                                                                                    <function symbol="+">
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                                                                                     <function symbol="+">
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                                                                                      <function symbol="+">
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                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
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                                                                                  </function>
                                                                                 </function>
                                                                                </function>
                                                                               </function>
                                                                               <constant symbol="1"/>
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                                                                              <constantatomformula symbol="A" definition="A"/>
                                                                              <constantatomformula symbol="A" definition="A"/>
                                                                             </formulalist>
                                                                             <formulalist>
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                                                                               <quantifiedformula type="exists">
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                                                                                 </constantatomformula>
                                                                                </conjunctiveformula>
                                                                               </quantifiedformula>
                                                                              </quantifiedformula>
                                                                             </formulalist>
                                                                            </sequent>
                                                                            <rule type="eql1" symbol="=:l1" param="(1 1 1)">
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                                                                              <formulalist>
                                                                               <constantatomformula symbol="=">
                                                                                <function symbol="f">
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                                                                                  <function symbol="+">
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                                                                                    <function symbol="+">
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                                                                                      <function symbol="+">
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                                                                                       <function symbol="+">
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                                                                                         <function symbol="+">
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                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="x_{109}"/>
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                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
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                                                                                </function>
                                                                                <constant symbol="1"/>
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                                                                               <constantatomformula symbol="A" definition="A"/>
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                                                                                <function symbol="f">
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                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
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                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
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                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
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                                                                                  </function>
                                                                                 </function>
                                                                                </function>
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                                                                               <constantatomformula symbol="A" definition="A"/>
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                                                                              <formulalist>
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                                                                                  </constantatomformula>
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                                                                                  </constantatomformula>
                                                                                 </conjunctiveformula>
                                                                                </quantifiedformula>
                                                                               </quantifiedformula>
                                                                              </formulalist>
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                                                                                     <variable symbol="q"/>
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                                                                                   </constantatomformula>
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                                                                                    <variable symbol="x_{112}"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
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                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{110}"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <variable symbol="x_{111}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <variable symbol="x_{109}"/>
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                                                                                   </function>
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                                                                                 </function>
                                                                                </constantatomformula>
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                                                                              <rule type="permr" symbol="\pi:r" param="">
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                                                                                   </function>
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                                                                                </sequent>
                                                                                <rule type="perml" symbol="\pi:l" param="">
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                                                                                    </function>
                                                                                   </constantatomformula>
                                                                                  </formulalist>
                                                                                 </sequent>
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                                                                                   <formulalist>
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                                                                                      </function>
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                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
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                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </constantatomformula>
                                                                                   </formulalist>
                                                                                  </sequent>
                                                                                  <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                                                   <sequent>
                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="A" definition="A"/>
                                                                                    </formulalist>
                                                                                    <formulalist>
                                                                                     <quantifiedformula type="exists">
                                                                                      <variable symbol="p"/>
                                                                                      <quantifiedformula type="exists">
                                                                                       <variable symbol="q"/>
                                                                                       <conjunctiveformula type="and">
                                                                                        <conjunctiveformula type="neg">
                                                                                         <constantatomformula symbol="=">
                                                                                          <variable symbol="p"/>
                                                                                          <variable symbol="q"/>
                                                                                         </constantatomformula>
                                                                                        </conjunctiveformula>
                                                                                        <constantatomformula symbol="=">
                                                                                         <function symbol="f">
                                                                                          <variable symbol="p"/>
                                                                                         </function>
                                                                                         <function symbol="f">
                                                                                          <variable symbol="q"/>
                                                                                         </function>
                                                                                        </constantatomformula>
                                                                                       </conjunctiveformula>
                                                                                      </quantifiedformula>
                                                                                     </quantifiedformula>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{112}"/>
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{110}"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <variable symbol="x_{111}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                          <variable symbol="x_{109}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{112}"/>
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{110}"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <variable symbol="x_{111}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                          <variable symbol="x_{109}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </formulalist>
                                                                                   </sequent>
                                                                                   <rule type="weakr" symbol="w:r">
                                                                                    <sequent>
                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="A" definition="A"/>
                                                                                     </formulalist>
                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{112}"/>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <variable symbol="x_{111}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{112}"/>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <variable symbol="x_{111}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </constantatomformula>
                                                                                      <quantifiedformula type="exists">
                                                                                       <variable symbol="p"/>
                                                                                       <quantifiedformula type="exists">
                                                                                        <variable symbol="q"/>
                                                                                        <conjunctiveformula type="and">
                                                                                         <conjunctiveformula type="neg">
                                                                                          <constantatomformula symbol="=">
                                                                                           <variable symbol="p"/>
                                                                                           <variable symbol="q"/>
                                                                                          </constantatomformula>
                                                                                         </conjunctiveformula>
                                                                                         <constantatomformula symbol="=">
                                                                                          <function symbol="f">
                                                                                           <variable symbol="p"/>
                                                                                          </function>
                                                                                          <function symbol="f">
                                                                                           <variable symbol="q"/>
                                                                                          </function>
                                                                                         </constantatomformula>
                                                                                        </conjunctiveformula>
                                                                                       </quantifiedformula>
                                                                                      </quantifiedformula>
                                                                                     </formulalist>
                                                                                    </sequent>
                                                                                    <rule type="perml" symbol="\pi:l" param="">
                                                                                     <sequent>
                                                                                      <formulalist>
                                                                                       <constantatomformula symbol="A" definition="A"/>
                                                                                      </formulalist>
                                                                                      <formulalist>
                                                                                       <constantatomformula symbol="=">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{112}"/>
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{110}"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <variable symbol="x_{111}"/>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <variable symbol="x_{109}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{112}"/>
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{110}"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <variable symbol="x_{111}"/>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <variable symbol="x_{109}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </constantatomformula>
                                                                                      </formulalist>
                                                                                     </sequent>
                                                                                     <rule type="weakl" symbol="w:l">
                                                                                      <sequent>
                                                                                       <formulalist>
                                                                                        <constantatomformula symbol="A" definition="A"/>
                                                                                       </formulalist>
                                                                                       <formulalist>
                                                                                        <constantatomformula symbol="=">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{112}"/>
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{111}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{112}"/>
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{111}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </constantatomformula>
                                                                                       </formulalist>
                                                                                      </sequent>
                                                                                      <rule type="axiom" symbol="axiom">
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                                                                                        <formulalist/>
                                                                                        <formulalist>
                                                                                         <constantatomformula symbol="=">
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                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{112}"/>
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{110}"/>
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <variable symbol="x_{111}"/>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                              <variable symbol="x_{109}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{112}"/>
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{110}"/>
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <variable symbol="x_{111}"/>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                              <variable symbol="x_{109}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </constantatomformula>
                                                                                        </formulalist>
                                                                                       </sequent>
                                                                                      </rule>
                                                                                     </rule>
                                                                                    </rule>
                                                                                   </rule>
                                                                                  </rule>
                                                                                  <rule type="permr" symbol="\pi:r" param="(1 2)">
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                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="A" definition="A"/>
                                                                                    </formulalist>
                                                                                    <formulalist>
                                                                                     <quantifiedformula type="exists">
                                                                                      <variable symbol="p"/>
                                                                                      <quantifiedformula type="exists">
                                                                                       <variable symbol="q"/>
                                                                                       <conjunctiveformula type="and">
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                                                                                         <constantatomformula symbol="=">
                                                                                          <variable symbol="p"/>
                                                                                          <variable symbol="q"/>
                                                                                         </constantatomformula>
                                                                                        </conjunctiveformula>
                                                                                        <constantatomformula symbol="=">
                                                                                         <function symbol="f">
                                                                                          <variable symbol="p"/>
                                                                                         </function>
                                                                                         <function symbol="f">
                                                                                          <variable symbol="q"/>
                                                                                         </function>
                                                                                        </constantatomformula>
                                                                                       </conjunctiveformula>
                                                                                      </quantifiedformula>
                                                                                     </quantifiedformula>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{112}"/>
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{110}"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <variable symbol="x_{111}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                          <variable symbol="x_{109}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                      <function symbol="+">
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                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{112}"/>
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{110}"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <variable symbol="x_{111}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                          <variable symbol="x_{109}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </formulalist>
                                                                                   </sequent>
                                                                                   <rule type="weakr" symbol="w:r">
                                                                                    <sequent>
                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="A" definition="A"/>
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                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="=">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{112}"/>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <variable symbol="x_{111}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{112}"/>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{110}"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <variable symbol="x_{111}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </constantatomformula>
                                                                                      <quantifiedformula type="exists">
                                                                                       <variable symbol="p"/>
                                                                                       <quantifiedformula type="exists">
                                                                                        <variable symbol="q"/>
                                                                                        <conjunctiveformula type="and">
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                                                                                          <constantatomformula symbol="=">
                                                                                           <variable symbol="p"/>
                                                                                           <variable symbol="q"/>
                                                                                          </constantatomformula>
                                                                                         </conjunctiveformula>
                                                                                         <constantatomformula symbol="=">
                                                                                          <function symbol="f">
                                                                                           <variable symbol="p"/>
                                                                                          </function>
                                                                                          <function symbol="f">
                                                                                           <variable symbol="q"/>
                                                                                          </function>
                                                                                         </constantatomformula>
                                                                                        </conjunctiveformula>
                                                                                       </quantifiedformula>
                                                                                      </quantifiedformula>
                                                                                     </formulalist>
                                                                                    </sequent>
                                                                                    <rule type="perml" symbol="\pi:l" param="">
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                                                                                      <formulalist>
                                                                                       <constantatomformula symbol="A" definition="A"/>
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                                                                                      <formulalist>
                                                                                       <constantatomformula symbol="=">
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                                                                                          <function symbol="+">
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                                                                                           <function symbol="+">
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                                                                                             <constant symbol="1"/>
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                                                                                              <variable symbol="x_{110}"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <variable symbol="x_{111}"/>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <variable symbol="x_{109}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <function symbol="+">
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                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{112}"/>
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{110}"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <variable symbol="x_{111}"/>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <variable symbol="x_{109}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </constantatomformula>
                                                                                      </formulalist>
                                                                                     </sequent>
                                                                                     <rule type="weakl" symbol="w:l">
                                                                                      <sequent>
                                                                                       <formulalist>
                                                                                        <constantatomformula symbol="A" definition="A"/>
                                                                                       </formulalist>
                                                                                       <formulalist>
                                                                                        <constantatomformula symbol="=">
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                                                                                          <constant symbol="1"/>
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                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{112}"/>
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{111}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{112}"/>
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{111}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </constantatomformula>
                                                                                       </formulalist>
                                                                                      </sequent>
                                                                                      <rule type="axiom" symbol="axiom">
                                                                                       <sequent>
                                                                                        <formulalist/>
                                                                                        <formulalist>
                                                                                         <constantatomformula symbol="=">
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                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{112}"/>
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{110}"/>
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <variable symbol="x_{111}"/>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                              <variable symbol="x_{109}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{112}"/>
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{110}"/>
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <variable symbol="x_{111}"/>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                              <variable symbol="x_{109}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </constantatomformula>
                                                                                        </formulalist>
                                                                                       </sequent>
                                                                                      </rule>
                                                                                     </rule>
                                                                                    </rule>
                                                                                   </rule>
                                                                                  </rule>
                                                                                 </rule>
                                                                                </rule>
                                                                               </rule>
                                                                              </rule>
                                                                             </rule>
                                                                             <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                                              <sequent>
                                                                               <formulalist>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <function symbol="+">
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{112}"/>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{110}"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <variable symbol="x_{111}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                       <variable symbol="x_{109}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </constantatomformula>
                                                                                <constantatomformula symbol="=">
                                                                                 <function symbol="f">
                                                                                  <function symbol="+">
                                                                                   <constant symbol="1"/>
                                                                                   <function symbol="+">
                                                                                    <variable symbol="x_{112}"/>
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
                                                                                       <variable symbol="x_{110}"/>
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <variable symbol="x_{111}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <variable symbol="x_{109}"/>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                 </function>
                                                                                 <constant symbol="1"/>
                                                                                </constantatomformula>
                                                                                <constantatomformula symbol="A" definition="A"/>
                                                                               </formulalist>
                                                                               <formulalist>
                                                                                <quantifiedformula type="exists">
                                                                                 <variable symbol="p"/>
                                                                                 <quantifiedformula type="exists">
                                                                                  <variable symbol="q"/>
                                                                                  <conjunctiveformula type="and">
                                                                                   <conjunctiveformula type="neg">
                                                                                    <constantatomformula symbol="=">
                                                                                     <variable symbol="p"/>
                                                                                     <variable symbol="q"/>
                                                                                    </constantatomformula>
                                                                                   </conjunctiveformula>
                                                                                   <constantatomformula symbol="=">
                                                                                    <function symbol="f">
                                                                                     <variable symbol="p"/>
                                                                                    </function>
                                                                                    <function symbol="f">
                                                                                     <variable symbol="q"/>
                                                                                    </function>
                                                                                   </constantatomformula>
                                                                                  </conjunctiveformula>
                                                                                 </quantifiedformula>
                                                                                </quantifiedformula>
                                                                               </formulalist>
                                                                              </sequent>
                                                                              <rule type="permr" symbol="\pi:r" param="">
                                                                               <sequent>
                                                                                <formulalist>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="f">
                                                                                   <function symbol="+">
                                                                                    <constant symbol="1"/>
                                                                                    <function symbol="+">
                                                                                     <variable symbol="x_{112}"/>
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{110}"/>
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <variable symbol="x_{111}"/>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <variable symbol="x_{109}"/>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </constantatomformula>
                                                                                 <constantatomformula symbol="=">
                                                                                  <function symbol="f">
                                                                                   <function symbol="+">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
                                                                                      <function symbol="+">
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                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
                                                                                         <constant symbol="1"/>
                                                                                         <function symbol="+">
                                                                                          <variable symbol="x_{110}"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <variable symbol="x_{111}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <variable symbol="x_{109}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </function>
                                                                                  </function>
                                                                                  <constant symbol="1"/>
                                                                                 </constantatomformula>
                                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                                </formulalist>
                                                                                <formulalist>
                                                                                 <quantifiedformula type="exists">
                                                                                  <variable symbol="p"/>
                                                                                  <quantifiedformula type="exists">
                                                                                   <variable symbol="q"/>
                                                                                   <conjunctiveformula type="and">
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                                                                                     <constantatomformula symbol="=">
                                                                                      <variable symbol="p"/>
                                                                                      <variable symbol="q"/>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                    <constantatomformula symbol="=">
                                                                                     <function symbol="f">
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                                                                                     <function symbol="f">
                                                                                      <variable symbol="q"/>
                                                                                     </function>
                                                                                    </constantatomformula>
                                                                                   </conjunctiveformula>
                                                                                  </quantifiedformula>
                                                                                 </quantifiedformula>
                                                                                </formulalist>
                                                                               </sequent>
                                                                               <rule type="perml" symbol="\pi:l" param="(1 2 3)">
                                                                                <sequent>
                                                                                 <formulalist>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="f">
                                                                                    <function symbol="+">
                                                                                     <constant symbol="1"/>
                                                                                     <function symbol="+">
                                                                                      <variable symbol="x_{112}"/>
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
                                                                                        <constant symbol="1"/>
                                                                                        <function symbol="+">
                                                                                         <variable symbol="x_{110}"/>
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <variable symbol="x_{111}"/>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                       </function>
                                                                                       <variable symbol="x_{109}"/>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </constantatomformula>
                                                                                  <constantatomformula symbol="=">
                                                                                   <function symbol="f">
                                                                                    <function symbol="+">
                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <constant symbol="1"/>
                                                                                       <function symbol="+">
                                                                                        <variable symbol="x_{112}"/>
                                                                                        <function symbol="+">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{110}"/>
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <variable symbol="x_{111}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <variable symbol="x_{109}"/>
                                                                                        </function>
                                                                                       </function>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                     <constant symbol="1"/>
                                                                                    </function>
                                                                                   </function>
                                                                                   <constant symbol="1"/>
                                                                                  </constantatomformula>
                                                                                  <constantatomformula symbol="A" definition="A"/>
                                                                                 </formulalist>
                                                                                 <formulalist>
                                                                                  <quantifiedformula type="exists">
                                                                                   <variable symbol="p"/>
                                                                                   <quantifiedformula type="exists">
                                                                                    <variable symbol="q"/>
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                                                                                      <constantatomformula symbol="=">
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                                                                                      </constantatomformula>
                                                                                     </conjunctiveformula>
                                                                                     <constantatomformula symbol="=">
                                                                                      <function symbol="f">
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                                                                                      <function symbol="f">
                                                                                       <variable symbol="q"/>
                                                                                      </function>
                                                                                     </constantatomformula>
                                                                                    </conjunctiveformula>
                                                                                   </quantifiedformula>
                                                                                  </quantifiedformula>
                                                                                 </formulalist>
                                                                                </sequent>
                                                                                <rule type="contrr" symbol="c:r" param="2">
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                                                                                  <formulalist>
                                                                                   <constantatomformula symbol="A" definition="A"/>
                                                                                   <constantatomformula symbol="=">
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                                                                                     <function symbol="+">
                                                                                      <constant symbol="1"/>
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                                                                                       <variable symbol="x_{112}"/>
                                                                                       <function symbol="+">
                                                                                        <function symbol="+">
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                                                                                         <function symbol="+">
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                                                                                          <function symbol="+">
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                                                                                           <function symbol="+">
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                                                                                            <variable symbol="x_{111}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <variable symbol="x_{109}"/>
                                                                                       </function>
                                                                                      </function>
                                                                                     </function>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </constantatomformula>
                                                                                   <constantatomformula symbol="=">
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                                                                                     <function symbol="+">
                                                                                      <function symbol="+">
                                                                                       <function symbol="+">
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                                                                                         <variable symbol="x_{112}"/>
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
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                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{110}"/>
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <variable symbol="x_{111}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                          <variable symbol="x_{109}"/>
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                                                                                        </function>
                                                                                       </function>
                                                                                       <constant symbol="1"/>
                                                                                      </function>
                                                                                      <constant symbol="1"/>
                                                                                     </function>
                                                                                    </function>
                                                                                    <constant symbol="1"/>
                                                                                   </constantatomformula>
                                                                                  </formulalist>
                                                                                  <formulalist>
                                                                                   <quantifiedformula type="exists">
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                                                                                    <quantifiedformula type="exists">
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                                                                                      </conjunctiveformula>
                                                                                      <constantatomformula symbol="=">
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                                                                                      </constantatomformula>
                                                                                     </conjunctiveformula>
                                                                                    </quantifiedformula>
                                                                                   </quantifiedformula>
                                                                                  </formulalist>
                                                                                 </sequent>
                                                                                 <rule type="permr" symbol="\pi:r" param="(1 2)">
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                                                                                   <formulalist>
                                                                                    <constantatomformula symbol="A" definition="A"/>
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                                                                                          </function>
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                                                                                      </function>
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                                                                                     <constant symbol="1"/>
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                                                                                       <function symbol="+">
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                                                                                             <function symbol="+">
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                                                                                               <variable symbol="x_{111}"/>
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                                                                                            </function>
                                                                                           </function>
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                                                                                       </constantatomformula>
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                                                                                       </constantatomformula>
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                                                                                     </quantifiedformula>
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                                                                                   </formulalist>
                                                                                  </sequent>
                                                                                  <rule type="weakr" symbol="w:r">
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                                                                                    <formulalist>
                                                                                     <constantatomformula symbol="A" definition="A"/>
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                                                                                          <function symbol="+">
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                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                          <variable symbol="x_{109}"/>
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                                                                                        </function>
                                                                                       </function>
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                                                                                      <constant symbol="1"/>
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                                                                                              </function>
                                                                                             </function>
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                                                                                          </function>
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                                                                                    <formulalist>
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                                                                                        </constantatomformula>
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                                                                                        </constantatomformula>
                                                                                       </conjunctiveformula>
                                                                                      </quantifiedformula>
                                                                                     </quantifiedformula>
                                                                                    </formulalist>
                                                                                   </sequent>
                                                                                   <rule type="weakl" symbol="w:l">
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                                                                                     <formulalist>
                                                                                      <constantatomformula symbol="A" definition="A"/>
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                                                                                       <function symbol="f">
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                                                                                          <variable symbol="x_{112}"/>
                                                                                          <function symbol="+">
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                                                                                               <variable symbol="x_{111}"/>
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                                                                                             </function>
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                                                                                           </function>
                                                                                           <variable symbol="x_{109}"/>
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                                                                                         </function>
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                                                                                       </function>
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                                                                                      <constantatomformula symbol="=">
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                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
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                                                                                           </function>
                                                                                          </function>
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                                                                                      </constantatomformula>
                                                                                     </formulalist>
                                                                                     <formulalist>
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                                                                                         </constantatomformula>
                                                                                        </conjunctiveformula>
                                                                                       </quantifiedformula>
                                                                                      </quantifiedformula>
                                                                                     </formulalist>
                                                                                    </sequent>
                                                                                    <rule type="existsr" symbol="\ex:r">
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                                                                                      <formulalist>
                                                                                       <constantatomformula symbol="=">
                                                                                        <function symbol="f">
                                                                                         <function symbol="+">
                                                                                          <constant symbol="1"/>
                                                                                          <function symbol="+">
                                                                                           <variable symbol="x_{112}"/>
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{110}"/>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <variable symbol="x_{111}"/>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <variable symbol="x_{109}"/>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </constantatomformula>
                                                                                       <constantatomformula symbol="=">
                                                                                        <function symbol="f">
                                                                                         <function symbol="+">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{112}"/>
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{110}"/>
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <variable symbol="x_{111}"/>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                              <variable symbol="x_{109}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                          <constant symbol="1"/>
                                                                                         </function>
                                                                                        </function>
                                                                                        <constant symbol="1"/>
                                                                                       </constantatomformula>
                                                                                      </formulalist>
                                                                                      <formulalist>
                                                                                       <quantifiedformula type="exists">
                                                                                        <variable symbol="p"/>
                                                                                        <quantifiedformula type="exists">
                                                                                         <variable symbol="q"/>
                                                                                         <conjunctiveformula type="and">
                                                                                          <conjunctiveformula type="neg">
                                                                                           <constantatomformula symbol="=">
                                                                                            <variable symbol="p"/>
                                                                                            <variable symbol="q"/>
                                                                                           </constantatomformula>
                                                                                          </conjunctiveformula>
                                                                                          <constantatomformula symbol="=">
                                                                                           <function symbol="f">
                                                                                            <variable symbol="p"/>
                                                                                           </function>
                                                                                           <function symbol="f">
                                                                                            <variable symbol="q"/>
                                                                                           </function>
                                                                                          </constantatomformula>
                                                                                         </conjunctiveformula>
                                                                                        </quantifiedformula>
                                                                                       </quantifiedformula>
                                                                                      </formulalist>
                                                                                     </sequent>
                                                                                     <rule type="existsr" symbol="\ex:r">
                                                                                      <sequent>
                                                                                       <formulalist>
                                                                                        <constantatomformula symbol="=">
                                                                                         <function symbol="f">
                                                                                          <function symbol="+">
                                                                                           <constant symbol="1"/>
                                                                                           <function symbol="+">
                                                                                            <variable symbol="x_{112}"/>
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{110}"/>
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <variable symbol="x_{111}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                             <variable symbol="x_{109}"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </constantatomformula>
                                                                                        <constantatomformula symbol="=">
                                                                                         <function symbol="f">
                                                                                          <function symbol="+">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{112}"/>
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{111}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                               <variable symbol="x_{109}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </function>
                                                                                         </function>
                                                                                         <constant symbol="1"/>
                                                                                        </constantatomformula>
                                                                                       </formulalist>
                                                                                       <formulalist>
                                                                                        <quantifiedformula type="exists">
                                                                                         <variable symbol="q"/>
                                                                                         <conjunctiveformula type="and">
                                                                                          <conjunctiveformula type="neg">
                                                                                           <constantatomformula symbol="=">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{112}"/>
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{111}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                               <variable symbol="x_{109}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <variable symbol="q"/>
                                                                                           </constantatomformula>
                                                                                          </conjunctiveformula>
                                                                                          <constantatomformula symbol="=">
                                                                                           <function symbol="f">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{112}"/>
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{111}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                               <variable symbol="x_{109}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <function symbol="f">
                                                                                            <variable symbol="q"/>
                                                                                           </function>
                                                                                          </constantatomformula>
                                                                                         </conjunctiveformula>
                                                                                        </quantifiedformula>
                                                                                       </formulalist>
                                                                                      </sequent>
                                                                                      <rule type="andr" symbol="\and:r">
                                                                                       <sequent>
                                                                                        <formulalist>
                                                                                         <constantatomformula symbol="=">
                                                                                          <function symbol="f">
                                                                                           <function symbol="+">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="+">
                                                                                             <variable symbol="x_{112}"/>
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{110}"/>
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <variable symbol="x_{111}"/>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                              <variable symbol="x_{109}"/>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                          </function>
                                                                                          <constant symbol="1"/>
                                                                                         </constantatomformula>
                                                                                         <constantatomformula symbol="=">
                                                                                          <function symbol="f">
                                                                                           <function symbol="+">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{112}"/>
                                                                                               <function symbol="+">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <function symbol="+">
                                                                                                  <variable symbol="x_{110}"/>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <variable symbol="x_{111}"/>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                                <variable symbol="x_{109}"/>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </function>
                                                                                          </function>
                                                                                          <constant symbol="1"/>
                                                                                         </constantatomformula>
                                                                                        </formulalist>
                                                                                        <formulalist>
                                                                                         <conjunctiveformula type="and">
                                                                                          <conjunctiveformula type="neg">
                                                                                           <constantatomformula symbol="=">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{112}"/>
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{111}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                               <variable symbol="x_{109}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{112}"/>
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <variable symbol="x_{110}"/>
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <function symbol="+">
                                                                                                     <constant symbol="1"/>
                                                                                                     <variable symbol="x_{111}"/>
                                                                                                    </function>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </constantatomformula>
                                                                                          </conjunctiveformula>
                                                                                          <constantatomformula symbol="=">
                                                                                           <function symbol="f">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{112}"/>
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{111}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                               <variable symbol="x_{109}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <function symbol="f">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{112}"/>
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <variable symbol="x_{110}"/>
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <function symbol="+">
                                                                                                     <constant symbol="1"/>
                                                                                                     <variable symbol="x_{111}"/>
                                                                                                    </function>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </constantatomformula>
                                                                                         </conjunctiveformula>
                                                                                        </formulalist>
                                                                                       </sequent>
                                                                                       <rule type="negr" symbol="\neg:r">
                                                                                        <sequent>
                                                                                         <formulalist/>
                                                                                         <formulalist>
                                                                                          <conjunctiveformula type="neg">
                                                                                           <constantatomformula symbol="=">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{112}"/>
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{111}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                               <variable symbol="x_{109}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{112}"/>
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <variable symbol="x_{110}"/>
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <function symbol="+">
                                                                                                     <constant symbol="1"/>
                                                                                                     <variable symbol="x_{111}"/>
                                                                                                    </function>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </constantatomformula>
                                                                                          </conjunctiveformula>
                                                                                         </formulalist>
                                                                                        </sequent>
                                                                                        <rule type="eql1" symbol="=:l1" param="(2)">
                                                                                         <sequent>
                                                                                          <formulalist>
                                                                                           <constantatomformula symbol="=">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{112}"/>
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{111}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                               <variable symbol="x_{109}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{112}"/>
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <variable symbol="x_{110}"/>
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <function symbol="+">
                                                                                                     <constant symbol="1"/>
                                                                                                     <variable symbol="x_{111}"/>
                                                                                                    </function>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </constantatomformula>
                                                                                          </formulalist>
                                                                                          <formulalist/>
                                                                                         </sequent>
                                                                                         <rule type="axiom" symbol="axiom">
                                                                                          <sequent>
                                                                                           <formulalist/>
                                                                                           <formulalist>
                                                                                            <constantatomformula symbol="=">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{112}"/>
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <variable symbol="x_{110}"/>
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <function symbol="+">
                                                                                                     <constant symbol="1"/>
                                                                                                     <variable symbol="x_{111}"/>
                                                                                                    </function>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <variable symbol="x_{112}"/>
                                                                                                 <function symbol="+">
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <function symbol="+">
                                                                                                    <variable symbol="x_{110}"/>
                                                                                                    <function symbol="+">
                                                                                                     <constant symbol="1"/>
                                                                                                     <function symbol="+">
                                                                                                      <constant symbol="1"/>
                                                                                                      <variable symbol="x_{111}"/>
                                                                                                     </function>
                                                                                                    </function>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                  <variable symbol="x_{109}"/>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                            </constantatomformula>
                                                                                           </formulalist>
                                                                                          </sequent>
                                                                                         </rule>
                                                                                         <rule type="axiom" symbol="axiom">
                                                                                          <sequent>
                                                                                           <formulalist>
                                                                                            <constantatomformula symbol="=">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{112}"/>
                                                                                               <function symbol="+">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <function symbol="+">
                                                                                                  <variable symbol="x_{110}"/>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <variable symbol="x_{111}"/>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                                <variable symbol="x_{109}"/>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{112}"/>
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <variable symbol="x_{110}"/>
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <function symbol="+">
                                                                                                     <constant symbol="1"/>
                                                                                                     <variable symbol="x_{111}"/>
                                                                                                    </function>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </constantatomformula>
                                                                                           </formulalist>
                                                                                           <formulalist/>
                                                                                          </sequent>
                                                                                         </rule>
                                                                                        </rule>
                                                                                       </rule>
                                                                                       <rule type="eqr2" symbol="=:r2" param="(1)">
                                                                                        <sequent>
                                                                                         <formulalist>
                                                                                          <constantatomformula symbol="=">
                                                                                           <function symbol="f">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{112}"/>
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{111}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                               <variable symbol="x_{109}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </constantatomformula>
                                                                                          <constantatomformula symbol="=">
                                                                                           <function symbol="f">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{112}"/>
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <variable symbol="x_{110}"/>
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <function symbol="+">
                                                                                                     <constant symbol="1"/>
                                                                                                     <variable symbol="x_{111}"/>
                                                                                                    </function>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                           <constant symbol="1"/>
                                                                                          </constantatomformula>
                                                                                         </formulalist>
                                                                                         <formulalist>
                                                                                          <constantatomformula symbol="=">
                                                                                           <function symbol="f">
                                                                                            <function symbol="+">
                                                                                             <constant symbol="1"/>
                                                                                             <function symbol="+">
                                                                                              <variable symbol="x_{112}"/>
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <variable symbol="x_{110}"/>
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <variable symbol="x_{111}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                               <variable symbol="x_{109}"/>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                           </function>
                                                                                           <function symbol="f">
                                                                                            <function symbol="+">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <constant symbol="1"/>
                                                                                               <function symbol="+">
                                                                                                <variable symbol="x_{112}"/>
                                                                                                <function symbol="+">
                                                                                                 <function symbol="+">
                                                                                                  <constant symbol="1"/>
                                                                                                  <function symbol="+">
                                                                                                   <variable symbol="x_{110}"/>
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <function symbol="+">
                                                                                                     <constant symbol="1"/>
                                                                                                     <variable symbol="x_{111}"/>
                                                                                                    </function>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                 <variable symbol="x_{109}"/>
                                                                                                </function>
                                                                                               </function>
                                                                                              </function>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </function>
                                                                                           </function>
                                                                                          </constantatomformula>
                                                                                         </formulalist>
                                                                                        </sequent>
                                                                                        <rule type="axiom" symbol="axiom">
                                                                                         <sequent>
                                                                                          <formulalist>
                                                                                           <constantatomformula symbol="=">
                                                                                            <function symbol="f">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{112}"/>
                                                                                               <function symbol="+">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <function symbol="+">
                                                                                                  <variable symbol="x_{110}"/>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <variable symbol="x_{111}"/>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                                <variable symbol="x_{109}"/>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </constantatomformula>
                                                                                          </formulalist>
                                                                                          <formulalist>
                                                                                           <constantatomformula symbol="=">
                                                                                            <function symbol="f">
                                                                                             <function symbol="+">
                                                                                              <constant symbol="1"/>
                                                                                              <function symbol="+">
                                                                                               <variable symbol="x_{112}"/>
                                                                                               <function symbol="+">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <function symbol="+">
                                                                                                  <variable symbol="x_{110}"/>
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <variable symbol="x_{111}"/>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                                <variable symbol="x_{109}"/>
                                                                                               </function>
                                                                                              </function>
                                                                                             </function>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </constantatomformula>
                                                                                          </formulalist>
                                                                                         </sequent>
                                                                                        </rule>
                                                                                        <rule type="eqr2" symbol="=:r2" param="(2)">
                                                                                         <sequent>
                                                                                          <formulalist>
                                                                                           <constantatomformula symbol="=">
                                                                                            <function symbol="f">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <variable symbol="x_{112}"/>
                                                                                                 <function symbol="+">
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <function symbol="+">
                                                                                                    <variable symbol="x_{110}"/>
                                                                                                    <function symbol="+">
                                                                                                     <constant symbol="1"/>
                                                                                                     <function symbol="+">
                                                                                                      <constant symbol="1"/>
                                                                                                      <variable symbol="x_{111}"/>
                                                                                                     </function>
                                                                                                    </function>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                  <variable symbol="x_{109}"/>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                            </function>
                                                                                            <constant symbol="1"/>
                                                                                           </constantatomformula>
                                                                                          </formulalist>
                                                                                          <formulalist>
                                                                                           <constantatomformula symbol="=">
                                                                                            <constant symbol="1"/>
                                                                                            <function symbol="f">
                                                                                             <function symbol="+">
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <constant symbol="1"/>
                                                                                                <function symbol="+">
                                                                                                 <variable symbol="x_{112}"/>
                                                                                                 <function symbol="+">
                                                                                                  <function symbol="+">
                                                                                                   <constant symbol="1"/>
                                                                                                   <function symbol="+">
                                                                                                    <variable symbol="x_{110}"/>
                                                                                                    <function symbol="+">
                                                                                                     <constant symbol="1"/>
                                                                                                     <function symbol="+">
                                                                                                      <constant symbol="1"/>
                                                                                                      <variable symbol="x_{111}"/>
                                                                                                     </function>
                                                                                                    </function>
                                                                                                   </function>
                                                                                                  </function>
                                                                                                  <variable symbol="x_{109}"/>
                                                                                                 </function>
                                                                                                </function>
                                                                                               </function>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                              <constant symbol="1"/>
                                                                                             </function>
                                                                                            </function>
                                                                                           </constantatomformula>
                                                                                          </formulalist>
                                                                                         </sequent>
                                                                                         <rule type="axiom" symbol="axiom">
                                                                                          <sequent>
                                                                                           <formulalist>
                                                                                            <constantatomformula symbol="=">
                                                                                             <function symbol="f">
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <function symbol="+">
                                                                                                  <variable symbol="x_{112}"/>
                                                                                                  <function symbol="+">
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <function symbol="+">
                                                                                                     <variable symbol="x_{110}"/>
                                                                                                     <function symbol="+">
                                                                                                      <constant symbol="1"/>
                                                                                                      <function symbol="+">
                                                                                                       <constant symbol="1"/>
                                                                                                       <variable symbol="x_{111}"/>
                                                                                                      </function>
                                                                                                     </function>
                                                                                                    </function>
                                                                                                   </function>
                                                                                                   <variable symbol="x_{109}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                                <constant symbol="1"/>
                                                                                               </function>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </constantatomformula>
                                                                                           </formulalist>
                                                                                           <formulalist>
                                                                                            <constantatomformula symbol="=">
                                                                                             <function symbol="f">
                                                                                              <function symbol="+">
                                                                                               <function symbol="+">
                                                                                                <function symbol="+">
                                                                                                 <constant symbol="1"/>
                                                                                                 <function symbol="+">
                                                                                                  <variable symbol="x_{112}"/>
                                                                                                  <function symbol="+">
                                                                                                   <function symbol="+">
                                                                                                    <constant symbol="1"/>
                                                                                                    <function symbol="+">
                                                                                                     <variable symbol="x_{110}"/>
                                                                                                     <function symbol="+">
                                                                                                      <constant symbol="1"/>
                                                                                                      <function symbol="+">
                                                                                                       <constant symbol="1"/>
                                                                                                       <variable symbol="x_{111}"/>
                                                                                                      </function>
                                                                                                     </function>
                                                                                                    </function>
                                                                                                   </function>
                                                                                                   <variable symbol="x_{109}"/>
                                                                                                  </function>
                                                                                                 </function>
                                                                                                </function>
                                                                                                <constant symbol="1"/>
                                                                                               </function>
                                                                                               <constant symbol="1"/>
                                                                                              </function>
                                                                                             </function>
                                                                                             <constant symbol="1"/>
                                                                                            </constantatomformula>
                                                                                           </formulalist>
                                                                                          </sequent>
                                                                                         </rule>
                                                                                         <rule type="axiom" symbol="axiom">
                                                                                          <sequent>
                                                                                           <formulalist/>
                                                                                           <formulalist>
                                                                                            <constantatomformula symbol="=">
                                                                                             <constant symbol="1"/>
                                                                                             <constant symbol="1"/>
                                                                                            </constantatomformula>
                                                                                           </formulalist>
                                                                                          </sequent>
                                                                                         </rule>
                                                                                        </rule>
                                                                                       </rule>
                                                                                      </rule>
                                                                                     </rule>
                                                                                    </rule>
                                                                                   </rule>
                                                                                  </rule>
                                                                                 </rule>
                                                                                </rule>
                                                                               </rule>
                                                                              </rule>
                                                                             </rule>
                                                                            </rule>
                                                                           </rule>
                                                                          </rule>
                                                                         </rule>
                                                                        </rule>
                                                                       </rule>
                                                                      </rule>
                                                                     </rule>
                                                                    </rule>
                                                                   </rule>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                   <rule type="perml" symbol="\pi:l" param="(1 2)">
                                    <sequent>
                                     <formulalist>
                                      <constantatomformula symbol="=">
                                       <function symbol="f">
                                        <function symbol="+">
                                         <function symbol="+">
                                          <variable symbol="x_{112}"/>
                                          <function symbol="+">
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
                                             <variable symbol="x_{110}"/>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <variable symbol="x_{111}"/>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                           <variable symbol="x_{109}"/>
                                          </function>
                                         </function>
                                         <constant symbol="1"/>
                                        </function>
                                       </function>
                                       <constant symbol="0"/>
                                      </constantatomformula>
                                      <constantatomformula symbol="=">
                                       <function symbol="f">
                                        <function symbol="+">
                                         <function symbol="+">
                                          <variable symbol="x_{112}"/>
                                          <function symbol="+">
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
                                             <variable symbol="x_{110}"/>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <variable symbol="x_{111}"/>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                           <variable symbol="x_{109}"/>
                                          </function>
                                         </function>
                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <constant symbol="1"/>
                                            </function>
                                           </function>
                                          </function>
                                         </function>
                                        </function>
                                       </function>
                                       <constant symbol="0"/>
                                      </constantatomformula>
                                      <constantatomformula symbol="A" definition="A"/>
                                     </formulalist>
                                     <formulalist>
                                      <quantifiedformula type="exists">
                                       <variable symbol="p"/>
                                       <quantifiedformula type="exists">
                                        <variable symbol="q"/>
                                        <conjunctiveformula type="and">
                                         <conjunctiveformula type="neg">
                                          <constantatomformula symbol="=">
                                           <variable symbol="p"/>
                                           <variable symbol="q"/>
                                          </constantatomformula>
                                         </conjunctiveformula>
                                         <constantatomformula symbol="=">
                                          <function symbol="f">
                                           <variable symbol="p"/>
                                          </function>
                                          <function symbol="f">
                                           <variable symbol="q"/>
                                          </function>
                                         </constantatomformula>
                                        </conjunctiveformula>
                                       </quantifiedformula>
                                      </quantifiedformula>
                                     </formulalist>
                                    </sequent>
                                    <rule type="contrr" symbol="c:r" param="2">
                                     <sequent>
                                      <formulalist>
                                       <constantatomformula symbol="=">
                                        <function symbol="f">
                                         <function symbol="+">
                                          <function symbol="+">
                                           <variable symbol="x_{112}"/>
                                           <function symbol="+">
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <variable symbol="x_{110}"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <variable symbol="x_{111}"/>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                            <variable symbol="x_{109}"/>
                                           </function>
                                          </function>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <constant symbol="1"/>
                                             </function>
                                            </function>
                                           </function>
                                          </function>
                                         </function>
                                        </function>
                                        <constant symbol="0"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="=">
                                        <function symbol="f">
                                         <function symbol="+">
                                          <function symbol="+">
                                           <variable symbol="x_{112}"/>
                                           <function symbol="+">
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <variable symbol="x_{110}"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <variable symbol="x_{111}"/>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                            <variable symbol="x_{109}"/>
                                           </function>
                                          </function>
                                          <constant symbol="1"/>
                                         </function>
                                        </function>
                                        <constant symbol="0"/>
                                       </constantatomformula>
                                       <constantatomformula symbol="A" definition="A"/>
                                      </formulalist>
                                      <formulalist>
                                       <quantifiedformula type="exists">
                                        <variable symbol="p"/>
                                        <quantifiedformula type="exists">
                                         <variable symbol="q"/>
                                         <conjunctiveformula type="and">
                                          <conjunctiveformula type="neg">
                                           <constantatomformula symbol="=">
                                            <variable symbol="p"/>
                                            <variable symbol="q"/>
                                           </constantatomformula>
                                          </conjunctiveformula>
                                          <constantatomformula symbol="=">
                                           <function symbol="f">
                                            <variable symbol="p"/>
                                           </function>
                                           <function symbol="f">
                                            <variable symbol="q"/>
                                           </function>
                                          </constantatomformula>
                                         </conjunctiveformula>
                                        </quantifiedformula>
                                       </quantifiedformula>
                                      </formulalist>
                                     </sequent>
                                     <rule type="permr" symbol="\pi:r" param="">
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                                       <formulalist>
                                        <constantatomformula symbol="=">
                                         <function symbol="f">
                                          <function symbol="+">
                                           <function symbol="+">
                                            <variable symbol="x_{112}"/>
                                            <function symbol="+">
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <variable symbol="x_{111}"/>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                             <variable symbol="x_{109}"/>
                                            </function>
                                           </function>
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                          </function>
                                         </function>
                                         <constant symbol="0"/>
                                        </constantatomformula>
                                        <constantatomformula symbol="=">
                                         <function symbol="f">
                                          <function symbol="+">
                                           <function symbol="+">
                                            <variable symbol="x_{112}"/>
                                            <function symbol="+">
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <variable symbol="x_{110}"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <variable symbol="x_{111}"/>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                             <variable symbol="x_{109}"/>
                                            </function>
                                           </function>
                                           <constant symbol="1"/>
                                          </function>
                                         </function>
                                         <constant symbol="0"/>
                                        </constantatomformula>
                                        <constantatomformula symbol="A" definition="A"/>
                                       </formulalist>
                                       <formulalist>
                                        <quantifiedformula type="exists">
                                         <variable symbol="p"/>
                                         <quantifiedformula type="exists">
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                                          <conjunctiveformula type="and">
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                                            <constantatomformula symbol="=">
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                                           <constantatomformula symbol="=">
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                                            <function symbol="f">
                                             <variable symbol="q"/>
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                                           </constantatomformula>
                                          </conjunctiveformula>
                                         </quantifiedformula>
                                        </quantifiedformula>
                                        <quantifiedformula type="exists">
                                         <variable symbol="p"/>
                                         <quantifiedformula type="exists">
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                                            <constantatomformula symbol="=">
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                                             <variable symbol="q"/>
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                                           </conjunctiveformula>
                                           <constantatomformula symbol="=">
                                            <function symbol="f">
                                             <variable symbol="p"/>
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                                            <function symbol="f">
                                             <variable symbol="q"/>
                                            </function>
                                           </constantatomformula>
                                          </conjunctiveformula>
                                         </quantifiedformula>
                                        </quantifiedformula>
                                       </formulalist>
                                      </sequent>
                                      <rule type="contrl" symbol="c:l" param="1,1,2">
                                       <sequent>
                                        <formulalist>
                                         <constantatomformula symbol="=">
                                          <function symbol="f">
                                           <function symbol="+">
                                            <function symbol="+">
                                             <variable symbol="x_{112}"/>
                                             <function symbol="+">
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <variable symbol="x_{110}"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <variable symbol="x_{111}"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                              <variable symbol="x_{109}"/>
                                             </function>
                                            </function>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                          </function>
                                          <constant symbol="0"/>
                                         </constantatomformula>
                                         <constantatomformula symbol="=">
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                                            <function symbol="+">
                                             <variable symbol="x_{112}"/>
                                             <function symbol="+">
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <variable symbol="x_{110}"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <variable symbol="x_{111}"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                              <variable symbol="x_{109}"/>
                                             </function>
                                            </function>
                                            <constant symbol="1"/>
                                           </function>
                                          </function>
                                          <constant symbol="0"/>
                                         </constantatomformula>
                                         <constantatomformula symbol="A" definition="A"/>
                                        </formulalist>
                                        <formulalist>
                                         <quantifiedformula type="exists">
                                          <variable symbol="p"/>
                                          <quantifiedformula type="exists">
                                           <variable symbol="q"/>
                                           <conjunctiveformula type="and">
                                            <conjunctiveformula type="neg">
                                             <constantatomformula symbol="=">
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                                            </conjunctiveformula>
                                            <constantatomformula symbol="=">
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                                              <variable symbol="p"/>
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                                             <function symbol="f">
                                              <variable symbol="q"/>
                                             </function>
                                            </constantatomformula>
                                           </conjunctiveformula>
                                          </quantifiedformula>
                                         </quantifiedformula>
                                         <quantifiedformula type="exists">
                                          <variable symbol="p"/>
                                          <quantifiedformula type="exists">
                                           <variable symbol="q"/>
                                           <conjunctiveformula type="and">
                                            <conjunctiveformula type="neg">
                                             <constantatomformula symbol="=">
                                              <variable symbol="p"/>
                                              <variable symbol="q"/>
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                                            </conjunctiveformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="f">
                                              <variable symbol="p"/>
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                                             <function symbol="f">
                                              <variable symbol="q"/>
                                             </function>
                                            </constantatomformula>
                                           </conjunctiveformula>
                                          </quantifiedformula>
                                         </quantifiedformula>
                                        </formulalist>
                                       </sequent>
                                       <rule type="perml" symbol="\pi:l" param="(2 3)">
                                        <sequent>
                                         <formulalist>
                                          <constantatomformula symbol="=">
                                           <function symbol="f">
                                            <function symbol="+">
                                             <function symbol="+">
                                              <variable symbol="x_{112}"/>
                                              <function symbol="+">
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <variable symbol="x_{110}"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{111}"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                               <variable symbol="x_{109}"/>
                                              </function>
                                             </function>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                           <constant symbol="0"/>
                                          </constantatomformula>
                                          <constantatomformula symbol="=">
                                           <function symbol="f">
                                            <function symbol="+">
                                             <function symbol="+">
                                              <variable symbol="x_{112}"/>
                                              <function symbol="+">
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <variable symbol="x_{110}"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{111}"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                               <variable symbol="x_{109}"/>
                                              </function>
                                             </function>
                                             <constant symbol="1"/>
                                            </function>
                                           </function>
                                           <constant symbol="0"/>
                                          </constantatomformula>
                                          <constantatomformula symbol="A" definition="A"/>
                                          <constantatomformula symbol="A" definition="A"/>
                                         </formulalist>
                                         <formulalist>
                                          <quantifiedformula type="exists">
                                           <variable symbol="p"/>
                                           <quantifiedformula type="exists">
                                            <variable symbol="q"/>
                                            <conjunctiveformula type="and">
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                                              <constantatomformula symbol="=">
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                                               <variable symbol="q"/>
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                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <variable symbol="p"/>
                                              </function>
                                              <function symbol="f">
                                               <variable symbol="q"/>
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                                          <quantifiedformula type="exists">
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                                           <quantifiedformula type="exists">
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                                               <variable symbol="q"/>
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                                             <constantatomformula symbol="=">
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                                               <variable symbol="p"/>
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                                              <function symbol="f">
                                               <variable symbol="q"/>
                                              </function>
                                             </constantatomformula>
                                            </conjunctiveformula>
                                           </quantifiedformula>
                                          </quantifiedformula>
                                         </formulalist>
                                        </sequent>
                                        <rule type="eql1" symbol="=:l1" param="(1 1 2)">
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                                          <formulalist>
                                           <constantatomformula symbol="=">
                                            <function symbol="f">
                                             <function symbol="+">
                                              <function symbol="+">
                                               <variable symbol="x_{112}"/>
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <variable symbol="x_{109}"/>
                                               </function>
                                              </function>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                            <constant symbol="0"/>
                                           </constantatomformula>
                                           <constantatomformula symbol="A" definition="A"/>
                                           <constantatomformula symbol="=">
                                            <function symbol="f">
                                             <function symbol="+">
                                              <function symbol="+">
                                               <variable symbol="x_{112}"/>
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <variable symbol="x_{109}"/>
                                               </function>
                                              </function>
                                              <constant symbol="1"/>
                                             </function>
                                            </function>
                                            <constant symbol="0"/>
                                           </constantatomformula>
                                           <constantatomformula symbol="A" definition="A"/>
                                          </formulalist>
                                          <formulalist>
                                           <quantifiedformula type="exists">
                                            <variable symbol="p"/>
                                            <quantifiedformula type="exists">
                                             <variable symbol="q"/>
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                                               <constantatomformula symbol="=">
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                                                <variable symbol="q"/>
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                                              </conjunctiveformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <variable symbol="p"/>
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                                               <function symbol="f">
                                                <variable symbol="q"/>
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                                              </constantatomformula>
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                                           <quantifiedformula type="exists">
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                                            <quantifiedformula type="exists">
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                                               <constantatomformula symbol="=">
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                                                <variable symbol="q"/>
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                                              </conjunctiveformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <variable symbol="p"/>
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                                               <function symbol="f">
                                                <variable symbol="q"/>
                                               </function>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                            </quantifiedformula>
                                           </quantifiedformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="contrr" symbol="c:r" param="2,1">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="A" definition="A"/>
                                           </formulalist>
                                           <formulalist>
                                            <quantifiedformula type="exists">
                                             <variable symbol="p"/>
                                             <quantifiedformula type="exists">
                                              <variable symbol="q"/>
                                              <conjunctiveformula type="and">
                                               <conjunctiveformula type="neg">
                                                <constantatomformula symbol="=">
                                                 <variable symbol="p"/>
                                                 <variable symbol="q"/>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <variable symbol="p"/>
                                                </function>
                                                <function symbol="f">
                                                 <variable symbol="q"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                             </quantifiedformula>
                                            </quantifiedformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="+">
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <constant symbol="1"/>
                                              </function>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                             </function>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="permr" symbol="\pi:r" param="">
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                                            <formulalist>
                                             <constantatomformula symbol="A" definition="A"/>
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                                            <formulalist>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
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                                                 <constantatomformula symbol="=">
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                                                  <variable symbol="q"/>
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                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
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                                                </constantatomformula>
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                                             </quantifiedformula>
                                             <quantifiedformula type="exists">
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                                                 <constantatomformula symbol="=">
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                                                <constantatomformula symbol="=">
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                                                  <variable symbol="p"/>
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                                                 <function symbol="f">
                                                  <variable symbol="q"/>
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                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <constantatomformula symbol="=">
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                                                <constant symbol="1"/>
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                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                              </function>
                                              <function symbol="+">
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                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="contrl" symbol="c:l" param="2">
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                                             <formulalist>
                                              <constantatomformula symbol="A" definition="A"/>
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                                             <formulalist>
                                              <quantifiedformula type="exists">
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                                               <quantifiedformula type="exists">
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                                                  <constantatomformula symbol="=">
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                                                 <constantatomformula symbol="=">
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                                                   <variable symbol="p"/>
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                                                  <function symbol="f">
                                                   <variable symbol="q"/>
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                                                 </constantatomformula>
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                                              <quantifiedformula type="exists">
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                                                  <constantatomformula symbol="=">
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                                                 <constantatomformula symbol="=">
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                                                   <variable symbol="p"/>
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                                                  <function symbol="f">
                                                   <variable symbol="q"/>
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                                              <constantatomformula symbol="=">
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                                                <function symbol="+">
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                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
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                                                 <function symbol="+">
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                                                   <constant symbol="1"/>
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                                            <rule type="perml" symbol="\pi:l" param="">
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                                              <formulalist>
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                                                  <constantatomformula symbol="=">
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                                                   <function symbol="f">
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                                                  <constantatomformula symbol="=">
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                                                   <function symbol="f">
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                                               <constantatomformula symbol="=">
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                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
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                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
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                                                    <constant symbol="1"/>
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                                                  </function>
                                                 </function>
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                                               </constantatomformula>
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                                             <rule type="eqr1" symbol="=:r1" param="(1)">
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                                               <formulalist>
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                                                <quantifiedformula type="exists">
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                                                   <constantatomformula symbol="=">
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                                                <quantifiedformula type="exists">
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                                                   <constantatomformula symbol="=">
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                                                    <function symbol="f">
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                                                   </constantatomformula>
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                                                <constantatomformula symbol="=">
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                                                   <constant symbol="1"/>
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                                                  <function symbol="+">
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                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
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                                                  </function>
                                                 </function>
                                                 <function symbol="+">
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                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
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                                                     <constant symbol="1"/>
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                                                   </function>
                                                  </function>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="permr" symbol="\pi:r" param="(1 2)">
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                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
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                                                 <quantifiedformula type="exists">
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                                                     <constantatomformula symbol="=">
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                                                    <constantatomformula symbol="=">
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                                                      <variable symbol="p"/>
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                                                     <function symbol="f">
                                                      <variable symbol="q"/>
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                                                    </constantatomformula>
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                                                  </quantifiedformula>
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                                                 <constantatomformula symbol="=">
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                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
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                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <function symbol="+">
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                                                    <function symbol="+">
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                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="weakr" symbol="w:r">
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                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
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                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
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                                                     <constant symbol="1"/>
                                                     <function symbol="+">
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                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
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                                                     </function>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
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                                                      <constantatomformula symbol="=">
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                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
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                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="perml" symbol="\pi:l" param="">
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                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
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                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="weakl" symbol="w:l">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="axiom" symbol="axiom">
                                                   <sequent>
                                                    <formulalist/>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                              <rule type="permr" symbol="\pi:r" param="(1 2)">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="weakr" symbol="w:r">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="perml" symbol="\pi:l" param="">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="weakl" symbol="w:l">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="axiom" symbol="axiom">
                                                   <sequent>
                                                    <formulalist/>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                         <rule type="perml" symbol="\pi:l" param="(1 2)">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="=">
                                             <function symbol="f">
                                              <function symbol="+">
                                               <function symbol="+">
                                                <variable symbol="x_{112}"/>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <variable symbol="x_{109}"/>
                                                </function>
                                               </function>
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <constant symbol="1"/>
                                                </function>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                             <constant symbol="0"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="f">
                                              <function symbol="+">
                                               <function symbol="+">
                                                <variable symbol="x_{112}"/>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <variable symbol="x_{109}"/>
                                                </function>
                                               </function>
                                               <constant symbol="1"/>
                                              </function>
                                             </function>
                                             <constant symbol="0"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="A" definition="A"/>
                                           </formulalist>
                                           <formulalist>
                                            <quantifiedformula type="exists">
                                             <variable symbol="p"/>
                                             <quantifiedformula type="exists">
                                              <variable symbol="q"/>
                                              <conjunctiveformula type="and">
                                               <conjunctiveformula type="neg">
                                                <constantatomformula symbol="=">
                                                 <variable symbol="p"/>
                                                 <variable symbol="q"/>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <variable symbol="p"/>
                                                </function>
                                                <function symbol="f">
                                                 <variable symbol="q"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                             </quantifiedformula>
                                            </quantifiedformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="perml" symbol="\pi:l" param="(1 2)">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <variable symbol="x_{112}"/>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <variable symbol="x_{109}"/>
                                                 </function>
                                                </function>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                              <constant symbol="0"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <variable symbol="x_{112}"/>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <variable symbol="x_{109}"/>
                                                 </function>
                                                </function>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                              <constant symbol="0"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="A" definition="A"/>
                                            </formulalist>
                                            <formulalist>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="contrr" symbol="c:r" param="2">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <variable symbol="x_{112}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{109}"/>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                               <constant symbol="0"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <variable symbol="x_{112}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{109}"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                               <constant symbol="0"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="A" definition="A"/>
                                             </formulalist>
                                             <formulalist>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="permr" symbol="\pi:r" param="">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{112}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <constant symbol="0"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{112}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                                <constant symbol="0"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="A" definition="A"/>
                                              </formulalist>
                                              <formulalist>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="contrl" symbol="c:l" param="1,1,2">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{112}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{112}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="perml" symbol="\pi:l" param="(2 3)">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="eql1" symbol="=:l1" param="(1 1)">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="contrr" symbol="c:r" param="2,1">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="permr" symbol="\pi:r" param="">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="contrl" symbol="c:l" param="2">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="perml" symbol="\pi:l" param="">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="eqr1" symbol="=:r1" param="(1)">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="weakr" symbol="w:r">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </constantatomformula>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
                                                            <conjunctiveformula type="neg">
                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="perml" symbol="\pi:l" param="">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="weakl" symbol="w:l">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="axiom" symbol="axiom">
                                                          <sequent>
                                                           <formulalist/>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                     <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="weakr" symbol="w:r">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </constantatomformula>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
                                                            <conjunctiveformula type="neg">
                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="perml" symbol="\pi:l" param="">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="weakl" symbol="w:l">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="axiom" symbol="axiom">
                                                          <sequent>
                                                           <formulalist/>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                                <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="contrr" symbol="c:r" param="2">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="permr" symbol="\pi:r" param="">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="contrl" symbol="c:l" param="1,1,2">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="0"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="0"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="perml" symbol="\pi:l" param="(2 3)">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="0"/>
                                                        </constantatomformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="0"/>
                                                        </constantatomformula>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="eql1" symbol="=:l1" param="(1 1 1)">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="0"/>
                                                         </constantatomformula>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
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                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
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                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
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                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="contrr" symbol="c:r" param="2,1">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
                                                            <conjunctiveformula type="and">
                                                             <conjunctiveformula type="neg">
                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="permr" symbol="\pi:r" param="">
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                                                          <formulalist>
                                                           <constantatomformula symbol="A" definition="A"/>
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                                                          <formulalist>
                                                           <quantifiedformula type="exists">
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                                                             <variable symbol="q"/>
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                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="p"/>
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                                                               <function symbol="f">
                                                                <variable symbol="q"/>
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                                                              </constantatomformula>
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                                                           </quantifiedformula>
                                                           <quantifiedformula type="exists">
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                                                               <constantatomformula symbol="=">
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                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
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                                                               <function symbol="f">
                                                                <variable symbol="q"/>
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                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                           <constantatomformula symbol="=">
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                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
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                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
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                                                                   <function symbol="+">
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                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
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                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
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                                                                  <function symbol="+">
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                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
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                                                             </function>
                                                             <function symbol="+">
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                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="contrl" symbol="c:l" param="2">
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                                                               <constantatomformula symbol="=">
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                                                               </constantatomformula>
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                                                               </constantatomformula>
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                                                            <constantatomformula symbol="=">
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                                                                   </function>
                                                                  </function>
                                                                 </function>
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                                                               </function>
                                                               <constant symbol="1"/>
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                                                              <constant symbol="1"/>
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                                                             <function symbol="+">
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                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
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                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
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                                                              </function>
                                                              <function symbol="+">
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                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="perml" symbol="\pi:l" param="">
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                                                            <formulalist>
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                                                             <constantatomformula symbol="A" definition="A"/>
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                                                                <constantatomformula symbol="=">
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                                                             <quantifiedformula type="exists">
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                                                                  </function>
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                                                               <constant symbol="1"/>
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                                                                  </function>
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                                                               </function>
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                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="eqr1" symbol="=:r1" param="(1)">
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                                                             <formulalist>
                                                              <constantatomformula symbol="A" definition="A"/>
                                                              <constantatomformula symbol="A" definition="A"/>
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                                                             <formulalist>
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                                                                 <constantatomformula symbol="=">
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                                                                 </constantatomformula>
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                                                               </quantifiedformula>
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                                                              <quantifiedformula type="exists">
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                                                                 <constantatomformula symbol="=">
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                                                                   <variable symbol="q"/>
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                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                               </quantifiedformula>
                                                              </quantifiedformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <function symbol="+">
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                                                                  <function symbol="+">
                                                                   <function symbol="+">
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                                                                     <function symbol="+">
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                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <function symbol="+">
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                                                                 <variable symbol="x_{112}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
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                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <variable symbol="x_{109}"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
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                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="A" definition="A"/>
                                                              </formulalist>
                                                              <formulalist>
                                                               <quantifiedformula type="exists">
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                                                                <quantifiedformula type="exists">
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                                                                  </conjunctiveformula>
                                                                  <constantatomformula symbol="=">
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                                                                   <function symbol="f">
                                                                    <variable symbol="q"/>
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                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                </quantifiedformula>
                                                               </quantifiedformula>
                                                               <constantatomformula symbol="=">
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                                                                  <variable symbol="x_{112}"/>
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                                                                   <function symbol="+">
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                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </formulalist>
                                                             </sequent>
                                                             <rule type="weakr" symbol="w:r">
                                                              <sequent>
                                                               <formulalist>
                                                                <constantatomformula symbol="A" definition="A"/>
                                                               </formulalist>
                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </constantatomformula>
                                                                <quantifiedformula type="exists">
                                                                 <variable symbol="p"/>
                                                                 <quantifiedformula type="exists">
                                                                  <variable symbol="q"/>
                                                                  <conjunctiveformula type="and">
                                                                   <conjunctiveformula type="neg">
                                                                    <constantatomformula symbol="=">
                                                                     <variable symbol="p"/>
                                                                     <variable symbol="q"/>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <variable symbol="p"/>
                                                                    </function>
                                                                    <function symbol="f">
                                                                     <variable symbol="q"/>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                 </quantifiedformula>
                                                                </quantifiedformula>
                                                               </formulalist>
                                                              </sequent>
                                                              <rule type="perml" symbol="\pi:l" param="">
                                                               <sequent>
                                                                <formulalist>
                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                </formulalist>
                                                                <formulalist>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </constantatomformula>
                                                                </formulalist>
                                                               </sequent>
                                                               <rule type="weakl" symbol="w:l">
                                                                <sequent>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="A" definition="A"/>
                                                                 </formulalist>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{112}"/>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="x_{109}"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </constantatomformula>
                                                                 </formulalist>
                                                                </sequent>
                                                                <rule type="axiom" symbol="axiom">
                                                                 <sequent>
                                                                  <formulalist/>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{112}"/>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="x_{109}"/>
                                                                      </function>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                            <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="A" definition="A"/>
                                                              </formulalist>
                                                              <formulalist>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="p"/>
                                                                <quantifiedformula type="exists">
                                                                 <variable symbol="q"/>
                                                                 <conjunctiveformula type="and">
                                                                  <conjunctiveformula type="neg">
                                                                   <constantatomformula symbol="=">
                                                                    <variable symbol="p"/>
                                                                    <variable symbol="q"/>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <variable symbol="p"/>
                                                                   </function>
                                                                   <function symbol="f">
                                                                    <variable symbol="q"/>
                                                                   </function>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                </quantifiedformula>
                                                               </quantifiedformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </constantatomformula>
                                                              </formulalist>
                                                             </sequent>
                                                             <rule type="weakr" symbol="w:r">
                                                              <sequent>
                                                               <formulalist>
                                                                <constantatomformula symbol="A" definition="A"/>
                                                               </formulalist>
                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
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                                                                  </function>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </constantatomformula>
                                                                <quantifiedformula type="exists">
                                                                 <variable symbol="p"/>
                                                                 <quantifiedformula type="exists">
                                                                  <variable symbol="q"/>
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                                                                    <constantatomformula symbol="=">
                                                                     <variable symbol="p"/>
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                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <variable symbol="p"/>
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                                                                    <function symbol="f">
                                                                     <variable symbol="q"/>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                 </quantifiedformula>
                                                                </quantifiedformula>
                                                               </formulalist>
                                                              </sequent>
                                                              <rule type="perml" symbol="\pi:l" param="">
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                                                                <formulalist>
                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                </formulalist>
                                                                <formulalist>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                 </constantatomformula>
                                                                </formulalist>
                                                               </sequent>
                                                               <rule type="weakl" symbol="w:l">
                                                                <sequent>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="A" definition="A"/>
                                                                 </formulalist>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
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                                                                     <variable symbol="x_{112}"/>
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                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
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                                                                    </function>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
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                                                                    </function>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                  </constantatomformula>
                                                                 </formulalist>
                                                                </sequent>
                                                                <rule type="axiom" symbol="axiom">
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                                                                  <formulalist/>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
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                                                                      <variable symbol="x_{112}"/>
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                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
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                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="x_{109}"/>
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                                                                     </function>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{112}"/>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="x_{109}"/>
                                                                      </function>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                       <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="0"/>
                                                          </constantatomformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
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                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="0"/>
                                                          </constantatomformula>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
                                                            <conjunctiveformula type="and">
                                                             <conjunctiveformula type="neg">
                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="permr" symbol="\pi:r" param="">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="0"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{112}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <variable symbol="x_{109}"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="0"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="p"/>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="q"/>
                                                             <conjunctiveformula type="and">
                                                              <conjunctiveformula type="neg">
                                                               <constantatomformula symbol="=">
                                                                <variable symbol="p"/>
                                                                <variable symbol="q"/>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="perml" symbol="\pi:l" param="(1 2 3)">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="0"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="0"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="p"/>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="q"/>
                                                              <conjunctiveformula type="and">
                                                               <conjunctiveformula type="neg">
                                                                <constantatomformula symbol="=">
                                                                 <variable symbol="p"/>
                                                                 <variable symbol="q"/>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <variable symbol="p"/>
                                                                </function>
                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </quantifiedformula>
                                                            </quantifiedformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="contrr" symbol="c:r" param="2">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="A" definition="A"/>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{112}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <variable symbol="x_{109}"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                            </formulalist>
                                                            <formulalist>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="p"/>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="q"/>
                                                               <conjunctiveformula type="and">
                                                                <conjunctiveformula type="neg">
                                                                 <constantatomformula symbol="=">
                                                                  <variable symbol="p"/>
                                                                  <variable symbol="q"/>
                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <variable symbol="p"/>
                                                                 </function>
                                                                 <function symbol="f">
                                                                  <variable symbol="q"/>
                                                                 </function>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                              </quantifiedformula>
                                                             </quantifiedformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="weakr" symbol="w:r">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="A" definition="A"/>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <constant symbol="0"/>
                                                              </constantatomformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="0"/>
                                                              </constantatomformula>
                                                             </formulalist>
                                                             <formulalist>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="p"/>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="q"/>
                                                                <conjunctiveformula type="and">
                                                                 <conjunctiveformula type="neg">
                                                                  <constantatomformula symbol="=">
                                                                   <variable symbol="p"/>
                                                                   <variable symbol="q"/>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <variable symbol="p"/>
                                                                  </function>
                                                                  <function symbol="f">
                                                                   <variable symbol="q"/>
                                                                  </function>
                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                               </quantifiedformula>
                                                              </quantifiedformula>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="p"/>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="q"/>
                                                                <conjunctiveformula type="and">
                                                                 <conjunctiveformula type="neg">
                                                                  <constantatomformula symbol="=">
                                                                   <variable symbol="p"/>
                                                                   <variable symbol="q"/>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <variable symbol="p"/>
                                                                  </function>
                                                                  <function symbol="f">
                                                                   <variable symbol="q"/>
                                                                  </function>
                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                               </quantifiedformula>
                                                              </quantifiedformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="weakl" symbol="w:l">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="A" definition="A"/>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="0"/>
                                                               </constantatomformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="0"/>
                                                               </constantatomformula>
                                                              </formulalist>
                                                              <formulalist>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="p"/>
                                                                <quantifiedformula type="exists">
                                                                 <variable symbol="q"/>
                                                                 <conjunctiveformula type="and">
                                                                  <conjunctiveformula type="neg">
                                                                   <constantatomformula symbol="=">
                                                                    <variable symbol="p"/>
                                                                    <variable symbol="q"/>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <variable symbol="p"/>
                                                                   </function>
                                                                   <function symbol="f">
                                                                    <variable symbol="q"/>
                                                                   </function>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                </quantifiedformula>
                                                               </quantifiedformula>
                                                              </formulalist>
                                                             </sequent>
                                                             <rule type="existsr" symbol="\ex:r">
                                                              <sequent>
                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="0"/>
                                                                </constantatomformula>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{112}"/>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="x_{109}"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="0"/>
                                                                </constantatomformula>
                                                               </formulalist>
                                                               <formulalist>
                                                                <quantifiedformula type="exists">
                                                                 <variable symbol="p"/>
                                                                 <quantifiedformula type="exists">
                                                                  <variable symbol="q"/>
                                                                  <conjunctiveformula type="and">
                                                                   <conjunctiveformula type="neg">
                                                                    <constantatomformula symbol="=">
                                                                     <variable symbol="p"/>
                                                                     <variable symbol="q"/>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <variable symbol="p"/>
                                                                    </function>
                                                                    <function symbol="f">
                                                                     <variable symbol="q"/>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                 </quantifiedformula>
                                                                </quantifiedformula>
                                                               </formulalist>
                                                              </sequent>
                                                              <rule type="existsr" symbol="\ex:r">
                                                               <sequent>
                                                                <formulalist>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="0"/>
                                                                 </constantatomformula>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="0"/>
                                                                 </constantatomformula>
                                                                </formulalist>
                                                                <formulalist>
                                                                 <quantifiedformula type="exists">
                                                                  <variable symbol="q"/>
                                                                  <conjunctiveformula type="and">
                                                                   <conjunctiveformula type="neg">
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <variable symbol="q"/>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="f">
                                                                     <variable symbol="q"/>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                 </quantifiedformula>
                                                                </formulalist>
                                                               </sequent>
                                                               <rule type="andr" symbol="\and:r">
                                                                <sequent>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{112}"/>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <variable symbol="x_{109}"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="0"/>
                                                                  </constantatomformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="0"/>
                                                                  </constantatomformula>
                                                                 </formulalist>
                                                                 <formulalist>
                                                                  <conjunctiveformula type="and">
                                                                   <conjunctiveformula type="neg">
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                 </formulalist>
                                                                </sequent>
                                                                <rule type="negr" symbol="\neg:r">
                                                                 <sequent>
                                                                  <formulalist/>
                                                                  <formulalist>
                                                                   <conjunctiveformula type="neg">
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                 <rule type="eql1" symbol="=:l1" param="(2)">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                   <formulalist/>
                                                                  </sequent>
                                                                  <rule type="axiom" symbol="axiom">
                                                                   <sequent>
                                                                    <formulalist/>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{112}"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{111}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                  </rule>
                                                                  <rule type="axiom" symbol="axiom">
                                                                   <sequent>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                    <formulalist/>
                                                                   </sequent>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                                <rule type="eqr2" symbol="=:r2" param="(1)">
                                                                 <sequent>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="0"/>
                                                                   </constantatomformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="0"/>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{112}"/>
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <variable symbol="x_{109}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{112}"/>
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                          <variable symbol="x_{109}"/>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                 <rule type="axiom" symbol="axiom">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="0"/>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{112}"/>
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <variable symbol="x_{109}"/>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="0"/>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                 </rule>
                                                                 <rule type="eqr2" symbol="=:r2" param="(2)">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{112}"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{111}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="0"/>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <constant symbol="0"/>
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{112}"/>
                                                                          <function symbol="+">
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <variable symbol="x_{110}"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <variable symbol="x_{111}"/>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                           <variable symbol="x_{109}"/>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                  <rule type="axiom" symbol="axiom">
                                                                   <sequent>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{112}"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <variable symbol="x_{111}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="0"/>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{112}"/>
                                                                           <function symbol="+">
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <variable symbol="x_{110}"/>
                                                                              <function symbol="+">
                                                                               <constant symbol="1"/>
                                                                               <function symbol="+">
                                                                                <constant symbol="1"/>
                                                                                <variable symbol="x_{111}"/>
                                                                               </function>
                                                                              </function>
                                                                             </function>
                                                                            </function>
                                                                            <variable symbol="x_{109}"/>
                                                                           </function>
                                                                          </function>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <constant symbol="1"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="0"/>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                  </rule>
                                                                  <rule type="axiom" symbol="axiom">
                                                                   <sequent>
                                                                    <formulalist/>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <constant symbol="0"/>
                                                                      <constant symbol="0"/>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                          </rule>
                         </rule>
                        </rule>
                        <rule type="contrr" symbol="c:r" param="2,1">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="f">
                             <function symbol="+">
                              <function symbol="+">
                               <variable symbol="x_{112}"/>
                               <function symbol="+">
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <function symbol="+">
                                  <variable symbol="x_{110}"/>
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <variable symbol="x_{111}"/>
                                   </function>
                                  </function>
                                 </function>
                                </function>
                                <variable symbol="x_{109}"/>
                               </function>
                              </function>
                              <function symbol="+">
                               <constant symbol="1"/>
                               <function symbol="+">
                                <constant symbol="1"/>
                                <constant symbol="1"/>
                               </function>
                              </function>
                             </function>
                            </function>
                            <constant symbol="0"/>
                           </constantatomformula>
                           <constantatomformula symbol="A" definition="A"/>
                          </formulalist>
                          <formulalist>
                           <quantifiedformula type="exists">
                            <variable symbol="p"/>
                            <quantifiedformula type="exists">
                             <variable symbol="q"/>
                             <conjunctiveformula type="and">
                              <conjunctiveformula type="neg">
                               <constantatomformula symbol="=">
                                <variable symbol="p"/>
                                <variable symbol="q"/>
                               </constantatomformula>
                              </conjunctiveformula>
                              <constantatomformula symbol="=">
                               <function symbol="f">
                                <variable symbol="p"/>
                               </function>
                               <function symbol="f">
                                <variable symbol="q"/>
                               </function>
                              </constantatomformula>
                             </conjunctiveformula>
                            </quantifiedformula>
                           </quantifiedformula>
                           <constantatomformula symbol="=">
                            <function symbol="f">
                             <function symbol="+">
                              <constant symbol="1"/>
                              <function symbol="+">
                               <variable symbol="x_{112}"/>
                               <function symbol="+">
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <function symbol="+">
                                  <variable symbol="x_{110}"/>
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <variable symbol="x_{111}"/>
                                   </function>
                                  </function>
                                 </function>
                                </function>
                                <variable symbol="x_{109}"/>
                               </function>
                              </function>
                             </function>
                            </function>
                            <constant symbol="1"/>
                           </constantatomformula>
                          </formulalist>
                         </sequent>
                         <rule type="permr" symbol="\pi:r" param="(2 3)">
                          <sequent>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="f">
                              <function symbol="+">
                               <function symbol="+">
                                <variable symbol="x_{112}"/>
                                <function symbol="+">
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <function symbol="+">
                                   <variable symbol="x_{110}"/>
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <function symbol="+">
                                     <constant symbol="1"/>
                                     <variable symbol="x_{111}"/>
                                    </function>
                                   </function>
                                  </function>
                                 </function>
                                 <variable symbol="x_{109}"/>
                                </function>
                               </function>
                               <function symbol="+">
                                <constant symbol="1"/>
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <constant symbol="1"/>
                                </function>
                               </function>
                              </function>
                             </function>
                             <constant symbol="0"/>
                            </constantatomformula>
                            <constantatomformula symbol="A" definition="A"/>
                           </formulalist>
                           <formulalist>
                            <quantifiedformula type="exists">
                             <variable symbol="p"/>
                             <quantifiedformula type="exists">
                              <variable symbol="q"/>
                              <conjunctiveformula type="and">
                               <conjunctiveformula type="neg">
                                <constantatomformula symbol="=">
                                 <variable symbol="p"/>
                                 <variable symbol="q"/>
                                </constantatomformula>
                               </conjunctiveformula>
                               <constantatomformula symbol="=">
                                <function symbol="f">
                                 <variable symbol="p"/>
                                </function>
                                <function symbol="f">
                                 <variable symbol="q"/>
                                </function>
                               </constantatomformula>
                              </conjunctiveformula>
                             </quantifiedformula>
                            </quantifiedformula>
                            <quantifiedformula type="exists">
                             <variable symbol="p"/>
                             <quantifiedformula type="exists">
                              <variable symbol="q"/>
                              <conjunctiveformula type="and">
                               <conjunctiveformula type="neg">
                                <constantatomformula symbol="=">
                                 <variable symbol="p"/>
                                 <variable symbol="q"/>
                                </constantatomformula>
                               </conjunctiveformula>
                               <constantatomformula symbol="=">
                                <function symbol="f">
                                 <variable symbol="p"/>
                                </function>
                                <function symbol="f">
                                 <variable symbol="q"/>
                                </function>
                               </constantatomformula>
                              </conjunctiveformula>
                             </quantifiedformula>
                            </quantifiedformula>
                            <constantatomformula symbol="=">
                             <function symbol="f">
                              <function symbol="+">
                               <constant symbol="1"/>
                               <function symbol="+">
                                <variable symbol="x_{112}"/>
                                <function symbol="+">
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <function symbol="+">
                                   <variable symbol="x_{110}"/>
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <function symbol="+">
                                     <constant symbol="1"/>
                                     <variable symbol="x_{111}"/>
                                    </function>
                                   </function>
                                  </function>
                                 </function>
                                 <variable symbol="x_{109}"/>
                                </function>
                               </function>
                              </function>
                             </function>
                             <constant symbol="1"/>
                            </constantatomformula>
                           </formulalist>
                          </sequent>
                          <rule type="contrl" symbol="c:l" param="1,2">
                           <sequent>
                            <formulalist>
                             <constantatomformula symbol="=">
                              <function symbol="f">
                               <function symbol="+">
                                <function symbol="+">
                                 <variable symbol="x_{112}"/>
                                 <function symbol="+">
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <function symbol="+">
                                    <variable symbol="x_{110}"/>
                                    <function symbol="+">
                                     <constant symbol="1"/>
                                     <function symbol="+">
                                      <constant symbol="1"/>
                                      <variable symbol="x_{111}"/>
                                     </function>
                                    </function>
                                   </function>
                                  </function>
                                  <variable symbol="x_{109}"/>
                                 </function>
                                </function>
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <constant symbol="1"/>
                                 </function>
                                </function>
                               </function>
                              </function>
                              <constant symbol="0"/>
                             </constantatomformula>
                             <constantatomformula symbol="A" definition="A"/>
                            </formulalist>
                            <formulalist>
                             <quantifiedformula type="exists">
                              <variable symbol="p"/>
                              <quantifiedformula type="exists">
                               <variable symbol="q"/>
                               <conjunctiveformula type="and">
                                <conjunctiveformula type="neg">
                                 <constantatomformula symbol="=">
                                  <variable symbol="p"/>
                                  <variable symbol="q"/>
                                 </constantatomformula>
                                </conjunctiveformula>
                                <constantatomformula symbol="=">
                                 <function symbol="f">
                                  <variable symbol="p"/>
                                 </function>
                                 <function symbol="f">
                                  <variable symbol="q"/>
                                 </function>
                                </constantatomformula>
                               </conjunctiveformula>
                              </quantifiedformula>
                             </quantifiedformula>
                             <constantatomformula symbol="=">
                              <function symbol="f">
                               <function symbol="+">
                                <constant symbol="1"/>
                                <function symbol="+">
                                 <variable symbol="x_{112}"/>
                                 <function symbol="+">
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <function symbol="+">
                                    <variable symbol="x_{110}"/>
                                    <function symbol="+">
                                     <constant symbol="1"/>
                                     <function symbol="+">
                                      <constant symbol="1"/>
                                      <variable symbol="x_{111}"/>
                                     </function>
                                    </function>
                                   </function>
                                  </function>
                                  <variable symbol="x_{109}"/>
                                 </function>
                                </function>
                               </function>
                              </function>
                              <constant symbol="1"/>
                             </constantatomformula>
                             <quantifiedformula type="exists">
                              <variable symbol="p"/>
                              <quantifiedformula type="exists">
                               <variable symbol="q"/>
                               <conjunctiveformula type="and">
                                <conjunctiveformula type="neg">
                                 <constantatomformula symbol="=">
                                  <variable symbol="p"/>
                                  <variable symbol="q"/>
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                                </conjunctiveformula>
                                <constantatomformula symbol="=">
                                 <function symbol="f">
                                  <variable symbol="p"/>
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                                 <function symbol="f">
                                  <variable symbol="q"/>
                                 </function>
                                </constantatomformula>
                               </conjunctiveformula>
                              </quantifiedformula>
                             </quantifiedformula>
                            </formulalist>
                           </sequent>
                           <rule type="perml" symbol="\pi:l" param="(1 2)">
                            <sequent>
                             <formulalist>
                              <constantatomformula symbol="=">
                               <function symbol="f">
                                <function symbol="+">
                                 <function symbol="+">
                                  <variable symbol="x_{112}"/>
                                  <function symbol="+">
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <function symbol="+">
                                     <variable symbol="x_{110}"/>
                                     <function symbol="+">
                                      <constant symbol="1"/>
                                      <function symbol="+">
                                       <constant symbol="1"/>
                                       <variable symbol="x_{111}"/>
                                      </function>
                                     </function>
                                    </function>
                                   </function>
                                   <variable symbol="x_{109}"/>
                                  </function>
                                 </function>
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <constant symbol="1"/>
                                  </function>
                                 </function>
                                </function>
                               </function>
                               <constant symbol="0"/>
                              </constantatomformula>
                              <constantatomformula symbol="A" definition="A"/>
                              <constantatomformula symbol="A" definition="A"/>
                             </formulalist>
                             <formulalist>
                              <quantifiedformula type="exists">
                               <variable symbol="p"/>
                               <quantifiedformula type="exists">
                                <variable symbol="q"/>
                                <conjunctiveformula type="and">
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                                  <constantatomformula symbol="=">
                                   <variable symbol="p"/>
                                   <variable symbol="q"/>
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                                 </conjunctiveformula>
                                 <constantatomformula symbol="=">
                                  <function symbol="f">
                                   <variable symbol="p"/>
                                  </function>
                                  <function symbol="f">
                                   <variable symbol="q"/>
                                  </function>
                                 </constantatomformula>
                                </conjunctiveformula>
                               </quantifiedformula>
                              </quantifiedformula>
                              <constantatomformula symbol="=">
                               <function symbol="f">
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <function symbol="+">
                                  <variable symbol="x_{112}"/>
                                  <function symbol="+">
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <function symbol="+">
                                     <variable symbol="x_{110}"/>
                                     <function symbol="+">
                                      <constant symbol="1"/>
                                      <function symbol="+">
                                       <constant symbol="1"/>
                                       <variable symbol="x_{111}"/>
                                      </function>
                                     </function>
                                    </function>
                                   </function>
                                   <variable symbol="x_{109}"/>
                                  </function>
                                 </function>
                                </function>
                               </function>
                               <constant symbol="1"/>
                              </constantatomformula>
                              <quantifiedformula type="exists">
                               <variable symbol="p"/>
                               <quantifiedformula type="exists">
                                <variable symbol="q"/>
                                <conjunctiveformula type="and">
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                                  <constantatomformula symbol="=">
                                   <variable symbol="p"/>
                                   <variable symbol="q"/>
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                                 </conjunctiveformula>
                                 <constantatomformula symbol="=">
                                  <function symbol="f">
                                   <variable symbol="p"/>
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                                  <function symbol="f">
                                   <variable symbol="q"/>
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                                 </constantatomformula>
                                </conjunctiveformula>
                               </quantifiedformula>
                              </quantifiedformula>
                             </formulalist>
                            </sequent>
                            <rule type="cut" symbol="cut">
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                              <formulalist>
                               <constantatomformula symbol="A" definition="A"/>
                               <constantatomformula symbol="=">
                                <function symbol="f">
                                 <function symbol="+">
                                  <function symbol="+">
                                   <variable symbol="x_{112}"/>
                                   <function symbol="+">
                                    <function symbol="+">
                                     <constant symbol="1"/>
                                     <function symbol="+">
                                      <variable symbol="x_{110}"/>
                                      <function symbol="+">
                                       <constant symbol="1"/>
                                       <function symbol="+">
                                        <constant symbol="1"/>
                                        <variable symbol="x_{111}"/>
                                       </function>
                                      </function>
                                     </function>
                                    </function>
                                    <variable symbol="x_{109}"/>
                                   </function>
                                  </function>
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <constant symbol="1"/>
                                   </function>
                                  </function>
                                 </function>
                                </function>
                                <constant symbol="0"/>
                               </constantatomformula>
                               <constantatomformula symbol="A" definition="A"/>
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                              <formulalist>
                               <quantifiedformula type="exists">
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                                <quantifiedformula type="exists">
                                 <variable symbol="q"/>
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                                    <variable symbol="q"/>
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                                  </conjunctiveformula>
                                  <constantatomformula symbol="=">
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                                    <variable symbol="p"/>
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                                   <function symbol="f">
                                    <variable symbol="q"/>
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                                 </conjunctiveformula>
                                </quantifiedformula>
                               </quantifiedformula>
                               <constantatomformula symbol="=">
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                                  <function symbol="+">
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                                   <function symbol="+">
                                    <function symbol="+">
                                     <constant symbol="1"/>
                                     <function symbol="+">
                                      <variable symbol="x_{110}"/>
                                      <function symbol="+">
                                       <constant symbol="1"/>
                                       <function symbol="+">
                                        <constant symbol="1"/>
                                        <variable symbol="x_{111}"/>
                                       </function>
                                      </function>
                                     </function>
                                    </function>
                                    <variable symbol="x_{109}"/>
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                                  </function>
                                 </function>
                                </function>
                                <constant symbol="1"/>
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                               <quantifiedformula type="exists">
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                                <quantifiedformula type="exists">
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                                    <variable symbol="q"/>
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                                  <constantatomformula symbol="=">
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                                    <variable symbol="p"/>
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                                   <function symbol="f">
                                    <variable symbol="q"/>
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                                  </constantatomformula>
                                 </conjunctiveformula>
                                </quantifiedformula>
                               </quantifiedformula>
                              </formulalist>
                             </sequent>
                             <rule type="permr" symbol="\pi:r" param="(2 3)">
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                               <formulalist>
                                <constantatomformula symbol="A" definition="A"/>
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                               <formulalist>
                                <quantifiedformula type="exists">
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                                 <quantifiedformula type="exists">
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                                    <constantatomformula symbol="=">
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                                   <constantatomformula symbol="=">
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                                     <variable symbol="p"/>
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                                    <function symbol="f">
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                                   </constantatomformula>
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                                 </quantifiedformula>
                                </quantifiedformula>
                                <constantatomformula symbol="=">
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                                   <function symbol="+">
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                                    <function symbol="+">
                                     <function symbol="+">
                                      <constant symbol="1"/>
                                      <function symbol="+">
                                       <variable symbol="x_{110}"/>
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                                        <function symbol="+">
                                         <constant symbol="1"/>
                                         <variable symbol="x_{111}"/>
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                                       </function>
                                      </function>
                                     </function>
                                     <variable symbol="x_{109}"/>
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                                   </function>
                                  </function>
                                 </function>
                                 <constant symbol="1"/>
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                                <constantatomformula symbol="=">
                                 <function symbol="f">
                                  <function symbol="+">
                                   <function symbol="+">
                                    <variable symbol="x_{112}"/>
                                    <function symbol="+">
                                     <function symbol="+">
                                      <constant symbol="1"/>
                                      <function symbol="+">
                                       <variable symbol="x_{110}"/>
                                       <function symbol="+">
                                        <constant symbol="1"/>
                                        <function symbol="+">
                                         <constant symbol="1"/>
                                         <variable symbol="x_{111}"/>
                                        </function>
                                       </function>
                                      </function>
                                     </function>
                                     <variable symbol="x_{109}"/>
                                    </function>
                                   </function>
                                   <constant symbol="1"/>
                                  </function>
                                 </function>
                                 <constant symbol="0"/>
                                </constantatomformula>
                               </formulalist>
                              </sequent>
                              <rule type="permr" symbol="\pi:r" param="(1 3 2)">
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                                <formulalist>
                                 <constantatomformula symbol="A" definition="A"/>
                                </formulalist>
                                <formulalist>
                                 <quantifiedformula type="exists">
                                  <variable symbol="p"/>
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                                    <constantatomformula symbol="=">
                                     <function symbol="f">
                                      <variable symbol="p"/>
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                                     <function symbol="f">
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                                    </constantatomformula>
                                   </conjunctiveformula>
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                                 </quantifiedformula>
                                 <constantatomformula symbol="=">
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                                    <function symbol="+">
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                                     <function symbol="+">
                                      <function symbol="+">
                                       <constant symbol="1"/>
                                       <function symbol="+">
                                        <variable symbol="x_{110}"/>
                                        <function symbol="+">
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                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <variable symbol="x_{111}"/>
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                                        </function>
                                       </function>
                                      </function>
                                      <variable symbol="x_{109}"/>
                                     </function>
                                    </function>
                                    <constant symbol="1"/>
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                                  </function>
                                  <constant symbol="0"/>
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                                 <constantatomformula symbol="=">
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                                    <function symbol="+">
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                                     <function symbol="+">
                                      <function symbol="+">
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                                       <function symbol="+">
                                        <variable symbol="x_{110}"/>
                                        <function symbol="+">
                                         <constant symbol="1"/>
                                         <function symbol="+">
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                                          <variable symbol="x_{111}"/>
                                         </function>
                                        </function>
                                       </function>
                                      </function>
                                      <variable symbol="x_{109}"/>
                                     </function>
                                    </function>
                                   </function>
                                  </function>
                                  <constant symbol="1"/>
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                                  <constantatomformula symbol="A" definition="A"/>
                                 </formulalist>
                                 <formulalist>
                                  <constantatomformula symbol="=">
                                   <function symbol="f">
                                    <function symbol="+">
                                     <function symbol="+">
                                      <variable symbol="x_{112}"/>
                                      <function symbol="+">
                                       <function symbol="+">
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                                        <function symbol="+">
                                         <variable symbol="x_{110}"/>
                                         <function symbol="+">
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                                       </function>
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                                     <constant symbol="1"/>
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                                   </function>
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                                  <constantatomformula symbol="=">
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                                     <function symbol="+">
                                      <variable symbol="x_{112}"/>
                                      <function symbol="+">
                                       <function symbol="+">
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                                        <function symbol="+">
                                         <variable symbol="x_{110}"/>
                                         <function symbol="+">
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                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <variable symbol="x_{111}"/>
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                                         </function>
                                        </function>
                                       </function>
                                       <variable symbol="x_{109}"/>
                                      </function>
                                     </function>
                                    </function>
                                   </function>
                                   <constant symbol="1"/>
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                                  <quantifiedformula type="exists">
                                   <variable symbol="p"/>
                                   <quantifiedformula type="exists">
                                    <variable symbol="q"/>
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                                      <constantatomformula symbol="=">
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                                       <variable symbol="q"/>
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                                     <constantatomformula symbol="=">
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                                  </quantifiedformula>
                                 </formulalist>
                                </sequent>
                                <rule type="contrr" symbol="c:r" param="1,1,2">
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                                  <formulalist>
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                                  <formulalist>
                                   <constantatomformula symbol="=">
                                    <function symbol="f">
                                     <function symbol="+">
                                      <function symbol="+">
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                                        <function symbol="+">
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                                        </function>
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                                   <constantatomformula symbol="=">
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                                      <function symbol="+">
                                       <variable symbol="x_{112}"/>
                                       <function symbol="+">
                                        <function symbol="+">
                                         <constant symbol="1"/>
                                         <function symbol="+">
                                          <variable symbol="x_{110}"/>
                                          <function symbol="+">
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                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <variable symbol="x_{111}"/>
                                           </function>
                                          </function>
                                         </function>
                                        </function>
                                        <variable symbol="x_{109}"/>
                                       </function>
                                      </function>
                                     </function>
                                    </function>
                                    <constant symbol="1"/>
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                                   <quantifiedformula type="exists">
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                                      <constantatomformula symbol="=">
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                                   </quantifiedformula>
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                                 </sequent>
                                 <rule type="weakr" symbol="w:r">
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                                   <formulalist>
                                    <constantatomformula symbol="A" definition="A"/>
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                                   <formulalist>
                                    <constantatomformula symbol="=">
                                     <function symbol="f">
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                                       <function symbol="+">
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                                        <function symbol="+">
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                                         </function>
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                                    <constantatomformula symbol="=">
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                                       <function symbol="+">
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                                        <function symbol="+">
                                         <function symbol="+">
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                                           <variable symbol="x_{110}"/>
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                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <variable symbol="x_{111}"/>
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                                           </function>
                                          </function>
                                         </function>
                                         <variable symbol="x_{109}"/>
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                                       </function>
                                      </function>
                                     </function>
                                     <constant symbol="1"/>
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                                       <constantatomformula symbol="=">
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                                    <quantifiedformula type="exists">
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                                       <constantatomformula symbol="=">
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                                  <rule type="weakr" symbol="w:r">
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                                    <formulalist>
                                     <constantatomformula symbol="A" definition="A"/>
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                                    <formulalist>
                                     <constantatomformula symbol="=">
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                                            <variable symbol="x_{110}"/>
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                                              <variable symbol="x_{111}"/>
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                                     <constantatomformula symbol="=">
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                                           </function>
                                          </function>
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                                   <rule type="defl" symbol="d:l" param="();[()];()">
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                                     <formulalist>
                                      <constantatomformula symbol="A" definition="A"/>
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                                     <formulalist>
                                      <constantatomformula symbol="=">
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                                       <constant symbol="1"/>
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                                    <rule type="foralll" symbol="\all:l">
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                                        <constantatomformula symbol="=">
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                                                <function symbol="+">
                                                 <variable symbol="x_{110}"/>
                                                 <function symbol="+">
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                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{111}"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                               <variable symbol="x_{109}"/>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
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                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
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                                  <function symbol="+">
                                   <function symbol="+">
                                    <variable symbol="x_{112}"/>
                                    <function symbol="+">
                                     <function symbol="+">
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                                      <function symbol="+">
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                                       <function symbol="+">
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                                        <function symbol="+">
                                         <constant symbol="1"/>
                                         <variable symbol="x_{111}"/>
                                        </function>
                                       </function>
                                      </function>
                                     </function>
                                     <variable symbol="x_{109}"/>
                                    </function>
                                   </function>
                                   <constant symbol="1"/>
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                                 </function>
                                 <constant symbol="0"/>
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                                <constantatomformula symbol="=">
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                                  <function symbol="+">
                                   <function symbol="+">
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                                    <function symbol="+">
                                     <function symbol="+">
                                      <constant symbol="1"/>
                                      <function symbol="+">
                                       <variable symbol="x_{110}"/>
                                       <function symbol="+">
                                        <constant symbol="1"/>
                                        <function symbol="+">
                                         <constant symbol="1"/>
                                         <variable symbol="x_{111}"/>
                                        </function>
                                       </function>
                                      </function>
                                     </function>
                                     <variable symbol="x_{109}"/>
                                    </function>
                                   </function>
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <function symbol="+">
                                     <constant symbol="1"/>
                                     <constant symbol="1"/>
                                    </function>
                                   </function>
                                  </function>
                                 </function>
                                 <constant symbol="0"/>
                                </constantatomformula>
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                                <quantifiedformula type="exists">
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                                 <quantifiedformula type="exists">
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                                   <constantatomformula symbol="=">
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                                    <function symbol="f">
                                     <variable symbol="q"/>
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                                   <function symbol="+">
                                    <function symbol="+">
                                     <variable symbol="x_{112}"/>
                                     <function symbol="+">
                                      <function symbol="+">
                                       <constant symbol="1"/>
                                       <function symbol="+">
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                                        <function symbol="+">
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                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <variable symbol="x_{111}"/>
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                                        </function>
                                       </function>
                                      </function>
                                      <variable symbol="x_{109}"/>
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                                    </function>
                                    <function symbol="+">
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                                     <function symbol="+">
                                      <constant symbol="1"/>
                                      <constant symbol="1"/>
                                     </function>
                                    </function>
                                   </function>
                                  </function>
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                                 <constantatomformula symbol="=">
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                                    <function symbol="+">
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                                     <function symbol="+">
                                      <function symbol="+">
                                       <constant symbol="1"/>
                                       <function symbol="+">
                                        <variable symbol="x_{110}"/>
                                        <function symbol="+">
                                         <constant symbol="1"/>
                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <variable symbol="x_{111}"/>
                                         </function>
                                        </function>
                                       </function>
                                      </function>
                                      <variable symbol="x_{109}"/>
                                     </function>
                                    </function>
                                    <constant symbol="1"/>
                                   </function>
                                  </function>
                                  <constant symbol="0"/>
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                                <formulalist>
                                 <quantifiedformula type="exists">
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                                  <quantifiedformula type="exists">
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                                      <variable symbol="q"/>
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                                    <constantatomformula symbol="=">
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                                      <variable symbol="p"/>
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                                     <function symbol="f">
                                      <variable symbol="q"/>
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                                   </conjunctiveformula>
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                                 <formulalist>
                                  <constantatomformula symbol="=">
                                   <function symbol="f">
                                    <function symbol="+">
                                     <function symbol="+">
                                      <variable symbol="x_{112}"/>
                                      <function symbol="+">
                                       <function symbol="+">
                                        <constant symbol="1"/>
                                        <function symbol="+">
                                         <variable symbol="x_{110}"/>
                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <variable symbol="x_{111}"/>
                                          </function>
                                         </function>
                                        </function>
                                       </function>
                                       <variable symbol="x_{109}"/>
                                      </function>
                                     </function>
                                     <function symbol="+">
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                                      <function symbol="+">
                                       <constant symbol="1"/>
                                       <constant symbol="1"/>
                                      </function>
                                     </function>
                                    </function>
                                   </function>
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                                  <constantatomformula symbol="=">
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                                     <function symbol="+">
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                                      <function symbol="+">
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                                        <function symbol="+">
                                         <variable symbol="x_{110}"/>
                                         <function symbol="+">
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                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <variable symbol="x_{111}"/>
                                          </function>
                                         </function>
                                        </function>
                                       </function>
                                       <variable symbol="x_{109}"/>
                                      </function>
                                     </function>
                                     <constant symbol="1"/>
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                                   </function>
                                   <constant symbol="0"/>
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                                 <formulalist>
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                                     <constantatomformula symbol="=">
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                                      <function symbol="+">
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                                       <function symbol="+">
                                        <function symbol="+">
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                                          <function symbol="+">
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                                         </function>
                                        </function>
                                        <variable symbol="x_{109}"/>
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                                      </function>
                                      <function symbol="+">
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                                       <function symbol="+">
                                        <constant symbol="1"/>
                                        <constant symbol="1"/>
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                                    </function>
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                                         <constant symbol="1"/>
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                                          <constant symbol="1"/>
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                                        <formulalist>
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                                            <constantatomformula symbol="=">
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                                             <function symbol="+">
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                                                <function symbol="+">
                                                 <variable symbol="x_{110}"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{111}"/>
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                                                </function>
                                               </function>
                                               <variable symbol="x_{109}"/>
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                                             </function>
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                                              <function symbol="+">
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                                               <constant symbol="1"/>
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                                              <constant symbol="1"/>
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                                             <function symbol="+">
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                                              <function symbol="+">
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                                                 <function symbol="+">
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                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{111}"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
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                                             </function>
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                                           <constantatomformula symbol="=">
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                                             <function symbol="+">
                                              <function symbol="+">
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                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <variable symbol="x_{109}"/>
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                                              </function>
                                              <function symbol="+">
                                               <function symbol="+">
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                                                <constant symbol="1"/>
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                                               <constant symbol="1"/>
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                                             </function>
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                                              <function symbol="+">
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                                               <function symbol="+">
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                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
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                                                <function symbol="+">
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                                                  <function symbol="+">
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                                                   <function symbol="+">
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                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
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                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
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                                                   </function>
                                                  </function>
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                                                  </function>
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                                              <function symbol="+">
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                                                <function symbol="+">
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                                                    </function>
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                                                  </function>
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                                                  </function>
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                                                     </function>
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                                                   </function>
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                                                   <constantatomformula symbol="=">
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                                                       </function>
                                                      </function>
                                                     </function>
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                                                   </function>
                                                   <function symbol="+">
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                                                  </function>
                                                  <constant symbol="1"/>
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                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{112}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
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                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="eqr1" symbol="=:r1" param="(1)">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{112}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="weakr" symbol="w:r">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </constantatomformula>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="perml" symbol="\pi:l" param="">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="weakl" symbol="w:l">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="axiom" symbol="axiom">
                                                    <sequent>
                                                     <formulalist/>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                               <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="weakr" symbol="w:r">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="perml" symbol="\pi:l" param="">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="weakl" symbol="w:l">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="axiom" symbol="axiom">
                                                    <sequent>
                                                     <formulalist/>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                          <rule type="perml" symbol="\pi:l" param="(1 2)">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <variable symbol="x_{112}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{109}"/>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                              <constant symbol="0"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <variable symbol="x_{112}"/>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <variable symbol="x_{109}"/>
                                                 </function>
                                                </function>
                                                <constant symbol="1"/>
                                               </function>
                                              </function>
                                              <constant symbol="0"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="A" definition="A"/>
                                            </formulalist>
                                            <formulalist>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="perml" symbol="\pi:l" param="(1 2)">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <variable symbol="x_{112}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{109}"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                               <constant symbol="0"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{112}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="1"/>
                                                </function>
                                               </function>
                                               <constant symbol="0"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="A" definition="A"/>
                                             </formulalist>
                                             <formulalist>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="contrr" symbol="c:r" param="2">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{112}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                                <constant symbol="0"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <variable symbol="x_{112}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="1"/>
                                                 </function>
                                                </function>
                                                <constant symbol="0"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="A" definition="A"/>
                                              </formulalist>
                                              <formulalist>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="permr" symbol="\pi:r" param="">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <variable symbol="x_{112}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </function>
                                                 </function>
                                                 <constant symbol="0"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="contrl" symbol="c:l" param="1,1,2">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{112}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                  </function>
                                                  <constant symbol="0"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="A" definition="A"/>
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                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="perml" symbol="\pi:l" param="(2 3)">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{112}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="0"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="eql1" symbol="=:l1" param="(1 1 1)">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="0"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="contrr" symbol="c:r" param="2,1">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{112}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="permr" symbol="\pi:r" param="">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="contrl" symbol="c:l" param="2">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="perml" symbol="\pi:l" param="">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="eqr1" symbol="=:r1" param="(1)">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                        </constantatomformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
                                                            <conjunctiveformula type="neg">
                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="weakr" symbol="w:r">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </constantatomformula>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
                                                            <conjunctiveformula type="and">
                                                             <conjunctiveformula type="neg">
                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="perml" symbol="\pi:l" param="">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="weakl" symbol="w:l">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="axiom" symbol="axiom">
                                                           <sequent>
                                                            <formulalist/>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{112}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <variable symbol="x_{109}"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                      <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
                                                            <conjunctiveformula type="neg">
                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="weakr" symbol="w:r">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </constantatomformula>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
                                                            <conjunctiveformula type="and">
                                                             <conjunctiveformula type="neg">
                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="perml" symbol="\pi:l" param="">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="weakl" symbol="w:l">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{112}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <variable symbol="x_{109}"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="axiom" symbol="axiom">
                                                           <sequent>
                                                            <formulalist/>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{112}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <variable symbol="x_{109}"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                 <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{112}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="0"/>
                                                    </constantatomformula>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="permr" symbol="\pi:r" param="">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{112}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="0"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="perml" symbol="\pi:l" param="(1 2 3)">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{112}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{109}"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{112}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
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                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="0"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
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                                                         <conjunctiveformula type="neg">
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                                                           <variable symbol="q"/>
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                                                         <constantatomformula symbol="=">
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                                                           <variable symbol="p"/>
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                                                          <function symbol="f">
                                                           <variable symbol="q"/>
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                                                         </constantatomformula>
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                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="contrr" symbol="c:r" param="2">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{112}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{109}"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="0"/>
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                                                       <constantatomformula symbol="=">
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                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{112}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
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                                                               <function symbol="+">
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                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <variable symbol="x_{109}"/>
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                                                            </function>
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                                                           <constant symbol="1"/>
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                                                          <constant symbol="1"/>
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                                                          <constantatomformula symbol="=">
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                                                     <rule type="weakr" symbol="w:r">
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                                                        <constantatomformula symbol="A" definition="A"/>
                                                        <constantatomformula symbol="=">
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                                                           <function symbol="+">
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                                                               <variable symbol="x_{110}"/>
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                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{109}"/>
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                                                           <constant symbol="1"/>
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                                                         </function>
                                                         <constant symbol="0"/>
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                                                        <constantatomformula symbol="=">
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                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{112}"/>
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                                                               <function symbol="+">
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                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
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                                                                   <variable symbol="x_{111}"/>
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                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <variable symbol="x_{109}"/>
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                                                             <constant symbol="1"/>
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                                                           <constant symbol="1"/>
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                                                         <constant symbol="0"/>
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                                                        <quantifiedformula type="exists">
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                                                           <constantatomformula symbol="=">
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                                                        <quantifiedformula type="exists">
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                                                           <constantatomformula symbol="=">
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                                                           </constantatomformula>
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                                                         </quantifiedformula>
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                                                      </sequent>
                                                      <rule type="weakl" symbol="w:l">
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                                                         <constantatomformula symbol="A" definition="A"/>
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                                                          <rule type="negr" symbol="\neg:r">
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                                                                <constant symbol="1"/>
                                                               </function>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="eql1" symbol="=:l1" param="(2)">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{112}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <variable symbol="x_{109}"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </formulalist>
                                                             <formulalist/>
                                                            </sequent>
                                                            <rule type="axiom" symbol="axiom">
                                                             <sequent>
                                                              <formulalist/>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </formulalist>
                                                             </sequent>
                                                            </rule>
                                                            <rule type="axiom" symbol="axiom">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </constantatomformula>
                                                              </formulalist>
                                                              <formulalist/>
                                                             </sequent>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                          <rule type="eqr2" symbol="=:r2" param="(1)">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{112}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <variable symbol="x_{109}"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="0"/>
                                                             </constantatomformula>
                                                            </formulalist>
                                                            <formulalist>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{112}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <variable symbol="x_{109}"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{112}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <variable symbol="x_{109}"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="axiom" symbol="axiom">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <constant symbol="0"/>
                                                              </constantatomformula>
                                                             </formulalist>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{112}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <variable symbol="x_{109}"/>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <constant symbol="0"/>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                           </rule>
                                                           <rule type="eqr2" symbol="=:r2" param="(2)">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <constant symbol="0"/>
                                                              </constantatomformula>
                                                             </formulalist>
                                                             <formulalist>
                                                              <constantatomformula symbol="=">
                                                               <constant symbol="0"/>
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{112}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="x_{109}"/>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="axiom" symbol="axiom">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="0"/>
                                                               </constantatomformula>
                                                              </formulalist>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{112}"/>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <variable symbol="x_{109}"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="0"/>
                                                               </constantatomformula>
                                                              </formulalist>
                                                             </sequent>
                                                            </rule>
                                                            <rule type="axiom" symbol="axiom">
                                                             <sequent>
                                                              <formulalist/>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <constant symbol="0"/>
                                                                <constant symbol="0"/>
                                                               </constantatomformula>
                                                              </formulalist>
                                                             </sequent>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                          </rule>
                         </rule>
                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                   <rule type="perml" symbol="\pi:l" param="(1 2)">
                    <sequent>
                     <formulalist>
                      <constantatomformula symbol="=">
                       <function symbol="f">
                        <function symbol="+">
                         <constant symbol="1"/>
                         <function symbol="+">
                          <variable symbol="x_{112}"/>
                          <function symbol="+">
                           <function symbol="+">
                            <constant symbol="1"/>
                            <function symbol="+">
                             <variable symbol="x_{110}"/>
                             <function symbol="+">
                              <constant symbol="1"/>
                              <function symbol="+">
                               <constant symbol="1"/>
                               <variable symbol="x_{111}"/>
                              </function>
                             </function>
                            </function>
                           </function>
                           <variable symbol="x_{109}"/>
                          </function>
                         </function>
                        </function>
                       </function>
                       <constant symbol="1"/>
                      </constantatomformula>
                      <constantatomformula symbol="=">
                       <function symbol="f">
                        <function symbol="+">
                         <variable symbol="x_{109}"/>
                         <function symbol="+">
                          <constant symbol="1"/>
                          <function symbol="+">
                           <variable symbol="x_{110}"/>
                           <function symbol="+">
                            <constant symbol="1"/>
                            <function symbol="+">
                             <constant symbol="1"/>
                             <variable symbol="x_{111}"/>
                            </function>
                           </function>
                          </function>
                         </function>
                        </function>
                       </function>
                       <constant symbol="1"/>
                      </constantatomformula>
                      <constantatomformula symbol="A" definition="A"/>
                     </formulalist>
                     <formulalist>
                      <quantifiedformula type="exists">
                       <variable symbol="p"/>
                       <quantifiedformula type="exists">
                        <variable symbol="q"/>
                        <conjunctiveformula type="and">
                         <conjunctiveformula type="neg">
                          <constantatomformula symbol="=">
                           <variable symbol="p"/>
                           <variable symbol="q"/>
                          </constantatomformula>
                         </conjunctiveformula>
                         <constantatomformula symbol="=">
                          <function symbol="f">
                           <variable symbol="p"/>
                          </function>
                          <function symbol="f">
                           <variable symbol="q"/>
                          </function>
                         </constantatomformula>
                        </conjunctiveformula>
                       </quantifiedformula>
                      </quantifiedformula>
                     </formulalist>
                    </sequent>
                    <rule type="perml" symbol="\pi:l" param="(1 2)">
                     <sequent>
                      <formulalist>
                       <constantatomformula symbol="=">
                        <function symbol="f">
                         <function symbol="+">
                          <variable symbol="x_{109}"/>
                          <function symbol="+">
                           <constant symbol="1"/>
                           <function symbol="+">
                            <variable symbol="x_{110}"/>
                            <function symbol="+">
                             <constant symbol="1"/>
                             <function symbol="+">
                              <constant symbol="1"/>
                              <variable symbol="x_{111}"/>
                             </function>
                            </function>
                           </function>
                          </function>
                         </function>
                        </function>
                        <constant symbol="1"/>
                       </constantatomformula>
                       <constantatomformula symbol="=">
                        <function symbol="f">
                         <function symbol="+">
                          <constant symbol="1"/>
                          <function symbol="+">
                           <variable symbol="x_{112}"/>
                           <function symbol="+">
                            <function symbol="+">
                             <constant symbol="1"/>
                             <function symbol="+">
                              <variable symbol="x_{110}"/>
                              <function symbol="+">
                               <constant symbol="1"/>
                               <function symbol="+">
                                <constant symbol="1"/>
                                <variable symbol="x_{111}"/>
                               </function>
                              </function>
                             </function>
                            </function>
                            <variable symbol="x_{109}"/>
                           </function>
                          </function>
                         </function>
                        </function>
                        <constant symbol="1"/>
                       </constantatomformula>
                       <constantatomformula symbol="A" definition="A"/>
                      </formulalist>
                      <formulalist>
                       <quantifiedformula type="exists">
                        <variable symbol="p"/>
                        <quantifiedformula type="exists">
                         <variable symbol="q"/>
                         <conjunctiveformula type="and">
                          <conjunctiveformula type="neg">
                           <constantatomformula symbol="=">
                            <variable symbol="p"/>
                            <variable symbol="q"/>
                           </constantatomformula>
                          </conjunctiveformula>
                          <constantatomformula symbol="=">
                           <function symbol="f">
                            <variable symbol="p"/>
                           </function>
                           <function symbol="f">
                            <variable symbol="q"/>
                           </function>
                          </constantatomformula>
                         </conjunctiveformula>
                        </quantifiedformula>
                       </quantifiedformula>
                      </formulalist>
                     </sequent>
                     <rule type="contrr" symbol="c:r" param="2">
                      <sequent>
                       <formulalist>
                        <constantatomformula symbol="=">
                         <function symbol="f">
                          <function symbol="+">
                           <constant symbol="1"/>
                           <function symbol="+">
                            <variable symbol="x_{112}"/>
                            <function symbol="+">
                             <function symbol="+">
                              <constant symbol="1"/>
                              <function symbol="+">
                               <variable symbol="x_{110}"/>
                               <function symbol="+">
                                <constant symbol="1"/>
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <variable symbol="x_{111}"/>
                                </function>
                               </function>
                              </function>
                             </function>
                             <variable symbol="x_{109}"/>
                            </function>
                           </function>
                          </function>
                         </function>
                         <constant symbol="1"/>
                        </constantatomformula>
                        <constantatomformula symbol="=">
                         <function symbol="f">
                          <function symbol="+">
                           <variable symbol="x_{109}"/>
                           <function symbol="+">
                            <constant symbol="1"/>
                            <function symbol="+">
                             <variable symbol="x_{110}"/>
                             <function symbol="+">
                              <constant symbol="1"/>
                              <function symbol="+">
                               <constant symbol="1"/>
                               <variable symbol="x_{111}"/>
                              </function>
                             </function>
                            </function>
                           </function>
                          </function>
                         </function>
                         <constant symbol="1"/>
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                        <constantatomformula symbol="A" definition="A"/>
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                         <quantifiedformula type="exists">
                          <variable symbol="q"/>
                          <conjunctiveformula type="and">
                           <conjunctiveformula type="neg">
                            <constantatomformula symbol="=">
                             <variable symbol="p"/>
                             <variable symbol="q"/>
                            </constantatomformula>
                           </conjunctiveformula>
                           <constantatomformula symbol="=">
                            <function symbol="f">
                             <variable symbol="p"/>
                            </function>
                            <function symbol="f">
                             <variable symbol="q"/>
                            </function>
                           </constantatomformula>
                          </conjunctiveformula>
                         </quantifiedformula>
                        </quantifiedformula>
                       </formulalist>
                      </sequent>
                      <rule type="permr" symbol="\pi:r" param="">
                       <sequent>
                        <formulalist>
                         <constantatomformula symbol="=">
                          <function symbol="f">
                           <function symbol="+">
                            <constant symbol="1"/>
                            <function symbol="+">
                             <variable symbol="x_{112}"/>
                             <function symbol="+">
                              <function symbol="+">
                               <constant symbol="1"/>
                               <function symbol="+">
                                <variable symbol="x_{110}"/>
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <variable symbol="x_{111}"/>
                                 </function>
                                </function>
                               </function>
                              </function>
                              <variable symbol="x_{109}"/>
                             </function>
                            </function>
                           </function>
                          </function>
                          <constant symbol="1"/>
                         </constantatomformula>
                         <constantatomformula symbol="=">
                          <function symbol="f">
                           <function symbol="+">
                            <variable symbol="x_{109}"/>
                            <function symbol="+">
                             <constant symbol="1"/>
                             <function symbol="+">
                              <variable symbol="x_{110}"/>
                              <function symbol="+">
                               <constant symbol="1"/>
                               <function symbol="+">
                                <constant symbol="1"/>
                                <variable symbol="x_{111}"/>
                               </function>
                              </function>
                             </function>
                            </function>
                           </function>
                          </function>
                          <constant symbol="1"/>
                         </constantatomformula>
                         <constantatomformula symbol="A" definition="A"/>
                        </formulalist>
                        <formulalist>
                         <quantifiedformula type="exists">
                          <variable symbol="p"/>
                          <quantifiedformula type="exists">
                           <variable symbol="q"/>
                           <conjunctiveformula type="and">
                            <conjunctiveformula type="neg">
                             <constantatomformula symbol="=">
                              <variable symbol="p"/>
                              <variable symbol="q"/>
                             </constantatomformula>
                            </conjunctiveformula>
                            <constantatomformula symbol="=">
                             <function symbol="f">
                              <variable symbol="p"/>
                             </function>
                             <function symbol="f">
                              <variable symbol="q"/>
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                         <quantifiedformula type="exists">
                          <variable symbol="p"/>
                          <quantifiedformula type="exists">
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                            <conjunctiveformula type="neg">
                             <constantatomformula symbol="=">
                              <variable symbol="p"/>
                              <variable symbol="q"/>
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                            </conjunctiveformula>
                            <constantatomformula symbol="=">
                             <function symbol="f">
                              <variable symbol="p"/>
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                             <function symbol="f">
                              <variable symbol="q"/>
                             </function>
                            </constantatomformula>
                           </conjunctiveformula>
                          </quantifiedformula>
                         </quantifiedformula>
                        </formulalist>
                       </sequent>
                       <rule type="contrl" symbol="c:l" param="1,1,2">
                        <sequent>
                         <formulalist>
                          <constantatomformula symbol="=">
                           <function symbol="f">
                            <function symbol="+">
                             <constant symbol="1"/>
                             <function symbol="+">
                              <variable symbol="x_{112}"/>
                              <function symbol="+">
                               <function symbol="+">
                                <constant symbol="1"/>
                                <function symbol="+">
                                 <variable symbol="x_{110}"/>
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <variable symbol="x_{111}"/>
                                  </function>
                                 </function>
                                </function>
                               </function>
                               <variable symbol="x_{109}"/>
                              </function>
                             </function>
                            </function>
                           </function>
                           <constant symbol="1"/>
                          </constantatomformula>
                          <constantatomformula symbol="=">
                           <function symbol="f">
                            <function symbol="+">
                             <variable symbol="x_{109}"/>
                             <function symbol="+">
                              <constant symbol="1"/>
                              <function symbol="+">
                               <variable symbol="x_{110}"/>
                               <function symbol="+">
                                <constant symbol="1"/>
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <variable symbol="x_{111}"/>
                                </function>
                               </function>
                              </function>
                             </function>
                            </function>
                           </function>
                           <constant symbol="1"/>
                          </constantatomformula>
                          <constantatomformula symbol="A" definition="A"/>
                         </formulalist>
                         <formulalist>
                          <quantifiedformula type="exists">
                           <variable symbol="p"/>
                           <quantifiedformula type="exists">
                            <variable symbol="q"/>
                            <conjunctiveformula type="and">
                             <conjunctiveformula type="neg">
                              <constantatomformula symbol="=">
                               <variable symbol="p"/>
                               <variable symbol="q"/>
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                             </conjunctiveformula>
                             <constantatomformula symbol="=">
                              <function symbol="f">
                               <variable symbol="p"/>
                              </function>
                              <function symbol="f">
                               <variable symbol="q"/>
                              </function>
                             </constantatomformula>
                            </conjunctiveformula>
                           </quantifiedformula>
                          </quantifiedformula>
                          <quantifiedformula type="exists">
                           <variable symbol="p"/>
                           <quantifiedformula type="exists">
                            <variable symbol="q"/>
                            <conjunctiveformula type="and">
                             <conjunctiveformula type="neg">
                              <constantatomformula symbol="=">
                               <variable symbol="p"/>
                               <variable symbol="q"/>
                              </constantatomformula>
                             </conjunctiveformula>
                             <constantatomformula symbol="=">
                              <function symbol="f">
                               <variable symbol="p"/>
                              </function>
                              <function symbol="f">
                               <variable symbol="q"/>
                              </function>
                             </constantatomformula>
                            </conjunctiveformula>
                           </quantifiedformula>
                          </quantifiedformula>
                         </formulalist>
                        </sequent>
                        <rule type="perml" symbol="\pi:l" param="(2 3)">
                         <sequent>
                          <formulalist>
                           <constantatomformula symbol="=">
                            <function symbol="f">
                             <function symbol="+">
                              <constant symbol="1"/>
                              <function symbol="+">
                               <variable symbol="x_{112}"/>
                               <function symbol="+">
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <function symbol="+">
                                  <variable symbol="x_{110}"/>
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <variable symbol="x_{111}"/>
                                   </function>
                                  </function>
                                 </function>
                                </function>
                                <variable symbol="x_{109}"/>
                               </function>
                              </function>
                             </function>
                            </function>
                            <constant symbol="1"/>
                           </constantatomformula>
                           <constantatomformula symbol="=">
                            <function symbol="f">
                             <function symbol="+">
                              <variable symbol="x_{109}"/>
                              <function symbol="+">
                               <constant symbol="1"/>
                               <function symbol="+">
                                <variable symbol="x_{110}"/>
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <variable symbol="x_{111}"/>
                                 </function>
                                </function>
                               </function>
                              </function>
                             </function>
                            </function>
                            <constant symbol="1"/>
                           </constantatomformula>
                           <constantatomformula symbol="A" definition="A"/>
                           <constantatomformula symbol="A" definition="A"/>
                          </formulalist>
                          <formulalist>
                           <quantifiedformula type="exists">
                            <variable symbol="p"/>
                            <quantifiedformula type="exists">
                             <variable symbol="q"/>
                             <conjunctiveformula type="and">
                              <conjunctiveformula type="neg">
                               <constantatomformula symbol="=">
                                <variable symbol="p"/>
                                <variable symbol="q"/>
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                              </conjunctiveformula>
                              <constantatomformula symbol="=">
                               <function symbol="f">
                                <variable symbol="p"/>
                               </function>
                               <function symbol="f">
                                <variable symbol="q"/>
                               </function>
                              </constantatomformula>
                             </conjunctiveformula>
                            </quantifiedformula>
                           </quantifiedformula>
                           <quantifiedformula type="exists">
                            <variable symbol="p"/>
                            <quantifiedformula type="exists">
                             <variable symbol="q"/>
                             <conjunctiveformula type="and">
                              <conjunctiveformula type="neg">
                               <constantatomformula symbol="=">
                                <variable symbol="p"/>
                                <variable symbol="q"/>
                               </constantatomformula>
                              </conjunctiveformula>
                              <constantatomformula symbol="=">
                               <function symbol="f">
                                <variable symbol="p"/>
                               </function>
                               <function symbol="f">
                                <variable symbol="q"/>
                               </function>
                              </constantatomformula>
                             </conjunctiveformula>
                            </quantifiedformula>
                           </quantifiedformula>
                          </formulalist>
                         </sequent>
                         <rule type="eql1" symbol="=:l1" param="(1 1)">
                          <sequent>
                           <formulalist>
                            <constantatomformula symbol="=">
                             <function symbol="f">
                              <function symbol="+">
                               <constant symbol="1"/>
                               <function symbol="+">
                                <variable symbol="x_{112}"/>
                                <function symbol="+">
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <function symbol="+">
                                   <variable symbol="x_{110}"/>
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <function symbol="+">
                                     <constant symbol="1"/>
                                     <variable symbol="x_{111}"/>
                                    </function>
                                   </function>
                                  </function>
                                 </function>
                                 <variable symbol="x_{109}"/>
                                </function>
                               </function>
                              </function>
                             </function>
                             <constant symbol="1"/>
                            </constantatomformula>
                            <constantatomformula symbol="A" definition="A"/>
                            <constantatomformula symbol="=">
                             <function symbol="f">
                              <function symbol="+">
                               <variable symbol="x_{109}"/>
                               <function symbol="+">
                                <constant symbol="1"/>
                                <function symbol="+">
                                 <variable symbol="x_{110}"/>
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <variable symbol="x_{111}"/>
                                  </function>
                                 </function>
                                </function>
                               </function>
                              </function>
                             </function>
                             <constant symbol="1"/>
                            </constantatomformula>
                            <constantatomformula symbol="A" definition="A"/>
                           </formulalist>
                           <formulalist>
                            <quantifiedformula type="exists">
                             <variable symbol="p"/>
                             <quantifiedformula type="exists">
                              <variable symbol="q"/>
                              <conjunctiveformula type="and">
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                                <constantatomformula symbol="=">
                                 <variable symbol="p"/>
                                 <variable symbol="q"/>
                                </constantatomformula>
                               </conjunctiveformula>
                               <constantatomformula symbol="=">
                                <function symbol="f">
                                 <variable symbol="p"/>
                                </function>
                                <function symbol="f">
                                 <variable symbol="q"/>
                                </function>
                               </constantatomformula>
                              </conjunctiveformula>
                             </quantifiedformula>
                            </quantifiedformula>
                            <quantifiedformula type="exists">
                             <variable symbol="p"/>
                             <quantifiedformula type="exists">
                              <variable symbol="q"/>
                              <conjunctiveformula type="and">
                               <conjunctiveformula type="neg">
                                <constantatomformula symbol="=">
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                                 <variable symbol="q"/>
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                               </conjunctiveformula>
                               <constantatomformula symbol="=">
                                <function symbol="f">
                                 <variable symbol="p"/>
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                                <function symbol="f">
                                 <variable symbol="q"/>
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                               </constantatomformula>
                              </conjunctiveformula>
                             </quantifiedformula>
                            </quantifiedformula>
                           </formulalist>
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                          <rule type="contrr" symbol="c:r" param="2,1">
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                            <formulalist>
                             <constantatomformula symbol="A" definition="A"/>
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                            <formulalist>
                             <quantifiedformula type="exists">
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                              <quantifiedformula type="exists">
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                                <constantatomformula symbol="=">
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                                 <function symbol="f">
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                             </quantifiedformula>
                             <constantatomformula symbol="=">
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                               <function symbol="+">
                                <constant symbol="1"/>
                                <variable symbol="x_{112}"/>
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                               <function symbol="+">
                                <function symbol="+">
                                 <constant symbol="1"/>
                                 <function symbol="+">
                                  <variable symbol="x_{110}"/>
                                  <function symbol="+">
                                   <constant symbol="1"/>
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <variable symbol="x_{111}"/>
                                   </function>
                                  </function>
                                 </function>
                                </function>
                                <variable symbol="x_{109}"/>
                               </function>
                              </function>
                              <function symbol="+">
                               <constant symbol="1"/>
                               <function symbol="+">
                                <variable symbol="x_{112}"/>
                                <function symbol="+">
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <function symbol="+">
                                   <variable symbol="x_{110}"/>
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <function symbol="+">
                                     <constant symbol="1"/>
                                     <variable symbol="x_{111}"/>
                                    </function>
                                   </function>
                                  </function>
                                 </function>
                                 <variable symbol="x_{109}"/>
                                </function>
                               </function>
                              </function>
                             </constantatomformula>
                            </formulalist>
                           </sequent>
                           <rule type="permr" symbol="\pi:r" param="">
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                             <formulalist>
                              <constantatomformula symbol="A" definition="A"/>
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                             <formulalist>
                              <quantifiedformula type="exists">
                               <variable symbol="p"/>
                               <quantifiedformula type="exists">
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                                 <constantatomformula symbol="=">
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                                  <function symbol="f">
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                              <quantifiedformula type="exists">
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                              </quantifiedformula>
                              <constantatomformula symbol="=">
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                                <function symbol="+">
                                 <function symbol="+">
                                  <constant symbol="1"/>
                                  <function symbol="+">
                                   <variable symbol="x_{110}"/>
                                   <function symbol="+">
                                    <constant symbol="1"/>
                                    <function symbol="+">
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                                     <variable symbol="x_{111}"/>
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                                   </function>
                                  </function>
                                 </function>
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                               </function>
                               <function symbol="+">
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                                <function symbol="+">
                                 <variable symbol="x_{112}"/>
                                 <function symbol="+">
                                  <function symbol="+">
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                                   <function symbol="+">
                                    <variable symbol="x_{110}"/>
                                    <function symbol="+">
                                     <constant symbol="1"/>
                                     <function symbol="+">
                                      <constant symbol="1"/>
                                      <variable symbol="x_{111}"/>
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                                    </function>
                                   </function>
                                  </function>
                                  <variable symbol="x_{109}"/>
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                                </function>
                               </function>
                              </constantatomformula>
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                            </sequent>
                            <rule type="contrl" symbol="c:l" param="2">
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                              <formulalist>
                               <constantatomformula symbol="A" definition="A"/>
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                              <formulalist>
                               <quantifiedformula type="exists">
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                                  <constantatomformula symbol="=">
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                               <quantifiedformula type="exists">
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                                  <constantatomformula symbol="=">
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                               <constantatomformula symbol="=">
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                                  <function symbol="+">
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                                   <function symbol="+">
                                    <variable symbol="x_{110}"/>
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                                     <function symbol="+">
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                                    </function>
                                   </function>
                                  </function>
                                  <variable symbol="x_{109}"/>
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                                </function>
                                <function symbol="+">
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                                 <function symbol="+">
                                  <variable symbol="x_{112}"/>
                                  <function symbol="+">
                                   <function symbol="+">
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                                    <function symbol="+">
                                     <variable symbol="x_{110}"/>
                                     <function symbol="+">
                                      <constant symbol="1"/>
                                      <function symbol="+">
                                       <constant symbol="1"/>
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                                     </function>
                                    </function>
                                   </function>
                                   <variable symbol="x_{109}"/>
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                                 </function>
                                </function>
                               </constantatomformula>
                              </formulalist>
                             </sequent>
                             <rule type="perml" symbol="\pi:l" param="">
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                               <formulalist>
                                <constantatomformula symbol="A" definition="A"/>
                                <constantatomformula symbol="A" definition="A"/>
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                               <formulalist>
                                <quantifiedformula type="exists">
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                                   <constantatomformula symbol="=">
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                                     <variable symbol="q"/>
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                                <quantifiedformula type="exists">
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                                   <constantatomformula symbol="=">
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                                    <function symbol="f">
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                                <constantatomformula symbol="=">
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                                  <function symbol="+">
                                   <function symbol="+">
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                                    <function symbol="+">
                                     <variable symbol="x_{110}"/>
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                                      <function symbol="+">
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                                     </function>
                                    </function>
                                   </function>
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                                 </function>
                                 <function symbol="+">
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                                  <function symbol="+">
                                   <variable symbol="x_{112}"/>
                                   <function symbol="+">
                                    <function symbol="+">
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                                     <function symbol="+">
                                      <variable symbol="x_{110}"/>
                                      <function symbol="+">
                                       <constant symbol="1"/>
                                       <function symbol="+">
                                        <constant symbol="1"/>
                                        <variable symbol="x_{111}"/>
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                                      </function>
                                     </function>
                                    </function>
                                    <variable symbol="x_{109}"/>
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                                  </function>
                                 </function>
                                </constantatomformula>
                               </formulalist>
                              </sequent>
                              <rule type="eqr1" symbol="=:r1" param="(1)">
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                                <formulalist>
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                                 <constantatomformula symbol="A" definition="A"/>
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                                <formulalist>
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                                       </quantifiedformula>
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                                         <function symbol="+">
                                          <function symbol="+">
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                                           <function symbol="+">
                                            <variable symbol="x_{110}"/>
                                            <function symbol="+">
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                                             <function symbol="+">
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                                            </function>
                                           </function>
                                          </function>
                                          <function symbol="+">
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                                           <variable symbol="x_{112}"/>
                                          </function>
                                         </function>
                                        </function>
                                        <function symbol="+">
                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <variable symbol="x_{112}"/>
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                                         <function symbol="+">
                                          <function symbol="+">
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                                           <function symbol="+">
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                                            <function symbol="+">
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                                          </function>
                                          <variable symbol="x_{109}"/>
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                                        </function>
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                                       <formulalist>
                                        <constantatomformula symbol="A" definition="A"/>
                                        <constantatomformula symbol="A" definition="A"/>
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                                       <formulalist>
                                        <quantifiedformula type="exists">
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                                           <constantatomformula symbol="=">
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                                           <constantatomformula symbol="=">
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                                           <function symbol="+">
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                                             <function symbol="+">
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                                           <function symbol="+">
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                                      <rule type="permr" symbol="\pi:r" param="(1 2)">
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                                             </function>
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                                           </function>
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                                            <function symbol="+">
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                                             </function>
                                            </function>
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                                         <formulalist>
                                          <constantatomformula symbol="=">
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                                           <function symbol="+">
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                                             <function symbol="+">
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                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                             <function symbol="+">
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <variable symbol="x_{112}"/>
                                              </function>
                                              <function symbol="+">
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <variable symbol="x_{110}"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{111}"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                               <variable symbol="x_{109}"/>
                                              </function>
                                             </function>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="eqr1" symbol="=:r1" param="(1)">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="A" definition="A"/>
                                             <constantatomformula symbol="A" definition="A"/>
                                            </formulalist>
                                            <formulalist>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="+">
                                               <variable symbol="x_{109}"/>
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <variable symbol="x_{112}"/>
                                                </function>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                              <function symbol="+">
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <variable symbol="x_{112}"/>
                                               </function>
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <variable symbol="x_{109}"/>
                                               </function>
                                              </function>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="permr" symbol="\pi:r" param="(1 2)">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="A" definition="A"/>
                                             </formulalist>
                                             <formulalist>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <variable symbol="x_{112}"/>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <variable symbol="x_{109}"/>
                                               </function>
                                               <function symbol="+">
                                                <variable symbol="x_{109}"/>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <variable symbol="x_{112}"/>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="weakr" symbol="w:r">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="A" definition="A"/>
                                              </formulalist>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{112}"/>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <variable symbol="x_{109}"/>
                                                </function>
                                                <function symbol="+">
                                                 <variable symbol="x_{109}"/>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{112}"/>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                               </constantatomformula>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="perml" symbol="\pi:l" param="">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{112}"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <variable symbol="x_{109}"/>
                                                 </function>
                                                 <function symbol="+">
                                                  <variable symbol="x_{109}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{112}"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="weakl" symbol="w:l">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{112}"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{109}"/>
                                                  </function>
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{112}"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="axiom" symbol="axiom">
                                                <sequent>
                                                 <formulalist/>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{112}"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{112}"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                           <rule type="contrr" symbol="c:r" param="2,1">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="A" definition="A"/>
                                             </formulalist>
                                             <formulalist>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <variable symbol="x_{112}"/>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <variable symbol="x_{109}"/>
                                               </function>
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <variable symbol="x_{112}"/>
                                                </function>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <variable symbol="x_{109}"/>
                                                </function>
                                               </function>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="permr" symbol="\pi:r" param="">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="A" definition="A"/>
                                              </formulalist>
                                              <formulalist>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{112}"/>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <variable symbol="x_{109}"/>
                                                </function>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <variable symbol="x_{112}"/>
                                                 </function>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <variable symbol="x_{109}"/>
                                                 </function>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="contrl" symbol="c:l" param="2">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{112}"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <variable symbol="x_{109}"/>
                                                 </function>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{112}"/>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{109}"/>
                                                  </function>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="perml" symbol="\pi:l" param="">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{112}"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <variable symbol="x_{109}"/>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{112}"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="eqr1" symbol="=:r1" param="(1)">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{112}"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{109}"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{112}"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{112}"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{112}"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{109}"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="weakr" symbol="w:r">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{112}"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{112}"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </constantatomformula>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="perml" symbol="\pi:l" param="">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{112}"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{112}"/>
                                                        </function>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="weakl" symbol="w:l">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{112}"/>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{112}"/>
                                                         </function>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="axiom" symbol="axiom">
                                                     <sequent>
                                                      <formulalist/>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{112}"/>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{112}"/>
                                                          </function>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                                <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{112}"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{112}"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{109}"/>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="weakr" symbol="w:r">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{112}"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{112}"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{109}"/>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="perml" symbol="\pi:l" param="">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{112}"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{112}"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{109}"/>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="weakl" symbol="w:l">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{112}"/>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{112}"/>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{109}"/>
                                                        </function>
                                                       </function>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="axiom" symbol="axiom">
                                                     <sequent>
                                                      <formulalist/>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{112}"/>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{112}"/>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{109}"/>
                                                         </function>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
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                                  <sequent>
                                   <formulalist>
                                    <constantatomformula symbol="=">
                                     <function symbol="f">
                                      <function symbol="+">
                                       <variable symbol="x_{109}"/>
                                       <function symbol="+">
                                        <function symbol="+">
                                         <constant symbol="1"/>
                                         <function symbol="+">
                                          <variable symbol="x_{110}"/>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <variable symbol="x_{111}"/>
                                           </function>
                                          </function>
                                         </function>
                                        </function>
                                        <function symbol="+">
                                         <constant symbol="1"/>
                                         <variable symbol="x_{112}"/>
                                        </function>
                                       </function>
                                      </function>
                                     </function>
                                     <constant symbol="1"/>
                                    </constantatomformula>
                                    <constantatomformula symbol="=">
                                     <function symbol="f">
                                      <function symbol="+">
                                       <variable symbol="x_{109}"/>
                                       <function symbol="+">
                                        <constant symbol="1"/>
                                        <function symbol="+">
                                         <variable symbol="x_{110}"/>
                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <variable symbol="x_{111}"/>
                                          </function>
                                         </function>
                                        </function>
                                       </function>
                                      </function>
                                     </function>
                                     <constant symbol="1"/>
                                    </constantatomformula>
                                    <constantatomformula symbol="A" definition="A"/>
                                   </formulalist>
                                   <formulalist>
                                    <quantifiedformula type="exists">
                                     <variable symbol="p"/>
                                     <quantifiedformula type="exists">
                                      <variable symbol="q"/>
                                      <conjunctiveformula type="and">
                                       <conjunctiveformula type="neg">
                                        <constantatomformula symbol="=">
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                                         <variable symbol="q"/>
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                                       <constantatomformula symbol="=">
                                        <function symbol="f">
                                         <variable symbol="p"/>
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                                        <function symbol="f">
                                         <variable symbol="q"/>
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                                       </constantatomformula>
                                      </conjunctiveformula>
                                     </quantifiedformula>
                                    </quantifiedformula>
                                   </formulalist>
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                                   <sequent>
                                    <formulalist>
                                     <constantatomformula symbol="=">
                                      <function symbol="f">
                                       <function symbol="+">
                                        <variable symbol="x_{109}"/>
                                        <function symbol="+">
                                         <constant symbol="1"/>
                                         <function symbol="+">
                                          <variable symbol="x_{110}"/>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <variable symbol="x_{111}"/>
                                           </function>
                                          </function>
                                         </function>
                                        </function>
                                       </function>
                                      </function>
                                      <constant symbol="1"/>
                                     </constantatomformula>
                                     <constantatomformula symbol="=">
                                      <function symbol="f">
                                       <function symbol="+">
                                        <variable symbol="x_{109}"/>
                                        <function symbol="+">
                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <function symbol="+">
                                           <variable symbol="x_{110}"/>
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <variable symbol="x_{111}"/>
                                            </function>
                                           </function>
                                          </function>
                                         </function>
                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <variable symbol="x_{112}"/>
                                         </function>
                                        </function>
                                       </function>
                                      </function>
                                      <constant symbol="1"/>
                                     </constantatomformula>
                                     <constantatomformula symbol="A" definition="A"/>
                                    </formulalist>
                                    <formulalist>
                                     <quantifiedformula type="exists">
                                      <variable symbol="p"/>
                                      <quantifiedformula type="exists">
                                       <variable symbol="q"/>
                                       <conjunctiveformula type="and">
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                                         <constantatomformula symbol="=">
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                                          <variable symbol="q"/>
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                                        <constantatomformula symbol="=">
                                         <function symbol="f">
                                          <variable symbol="p"/>
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                                         <function symbol="f">
                                          <variable symbol="q"/>
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                                        </constantatomformula>
                                       </conjunctiveformula>
                                      </quantifiedformula>
                                     </quantifiedformula>
                                    </formulalist>
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                                    <sequent>
                                     <formulalist>
                                      <constantatomformula symbol="=">
                                       <function symbol="f">
                                        <function symbol="+">
                                         <variable symbol="x_{109}"/>
                                         <function symbol="+">
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <function symbol="+">
                                            <variable symbol="x_{110}"/>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <variable symbol="x_{111}"/>
                                             </function>
                                            </function>
                                           </function>
                                          </function>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <variable symbol="x_{112}"/>
                                          </function>
                                         </function>
                                        </function>
                                       </function>
                                       <constant symbol="1"/>
                                      </constantatomformula>
                                      <constantatomformula symbol="=">
                                       <function symbol="f">
                                        <function symbol="+">
                                         <variable symbol="x_{109}"/>
                                         <function symbol="+">
                                          <constant symbol="1"/>
                                          <function symbol="+">
                                           <variable symbol="x_{110}"/>
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <variable symbol="x_{111}"/>
                                            </function>
                                           </function>
                                          </function>
                                         </function>
                                        </function>
                                       </function>
                                       <constant symbol="1"/>
                                      </constantatomformula>
                                      <constantatomformula symbol="A" definition="A"/>
                                     </formulalist>
                                     <formulalist>
                                      <quantifiedformula type="exists">
                                       <variable symbol="p"/>
                                       <quantifiedformula type="exists">
                                        <variable symbol="q"/>
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                                          <constantatomformula symbol="=">
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                                           <variable symbol="q"/>
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                                         <constantatomformula symbol="=">
                                          <function symbol="f">
                                           <variable symbol="p"/>
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                                          <function symbol="f">
                                           <variable symbol="q"/>
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                                         </constantatomformula>
                                        </conjunctiveformula>
                                       </quantifiedformula>
                                      </quantifiedformula>
                                     </formulalist>
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                                    <rule type="permr" symbol="\pi:r" param="">
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                                      <formulalist>
                                       <constantatomformula symbol="=">
                                        <function symbol="f">
                                         <function symbol="+">
                                          <variable symbol="x_{109}"/>
                                          <function symbol="+">
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <function symbol="+">
                                             <variable symbol="x_{110}"/>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <variable symbol="x_{111}"/>
                                              </function>
                                             </function>
                                            </function>
                                           </function>
                                           <function symbol="+">
                                            <constant symbol="1"/>
                                            <variable symbol="x_{112}"/>
                                           </function>
                                          </function>
                                         </function>
                                        </function>
                                        <constant symbol="1"/>
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                                       <constantatomformula symbol="=">
                                        <function symbol="f">
                                         <function symbol="+">
                                          <variable symbol="x_{109}"/>
                                          <function symbol="+">
                                           <constant symbol="1"/>
                                           <function symbol="+">
                                            <variable symbol="x_{110}"/>
                                            <function symbol="+">
                                             <constant symbol="1"/>
                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <variable symbol="x_{111}"/>
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                                            </function>
                                           </function>
                                          </function>
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                                        </function>
                                        <constant symbol="1"/>
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                                       <constantatomformula symbol="A" definition="A"/>
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                                      <formulalist>
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                                          <function symbol="+">
                                           <variable symbol="x_{109}"/>
                                           <function symbol="+">
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                                              <variable symbol="x_{110}"/>
                                              <function symbol="+">
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                                               <function symbol="+">
                                                <constant symbol="1"/>
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                                              </function>
                                             </function>
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                                            <function symbol="+">
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                                           </function>
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                                        <constantatomformula symbol="=">
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                                             <function symbol="+">
                                              <constant symbol="1"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <variable symbol="x_{111}"/>
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                                            </function>
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                                                <function symbol="+">
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                                             </function>
                                             <function symbol="+">
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                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <variable symbol="x_{111}"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <variable symbol="x_{112}"/>
                                              </function>
                                             </function>
                                            </constantatomformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="contrl" symbol="c:l" param="2">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="A" definition="A"/>
                                            </formulalist>
                                            <formulalist>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="+">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <constant symbol="1"/>
                                               </function>
                                               <variable symbol="x_{112}"/>
                                              </function>
                                              <function symbol="+">
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <variable symbol="x_{110}"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{111}"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <variable symbol="x_{112}"/>
                                               </function>
                                              </function>
                                             </constantatomformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="perml" symbol="\pi:l" param="">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="A" definition="A"/>
                                              <constantatomformula symbol="A" definition="A"/>
                                             </formulalist>
                                             <formulalist>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <constant symbol="1"/>
                                                </function>
                                                <variable symbol="x_{112}"/>
                                               </function>
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <variable symbol="x_{112}"/>
                                                </function>
                                               </function>
                                              </constantatomformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="eqr1" symbol="=:r1" param="(1)">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="A" definition="A"/>
                                               <constantatomformula symbol="A" definition="A"/>
                                              </formulalist>
                                              <formulalist>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="1"/>
                                                 </function>
                                                 <variable symbol="x_{112}"/>
                                                </function>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <variable symbol="x_{112}"/>
                                                 </function>
                                                </function>
                                               </constantatomformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="permr" symbol="\pi:r" param="(1 2)">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{112}"/>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </function>
                                                  <variable symbol="x_{112}"/>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="weakr" symbol="w:r">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{112}"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <variable symbol="x_{112}"/>
                                                  </function>
                                                 </constantatomformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="perml" symbol="\pi:l" param="">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{112}"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <variable symbol="x_{112}"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="weakl" symbol="w:l">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{112}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <variable symbol="x_{112}"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="axiom" symbol="axiom">
                                                  <sequent>
                                                   <formulalist/>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{112}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <variable symbol="x_{112}"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                             <rule type="permr" symbol="\pi:r" param="(1 2)">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{112}"/>
                                                  </function>
                                                 </function>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{112}"/>
                                                  </function>
                                                 </function>
                                                </constantatomformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="weakr" symbol="w:r">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{112}"/>
                                                   </function>
                                                  </function>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{112}"/>
                                                   </function>
                                                  </function>
                                                 </constantatomformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="perml" symbol="\pi:l" param="">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{112}"/>
                                                    </function>
                                                   </function>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{112}"/>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="weakl" symbol="w:l">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{112}"/>
                                                     </function>
                                                    </function>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{112}"/>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="axiom" symbol="axiom">
                                                  <sequent>
                                                   <formulalist/>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{112}"/>
                                                      </function>
                                                     </function>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{112}"/>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                        <rule type="perml" symbol="\pi:l" param="(1 2)">
                                         <sequent>
                                          <formulalist>
                                           <constantatomformula symbol="=">
                                            <function symbol="f">
                                             <function symbol="+">
                                              <variable symbol="x_{109}"/>
                                              <function symbol="+">
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <constant symbol="1"/>
                                               </function>
                                               <variable symbol="x_{112}"/>
                                              </function>
                                             </function>
                                            </function>
                                            <constant symbol="1"/>
                                           </constantatomformula>
                                           <constantatomformula symbol="=">
                                            <function symbol="f">
                                             <function symbol="+">
                                              <variable symbol="x_{109}"/>
                                              <function symbol="+">
                                               <constant symbol="1"/>
                                               <function symbol="+">
                                                <variable symbol="x_{110}"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <variable symbol="x_{111}"/>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                            </function>
                                            <constant symbol="1"/>
                                           </constantatomformula>
                                           <constantatomformula symbol="A" definition="A"/>
                                          </formulalist>
                                          <formulalist>
                                           <quantifiedformula type="exists">
                                            <variable symbol="p"/>
                                            <quantifiedformula type="exists">
                                             <variable symbol="q"/>
                                             <conjunctiveformula type="and">
                                              <conjunctiveformula type="neg">
                                               <constantatomformula symbol="=">
                                                <variable symbol="p"/>
                                                <variable symbol="q"/>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <variable symbol="p"/>
                                               </function>
                                               <function symbol="f">
                                                <variable symbol="q"/>
                                               </function>
                                              </constantatomformula>
                                             </conjunctiveformula>
                                            </quantifiedformula>
                                           </quantifiedformula>
                                          </formulalist>
                                         </sequent>
                                         <rule type="perml" symbol="\pi:l" param="(1 2)">
                                          <sequent>
                                           <formulalist>
                                            <constantatomformula symbol="=">
                                             <function symbol="f">
                                              <function symbol="+">
                                               <variable symbol="x_{109}"/>
                                               <function symbol="+">
                                                <constant symbol="1"/>
                                                <function symbol="+">
                                                 <variable symbol="x_{110}"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <variable symbol="x_{111}"/>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                             </function>
                                             <constant symbol="1"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="=">
                                             <function symbol="f">
                                              <function symbol="+">
                                               <variable symbol="x_{109}"/>
                                               <function symbol="+">
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <constant symbol="1"/>
                                                </function>
                                                <variable symbol="x_{112}"/>
                                               </function>
                                              </function>
                                             </function>
                                             <constant symbol="1"/>
                                            </constantatomformula>
                                            <constantatomformula symbol="A" definition="A"/>
                                           </formulalist>
                                           <formulalist>
                                            <quantifiedformula type="exists">
                                             <variable symbol="p"/>
                                             <quantifiedformula type="exists">
                                              <variable symbol="q"/>
                                              <conjunctiveformula type="and">
                                               <conjunctiveformula type="neg">
                                                <constantatomformula symbol="=">
                                                 <variable symbol="p"/>
                                                 <variable symbol="q"/>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <variable symbol="p"/>
                                                </function>
                                                <function symbol="f">
                                                 <variable symbol="q"/>
                                                </function>
                                               </constantatomformula>
                                              </conjunctiveformula>
                                             </quantifiedformula>
                                            </quantifiedformula>
                                           </formulalist>
                                          </sequent>
                                          <rule type="contrr" symbol="c:r" param="2">
                                           <sequent>
                                            <formulalist>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <function symbol="+">
                                                <variable symbol="x_{109}"/>
                                                <function symbol="+">
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="1"/>
                                                 </function>
                                                 <variable symbol="x_{112}"/>
                                                </function>
                                               </function>
                                              </function>
                                              <constant symbol="1"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="=">
                                              <function symbol="f">
                                               <function symbol="+">
                                                <variable symbol="x_{109}"/>
                                                <function symbol="+">
                                                 <constant symbol="1"/>
                                                 <function symbol="+">
                                                  <variable symbol="x_{110}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <variable symbol="x_{111}"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                              </function>
                                              <constant symbol="1"/>
                                             </constantatomformula>
                                             <constantatomformula symbol="A" definition="A"/>
                                            </formulalist>
                                            <formulalist>
                                             <quantifiedformula type="exists">
                                              <variable symbol="p"/>
                                              <quantifiedformula type="exists">
                                               <variable symbol="q"/>
                                               <conjunctiveformula type="and">
                                                <conjunctiveformula type="neg">
                                                 <constantatomformula symbol="=">
                                                  <variable symbol="p"/>
                                                  <variable symbol="q"/>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <variable symbol="p"/>
                                                 </function>
                                                 <function symbol="f">
                                                  <variable symbol="q"/>
                                                 </function>
                                                </constantatomformula>
                                               </conjunctiveformula>
                                              </quantifiedformula>
                                             </quantifiedformula>
                                            </formulalist>
                                           </sequent>
                                           <rule type="permr" symbol="\pi:r" param="">
                                            <sequent>
                                             <formulalist>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <variable symbol="x_{109}"/>
                                                 <function symbol="+">
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </function>
                                                  <variable symbol="x_{112}"/>
                                                 </function>
                                                </function>
                                               </function>
                                               <constant symbol="1"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="=">
                                               <function symbol="f">
                                                <function symbol="+">
                                                 <variable symbol="x_{109}"/>
                                                 <function symbol="+">
                                                  <constant symbol="1"/>
                                                  <function symbol="+">
                                                   <variable symbol="x_{110}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <variable symbol="x_{111}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                               </function>
                                               <constant symbol="1"/>
                                              </constantatomformula>
                                              <constantatomformula symbol="A" definition="A"/>
                                             </formulalist>
                                             <formulalist>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                              <quantifiedformula type="exists">
                                               <variable symbol="p"/>
                                               <quantifiedformula type="exists">
                                                <variable symbol="q"/>
                                                <conjunctiveformula type="and">
                                                 <conjunctiveformula type="neg">
                                                  <constantatomformula symbol="=">
                                                   <variable symbol="p"/>
                                                   <variable symbol="q"/>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <variable symbol="p"/>
                                                  </function>
                                                  <function symbol="f">
                                                   <variable symbol="q"/>
                                                  </function>
                                                 </constantatomformula>
                                                </conjunctiveformula>
                                               </quantifiedformula>
                                              </quantifiedformula>
                                             </formulalist>
                                            </sequent>
                                            <rule type="contrl" symbol="c:l" param="1,1,2">
                                             <sequent>
                                              <formulalist>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <variable symbol="x_{109}"/>
                                                  <function symbol="+">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </function>
                                                   <variable symbol="x_{112}"/>
                                                  </function>
                                                 </function>
                                                </function>
                                                <constant symbol="1"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="=">
                                                <function symbol="f">
                                                 <function symbol="+">
                                                  <variable symbol="x_{109}"/>
                                                  <function symbol="+">
                                                   <constant symbol="1"/>
                                                   <function symbol="+">
                                                    <variable symbol="x_{110}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <variable symbol="x_{111}"/>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                </function>
                                                <constant symbol="1"/>
                                               </constantatomformula>
                                               <constantatomformula symbol="A" definition="A"/>
                                              </formulalist>
                                              <formulalist>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                               <quantifiedformula type="exists">
                                                <variable symbol="p"/>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="q"/>
                                                 <conjunctiveformula type="and">
                                                  <conjunctiveformula type="neg">
                                                   <constantatomformula symbol="=">
                                                    <variable symbol="p"/>
                                                    <variable symbol="q"/>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <variable symbol="p"/>
                                                   </function>
                                                   <function symbol="f">
                                                    <variable symbol="q"/>
                                                   </function>
                                                  </constantatomformula>
                                                 </conjunctiveformula>
                                                </quantifiedformula>
                                               </quantifiedformula>
                                              </formulalist>
                                             </sequent>
                                             <rule type="perml" symbol="\pi:l" param="(2 3)">
                                              <sequent>
                                               <formulalist>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </function>
                                                    <variable symbol="x_{112}"/>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <constant symbol="1"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="=">
                                                 <function symbol="f">
                                                  <function symbol="+">
                                                   <variable symbol="x_{109}"/>
                                                   <function symbol="+">
                                                    <constant symbol="1"/>
                                                    <function symbol="+">
                                                     <variable symbol="x_{110}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <variable symbol="x_{111}"/>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                 </function>
                                                 <constant symbol="1"/>
                                                </constantatomformula>
                                                <constantatomformula symbol="A" definition="A"/>
                                                <constantatomformula symbol="A" definition="A"/>
                                               </formulalist>
                                               <formulalist>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                                <quantifiedformula type="exists">
                                                 <variable symbol="p"/>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="q"/>
                                                  <conjunctiveformula type="and">
                                                   <conjunctiveformula type="neg">
                                                    <constantatomformula symbol="=">
                                                     <variable symbol="p"/>
                                                     <variable symbol="q"/>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <variable symbol="p"/>
                                                    </function>
                                                    <function symbol="f">
                                                     <variable symbol="q"/>
                                                    </function>
                                                   </constantatomformula>
                                                  </conjunctiveformula>
                                                 </quantifiedformula>
                                                </quantifiedformula>
                                               </formulalist>
                                              </sequent>
                                              <rule type="eql1" symbol="=:l1" param="(1 1)">
                                               <sequent>
                                                <formulalist>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <variable symbol="x_{112}"/>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="1"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                 <constantatomformula symbol="=">
                                                  <function symbol="f">
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <constant symbol="1"/>
                                                     <function symbol="+">
                                                      <variable symbol="x_{110}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <variable symbol="x_{111}"/>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                  </function>
                                                  <constant symbol="1"/>
                                                 </constantatomformula>
                                                 <constantatomformula symbol="A" definition="A"/>
                                                </formulalist>
                                                <formulalist>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                 <quantifiedformula type="exists">
                                                  <variable symbol="p"/>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="q"/>
                                                   <conjunctiveformula type="and">
                                                    <conjunctiveformula type="neg">
                                                     <constantatomformula symbol="=">
                                                      <variable symbol="p"/>
                                                      <variable symbol="q"/>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <variable symbol="p"/>
                                                     </function>
                                                     <function symbol="f">
                                                      <variable symbol="q"/>
                                                     </function>
                                                    </constantatomformula>
                                                   </conjunctiveformula>
                                                  </quantifiedformula>
                                                 </quantifiedformula>
                                                </formulalist>
                                               </sequent>
                                               <rule type="contrr" symbol="c:r" param="2,1">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="+">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                    </function>
                                                    <variable symbol="x_{112}"/>
                                                   </function>
                                                   <function symbol="+">
                                                    <variable symbol="x_{109}"/>
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </function>
                                                     <variable symbol="x_{112}"/>
                                                    </function>
                                                   </function>
                                                  </constantatomformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="permr" symbol="\pi:r" param="">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{112}"/>
                                                    </function>
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                      <variable symbol="x_{112}"/>
                                                     </function>
                                                    </function>
                                                   </constantatomformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="contrl" symbol="c:l" param="2">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{112}"/>
                                                     </function>
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                       <variable symbol="x_{112}"/>
                                                      </function>
                                                     </function>
                                                    </constantatomformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="perml" symbol="\pi:l" param="">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{112}"/>
                                                      </function>
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                        <variable symbol="x_{112}"/>
                                                       </function>
                                                      </function>
                                                     </constantatomformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="eqr1" symbol="=:r1" param="(1)">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{112}"/>
                                                       </function>
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                         <variable symbol="x_{112}"/>
                                                        </function>
                                                       </function>
                                                      </constantatomformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <variable symbol="x_{112}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{112}"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="weakr" symbol="w:r">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <variable symbol="x_{112}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{112}"/>
                                                         </function>
                                                        </constantatomformula>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="perml" symbol="\pi:l" param="">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <variable symbol="x_{112}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{112}"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="weakl" symbol="w:l">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <variable symbol="x_{112}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                            <variable symbol="x_{112}"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="axiom" symbol="axiom">
                                                         <sequent>
                                                          <formulalist/>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <variable symbol="x_{112}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                             <variable symbol="x_{112}"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                    <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <variable symbol="x_{112}"/>
                                                         </function>
                                                        </function>
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <variable symbol="x_{112}"/>
                                                         </function>
                                                        </function>
                                                       </constantatomformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="weakr" symbol="w:r">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <variable symbol="x_{112}"/>
                                                          </function>
                                                         </function>
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <variable symbol="x_{112}"/>
                                                          </function>
                                                         </function>
                                                        </constantatomformula>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="perml" symbol="\pi:l" param="">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <variable symbol="x_{112}"/>
                                                           </function>
                                                          </function>
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <variable symbol="x_{112}"/>
                                                           </function>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="weakl" symbol="w:l">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <variable symbol="x_{112}"/>
                                                            </function>
                                                           </function>
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <variable symbol="x_{112}"/>
                                                            </function>
                                                           </function>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="axiom" symbol="axiom">
                                                         <sequent>
                                                          <formulalist/>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <variable symbol="x_{112}"/>
                                                             </function>
                                                            </function>
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <variable symbol="x_{112}"/>
                                                             </function>
                                                            </function>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                               <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                <sequent>
                                                 <formulalist>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </function>
                                                     </function>
                                                     <variable symbol="x_{112}"/>
                                                    </function>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="=">
                                                   <function symbol="f">
                                                    <function symbol="+">
                                                     <variable symbol="x_{109}"/>
                                                     <function symbol="+">
                                                      <constant symbol="1"/>
                                                      <function symbol="+">
                                                       <variable symbol="x_{110}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <variable symbol="x_{111}"/>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                   </function>
                                                   <constant symbol="1"/>
                                                  </constantatomformula>
                                                  <constantatomformula symbol="A" definition="A"/>
                                                 </formulalist>
                                                 <formulalist>
                                                  <quantifiedformula type="exists">
                                                   <variable symbol="p"/>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="q"/>
                                                    <conjunctiveformula type="and">
                                                     <conjunctiveformula type="neg">
                                                      <constantatomformula symbol="=">
                                                       <variable symbol="p"/>
                                                       <variable symbol="q"/>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <variable symbol="p"/>
                                                      </function>
                                                      <function symbol="f">
                                                       <variable symbol="q"/>
                                                      </function>
                                                     </constantatomformula>
                                                    </conjunctiveformula>
                                                   </quantifiedformula>
                                                  </quantifiedformula>
                                                 </formulalist>
                                                </sequent>
                                                <rule type="perml" symbol="\pi:l" param="(1 2)">
                                                 <sequent>
                                                  <formulalist>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <variable symbol="x_{109}"/>
                                                      <function symbol="+">
                                                       <constant symbol="1"/>
                                                       <function symbol="+">
                                                        <variable symbol="x_{110}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <variable symbol="x_{111}"/>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="=">
                                                    <function symbol="f">
                                                     <function symbol="+">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </function>
                                                      </function>
                                                      <variable symbol="x_{112}"/>
                                                     </function>
                                                    </function>
                                                    <constant symbol="1"/>
                                                   </constantatomformula>
                                                   <constantatomformula symbol="A" definition="A"/>
                                                  </formulalist>
                                                  <formulalist>
                                                   <quantifiedformula type="exists">
                                                    <variable symbol="p"/>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="q"/>
                                                     <conjunctiveformula type="and">
                                                      <conjunctiveformula type="neg">
                                                       <constantatomformula symbol="=">
                                                        <variable symbol="p"/>
                                                        <variable symbol="q"/>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <variable symbol="p"/>
                                                       </function>
                                                       <function symbol="f">
                                                        <variable symbol="q"/>
                                                       </function>
                                                      </constantatomformula>
                                                     </conjunctiveformula>
                                                    </quantifiedformula>
                                                   </quantifiedformula>
                                                  </formulalist>
                                                 </sequent>
                                                 <rule type="contrr" symbol="c:r" param="2">
                                                  <sequent>
                                                   <formulalist>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </function>
                                                       </function>
                                                       <variable symbol="x_{112}"/>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </constantatomformula>
                                                    <constantatomformula symbol="=">
                                                     <function symbol="f">
                                                      <function symbol="+">
                                                       <variable symbol="x_{109}"/>
                                                       <function symbol="+">
                                                        <constant symbol="1"/>
                                                        <function symbol="+">
                                                         <variable symbol="x_{110}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <variable symbol="x_{111}"/>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                     </function>
                                                     <constant symbol="1"/>
                                                    </constantatomformula>
                                                    <constantatomformula symbol="A" definition="A"/>
                                                   </formulalist>
                                                   <formulalist>
                                                    <quantifiedformula type="exists">
                                                     <variable symbol="p"/>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="q"/>
                                                      <conjunctiveformula type="and">
                                                       <conjunctiveformula type="neg">
                                                        <constantatomformula symbol="=">
                                                         <variable symbol="p"/>
                                                         <variable symbol="q"/>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <variable symbol="p"/>
                                                        </function>
                                                        <function symbol="f">
                                                         <variable symbol="q"/>
                                                        </function>
                                                       </constantatomformula>
                                                      </conjunctiveformula>
                                                     </quantifiedformula>
                                                    </quantifiedformula>
                                                   </formulalist>
                                                  </sequent>
                                                  <rule type="permr" symbol="\pi:r" param="">
                                                   <sequent>
                                                    <formulalist>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </function>
                                                        </function>
                                                        <variable symbol="x_{112}"/>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="=">
                                                      <function symbol="f">
                                                       <function symbol="+">
                                                        <variable symbol="x_{109}"/>
                                                        <function symbol="+">
                                                         <constant symbol="1"/>
                                                         <function symbol="+">
                                                          <variable symbol="x_{110}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <variable symbol="x_{111}"/>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                      </function>
                                                      <constant symbol="1"/>
                                                     </constantatomformula>
                                                     <constantatomformula symbol="A" definition="A"/>
                                                    </formulalist>
                                                    <formulalist>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                     <quantifiedformula type="exists">
                                                      <variable symbol="p"/>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="q"/>
                                                       <conjunctiveformula type="and">
                                                        <conjunctiveformula type="neg">
                                                         <constantatomformula symbol="=">
                                                          <variable symbol="p"/>
                                                          <variable symbol="q"/>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <variable symbol="p"/>
                                                         </function>
                                                         <function symbol="f">
                                                          <variable symbol="q"/>
                                                         </function>
                                                        </constantatomformula>
                                                       </conjunctiveformula>
                                                      </quantifiedformula>
                                                     </quantifiedformula>
                                                    </formulalist>
                                                   </sequent>
                                                   <rule type="contrl" symbol="c:l" param="1,1,2">
                                                    <sequent>
                                                     <formulalist>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                         </function>
                                                         <variable symbol="x_{112}"/>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="=">
                                                       <function symbol="f">
                                                        <function symbol="+">
                                                         <variable symbol="x_{109}"/>
                                                         <function symbol="+">
                                                          <constant symbol="1"/>
                                                          <function symbol="+">
                                                           <variable symbol="x_{110}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <variable symbol="x_{111}"/>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                       </function>
                                                       <constant symbol="1"/>
                                                      </constantatomformula>
                                                      <constantatomformula symbol="A" definition="A"/>
                                                     </formulalist>
                                                     <formulalist>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                      <quantifiedformula type="exists">
                                                       <variable symbol="p"/>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="q"/>
                                                        <conjunctiveformula type="and">
                                                         <conjunctiveformula type="neg">
                                                          <constantatomformula symbol="=">
                                                           <variable symbol="p"/>
                                                           <variable symbol="q"/>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <variable symbol="p"/>
                                                          </function>
                                                          <function symbol="f">
                                                           <variable symbol="q"/>
                                                          </function>
                                                         </constantatomformula>
                                                        </conjunctiveformula>
                                                       </quantifiedformula>
                                                      </quantifiedformula>
                                                     </formulalist>
                                                    </sequent>
                                                    <rule type="perml" symbol="\pi:l" param="(2 3)">
                                                     <sequent>
                                                      <formulalist>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                          <variable symbol="x_{112}"/>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="=">
                                                        <function symbol="f">
                                                         <function symbol="+">
                                                          <variable symbol="x_{109}"/>
                                                          <function symbol="+">
                                                           <constant symbol="1"/>
                                                           <function symbol="+">
                                                            <variable symbol="x_{110}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <variable symbol="x_{111}"/>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                        </function>
                                                        <constant symbol="1"/>
                                                       </constantatomformula>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                       <constantatomformula symbol="A" definition="A"/>
                                                      </formulalist>
                                                      <formulalist>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                       <quantifiedformula type="exists">
                                                        <variable symbol="p"/>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="q"/>
                                                         <conjunctiveformula type="and">
                                                          <conjunctiveformula type="neg">
                                                           <constantatomformula symbol="=">
                                                            <variable symbol="p"/>
                                                            <variable symbol="q"/>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <variable symbol="p"/>
                                                           </function>
                                                           <function symbol="f">
                                                            <variable symbol="q"/>
                                                           </function>
                                                          </constantatomformula>
                                                         </conjunctiveformula>
                                                        </quantifiedformula>
                                                       </quantifiedformula>
                                                      </formulalist>
                                                     </sequent>
                                                     <rule type="eql1" symbol="=:l1" param="(1 1 1)">
                                                      <sequent>
                                                       <formulalist>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                           <variable symbol="x_{112}"/>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </constantatomformula>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                        <constantatomformula symbol="=">
                                                         <function symbol="f">
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <constant symbol="1"/>
                                                            <function symbol="+">
                                                             <variable symbol="x_{110}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <variable symbol="x_{111}"/>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                         </function>
                                                         <constant symbol="1"/>
                                                        </constantatomformula>
                                                        <constantatomformula symbol="A" definition="A"/>
                                                       </formulalist>
                                                       <formulalist>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                        <quantifiedformula type="exists">
                                                         <variable symbol="p"/>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="q"/>
                                                          <conjunctiveformula type="and">
                                                           <conjunctiveformula type="neg">
                                                            <constantatomformula symbol="=">
                                                             <variable symbol="p"/>
                                                             <variable symbol="q"/>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <variable symbol="p"/>
                                                            </function>
                                                            <function symbol="f">
                                                             <variable symbol="q"/>
                                                            </function>
                                                           </constantatomformula>
                                                          </conjunctiveformula>
                                                         </quantifiedformula>
                                                        </quantifiedformula>
                                                       </formulalist>
                                                      </sequent>
                                                      <rule type="contrr" symbol="c:r" param="2,1">
                                                       <sequent>
                                                        <formulalist>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
                                                            <conjunctiveformula type="neg">
                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <variable symbol="p"/>
                                                             </function>
                                                             <function symbol="f">
                                                              <variable symbol="q"/>
                                                             </function>
                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="+">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </function>
                                                          <function symbol="+">
                                                           <variable symbol="x_{109}"/>
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                          </function>
                                                         </constantatomformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="permr" symbol="\pi:r" param="">
                                                        <sequent>
                                                         <formulalist>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
                                                            <conjunctiveformula type="and">
                                                             <conjunctiveformula type="neg">
                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
                                                            <conjunctiveformula type="and">
                                                             <conjunctiveformula type="neg">
                                                              <constantatomformula symbol="=">
                                                               <variable symbol="p"/>
                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </function>
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                           </function>
                                                          </constantatomformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="contrl" symbol="c:l" param="2">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="p"/>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="q"/>
                                                             <conjunctiveformula type="and">
                                                              <conjunctiveformula type="neg">
                                                               <constantatomformula symbol="=">
                                                                <variable symbol="p"/>
                                                                <variable symbol="q"/>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="p"/>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="q"/>
                                                             <conjunctiveformula type="and">
                                                              <conjunctiveformula type="neg">
                                                               <constantatomformula symbol="=">
                                                                <variable symbol="p"/>
                                                                <variable symbol="q"/>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                            </function>
                                                           </constantatomformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="perml" symbol="\pi:l" param="">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                           </formulalist>
                                                           <formulalist>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="p"/>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="q"/>
                                                              <conjunctiveformula type="and">
                                                               <conjunctiveformula type="neg">
                                                                <constantatomformula symbol="=">
                                                                 <variable symbol="p"/>
                                                                 <variable symbol="q"/>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <variable symbol="p"/>
                                                                </function>
                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </quantifiedformula>
                                                            </quantifiedformula>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="p"/>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="q"/>
                                                              <conjunctiveformula type="and">
                                                               <conjunctiveformula type="neg">
                                                                <constantatomformula symbol="=">
                                                                 <variable symbol="p"/>
                                                                 <variable symbol="q"/>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <variable symbol="p"/>
                                                                </function>
                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </quantifiedformula>
                                                            </quantifiedformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                             </function>
                                                            </constantatomformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="eqr1" symbol="=:r1" param="(1)">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="A" definition="A"/>
                                                             <constantatomformula symbol="A" definition="A"/>
                                                            </formulalist>
                                                            <formulalist>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="p"/>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="q"/>
                                                               <conjunctiveformula type="and">
                                                                <conjunctiveformula type="neg">
                                                                 <constantatomformula symbol="=">
                                                                  <variable symbol="p"/>
                                                                  <variable symbol="q"/>
                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <variable symbol="p"/>
                                                                 </function>
                                                                 <function symbol="f">
                                                                  <variable symbol="q"/>
                                                                 </function>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                              </quantifiedformula>
                                                             </quantifiedformula>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="p"/>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="q"/>
                                                               <conjunctiveformula type="and">
                                                                <conjunctiveformula type="neg">
                                                                 <constantatomformula symbol="=">
                                                                  <variable symbol="p"/>
                                                                  <variable symbol="q"/>
                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <variable symbol="p"/>
                                                                 </function>
                                                                 <function symbol="f">
                                                                  <variable symbol="q"/>
                                                                 </function>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                              </quantifiedformula>
                                                             </quantifiedformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </function>
                                                             </constantatomformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="A" definition="A"/>
                                                             </formulalist>
                                                             <formulalist>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="p"/>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="q"/>
                                                                <conjunctiveformula type="and">
                                                                 <conjunctiveformula type="neg">
                                                                  <constantatomformula symbol="=">
                                                                   <variable symbol="p"/>
                                                                   <variable symbol="q"/>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <variable symbol="p"/>
                                                                  </function>
                                                                  <function symbol="f">
                                                                   <variable symbol="q"/>
                                                                  </function>
                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                               </quantifiedformula>
                                                              </quantifiedformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="weakr" symbol="w:r">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="A" definition="A"/>
                                                              </formulalist>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </constantatomformula>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="p"/>
                                                                <quantifiedformula type="exists">
                                                                 <variable symbol="q"/>
                                                                 <conjunctiveformula type="and">
                                                                  <conjunctiveformula type="neg">
                                                                   <constantatomformula symbol="=">
                                                                    <variable symbol="p"/>
                                                                    <variable symbol="q"/>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <variable symbol="p"/>
                                                                   </function>
                                                                   <function symbol="f">
                                                                    <variable symbol="q"/>
                                                                   </function>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                </quantifiedformula>
                                                               </quantifiedformula>
                                                              </formulalist>
                                                             </sequent>
                                                             <rule type="perml" symbol="\pi:l" param="">
                                                              <sequent>
                                                               <formulalist>
                                                                <constantatomformula symbol="A" definition="A"/>
                                                               </formulalist>
                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{109}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </constantatomformula>
                                                               </formulalist>
                                                              </sequent>
                                                              <rule type="weakl" symbol="w:l">
                                                               <sequent>
                                                                <formulalist>
                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                </formulalist>
                                                                <formulalist>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{109}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{109}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </constantatomformula>
                                                                </formulalist>
                                                               </sequent>
                                                               <rule type="axiom" symbol="axiom">
                                                                <sequent>
                                                                 <formulalist/>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{109}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                   </function>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{109}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </constantatomformula>
                                                                 </formulalist>
                                                                </sequent>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                           <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="A" definition="A"/>
                                                             </formulalist>
                                                             <formulalist>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="p"/>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="q"/>
                                                                <conjunctiveformula type="and">
                                                                 <conjunctiveformula type="neg">
                                                                  <constantatomformula symbol="=">
                                                                   <variable symbol="p"/>
                                                                   <variable symbol="q"/>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <variable symbol="p"/>
                                                                  </function>
                                                                  <function symbol="f">
                                                                   <variable symbol="q"/>
                                                                  </function>
                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                               </quantifiedformula>
                                                              </quantifiedformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                               </function>
                                                              </constantatomformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="weakr" symbol="w:r">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="A" definition="A"/>
                                                              </formulalist>
                                                              <formulalist>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                </function>
                                                               </constantatomformula>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="p"/>
                                                                <quantifiedformula type="exists">
                                                                 <variable symbol="q"/>
                                                                 <conjunctiveformula type="and">
                                                                  <conjunctiveformula type="neg">
                                                                   <constantatomformula symbol="=">
                                                                    <variable symbol="p"/>
                                                                    <variable symbol="q"/>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <variable symbol="p"/>
                                                                   </function>
                                                                   <function symbol="f">
                                                                    <variable symbol="q"/>
                                                                   </function>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                </quantifiedformula>
                                                               </quantifiedformula>
                                                              </formulalist>
                                                             </sequent>
                                                             <rule type="perml" symbol="\pi:l" param="">
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                                                               <formulalist>
                                                                <constantatomformula symbol="A" definition="A"/>
                                                               </formulalist>
                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                 </function>
                                                                </constantatomformula>
                                                               </formulalist>
                                                              </sequent>
                                                              <rule type="weakl" symbol="w:l">
                                                               <sequent>
                                                                <formulalist>
                                                                 <constantatomformula symbol="A" definition="A"/>
                                                                </formulalist>
                                                                <formulalist>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{109}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{109}"/>
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                  </function>
                                                                 </constantatomformula>
                                                                </formulalist>
                                                               </sequent>
                                                               <rule type="axiom" symbol="axiom">
                                                                <sequent>
                                                                 <formulalist/>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{109}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
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                                                                   </function>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{109}"/>
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
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                                                                   </function>
                                                                  </constantatomformula>
                                                                 </formulalist>
                                                                </sequent>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                      <rule type="perml" symbol="\pi:l" param="(1 2)">
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                                                        <formulalist>
                                                         <constantatomformula symbol="=">
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                                                           <function symbol="+">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </function>
                                                            <variable symbol="x_{112}"/>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </constantatomformula>
                                                         <constantatomformula symbol="=">
                                                          <function symbol="f">
                                                           <function symbol="+">
                                                            <variable symbol="x_{109}"/>
                                                            <function symbol="+">
                                                             <constant symbol="1"/>
                                                             <function symbol="+">
                                                              <variable symbol="x_{110}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <variable symbol="x_{111}"/>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                          </function>
                                                          <constant symbol="1"/>
                                                         </constantatomformula>
                                                         <constantatomformula symbol="A" definition="A"/>
                                                        </formulalist>
                                                        <formulalist>
                                                         <quantifiedformula type="exists">
                                                          <variable symbol="p"/>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="q"/>
                                                           <conjunctiveformula type="and">
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                                                             <constantatomformula symbol="=">
                                                              <variable symbol="p"/>
                                                              <variable symbol="q"/>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                            <constantatomformula symbol="=">
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                                                             <function symbol="f">
                                                              <variable symbol="q"/>
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                                                            </constantatomformula>
                                                           </conjunctiveformula>
                                                          </quantifiedformula>
                                                         </quantifiedformula>
                                                        </formulalist>
                                                       </sequent>
                                                       <rule type="permr" symbol="\pi:r" param="">
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                                                         <formulalist>
                                                          <constantatomformula symbol="=">
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                                                            <function symbol="+">
                                                             <variable symbol="x_{109}"/>
                                                             <function symbol="+">
                                                              <constant symbol="1"/>
                                                              <function symbol="+">
                                                               <variable symbol="x_{110}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <variable symbol="x_{111}"/>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                           </function>
                                                           <constant symbol="1"/>
                                                          </constantatomformula>
                                                          <constantatomformula symbol="=">
                                                           <function symbol="f">
                                                            <function symbol="+">
                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </function>
                                                             <variable symbol="x_{112}"/>
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                                                           </function>
                                                           <constant symbol="1"/>
                                                          </constantatomformula>
                                                          <constantatomformula symbol="A" definition="A"/>
                                                         </formulalist>
                                                         <formulalist>
                                                          <quantifiedformula type="exists">
                                                           <variable symbol="p"/>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="q"/>
                                                            <conjunctiveformula type="and">
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                                                              <constantatomformula symbol="=">
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                                                               <variable symbol="q"/>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <variable symbol="p"/>
                                                              </function>
                                                              <function symbol="f">
                                                               <variable symbol="q"/>
                                                              </function>
                                                             </constantatomformula>
                                                            </conjunctiveformula>
                                                           </quantifiedformula>
                                                          </quantifiedformula>
                                                         </formulalist>
                                                        </sequent>
                                                        <rule type="perml" symbol="\pi:l" param="(1 2 3)">
                                                         <sequent>
                                                          <formulalist>
                                                           <constantatomformula symbol="=">
                                                            <function symbol="f">
                                                             <function symbol="+">
                                                              <variable symbol="x_{109}"/>
                                                              <function symbol="+">
                                                               <constant symbol="1"/>
                                                               <function symbol="+">
                                                                <variable symbol="x_{110}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <variable symbol="x_{111}"/>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="=">
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                                                             <function symbol="+">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
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                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </function>
                                                              <variable symbol="x_{112}"/>
                                                             </function>
                                                            </function>
                                                            <constant symbol="1"/>
                                                           </constantatomformula>
                                                           <constantatomformula symbol="A" definition="A"/>
                                                          </formulalist>
                                                          <formulalist>
                                                           <quantifiedformula type="exists">
                                                            <variable symbol="p"/>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="q"/>
                                                             <conjunctiveformula type="and">
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                                                               <constantatomformula symbol="=">
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                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <variable symbol="p"/>
                                                               </function>
                                                               <function symbol="f">
                                                                <variable symbol="q"/>
                                                               </function>
                                                              </constantatomformula>
                                                             </conjunctiveformula>
                                                            </quantifiedformula>
                                                           </quantifiedformula>
                                                          </formulalist>
                                                         </sequent>
                                                         <rule type="contrr" symbol="c:r" param="2">
                                                          <sequent>
                                                           <formulalist>
                                                            <constantatomformula symbol="A" definition="A"/>
                                                            <constantatomformula symbol="=">
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                                                              <function symbol="+">
                                                               <variable symbol="x_{109}"/>
                                                               <function symbol="+">
                                                                <constant symbol="1"/>
                                                                <function symbol="+">
                                                                 <variable symbol="x_{110}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <variable symbol="x_{111}"/>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </constantatomformula>
                                                            <constantatomformula symbol="=">
                                                             <function symbol="f">
                                                              <function symbol="+">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </function>
                                                               <variable symbol="x_{112}"/>
                                                              </function>
                                                             </function>
                                                             <constant symbol="1"/>
                                                            </constantatomformula>
                                                           </formulalist>
                                                           <formulalist>
                                                            <quantifiedformula type="exists">
                                                             <variable symbol="p"/>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="q"/>
                                                              <conjunctiveformula type="and">
                                                               <conjunctiveformula type="neg">
                                                                <constantatomformula symbol="=">
                                                                 <variable symbol="p"/>
                                                                 <variable symbol="q"/>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <variable symbol="p"/>
                                                                </function>
                                                                <function symbol="f">
                                                                 <variable symbol="q"/>
                                                                </function>
                                                               </constantatomformula>
                                                              </conjunctiveformula>
                                                             </quantifiedformula>
                                                            </quantifiedformula>
                                                           </formulalist>
                                                          </sequent>
                                                          <rule type="permr" symbol="\pi:r" param="(1 2)">
                                                           <sequent>
                                                            <formulalist>
                                                             <constantatomformula symbol="A" definition="A"/>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <variable symbol="x_{109}"/>
                                                                <function symbol="+">
                                                                 <constant symbol="1"/>
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{110}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <variable symbol="x_{111}"/>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </constantatomformula>
                                                             <constantatomformula symbol="=">
                                                              <function symbol="f">
                                                               <function symbol="+">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </function>
                                                                <variable symbol="x_{112}"/>
                                                               </function>
                                                              </function>
                                                              <constant symbol="1"/>
                                                             </constantatomformula>
                                                            </formulalist>
                                                            <formulalist>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="p"/>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="q"/>
                                                               <conjunctiveformula type="and">
                                                                <conjunctiveformula type="neg">
                                                                 <constantatomformula symbol="=">
                                                                  <variable symbol="p"/>
                                                                  <variable symbol="q"/>
                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <variable symbol="p"/>
                                                                 </function>
                                                                 <function symbol="f">
                                                                  <variable symbol="q"/>
                                                                 </function>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                              </quantifiedformula>
                                                             </quantifiedformula>
                                                             <quantifiedformula type="exists">
                                                              <variable symbol="p"/>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="q"/>
                                                               <conjunctiveformula type="and">
                                                                <conjunctiveformula type="neg">
                                                                 <constantatomformula symbol="=">
                                                                  <variable symbol="p"/>
                                                                  <variable symbol="q"/>
                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <variable symbol="p"/>
                                                                 </function>
                                                                 <function symbol="f">
                                                                  <variable symbol="q"/>
                                                                 </function>
                                                                </constantatomformula>
                                                               </conjunctiveformula>
                                                              </quantifiedformula>
                                                             </quantifiedformula>
                                                            </formulalist>
                                                           </sequent>
                                                           <rule type="weakr" symbol="w:r">
                                                            <sequent>
                                                             <formulalist>
                                                              <constantatomformula symbol="A" definition="A"/>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <variable symbol="x_{109}"/>
                                                                 <function symbol="+">
                                                                  <constant symbol="1"/>
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{110}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <variable symbol="x_{111}"/>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </constantatomformula>
                                                              <constantatomformula symbol="=">
                                                               <function symbol="f">
                                                                <function symbol="+">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{109}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </function>
                                                                 <variable symbol="x_{112}"/>
                                                                </function>
                                                               </function>
                                                               <constant symbol="1"/>
                                                              </constantatomformula>
                                                             </formulalist>
                                                             <formulalist>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="p"/>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="q"/>
                                                                <conjunctiveformula type="and">
                                                                 <conjunctiveformula type="neg">
                                                                  <constantatomformula symbol="=">
                                                                   <variable symbol="p"/>
                                                                   <variable symbol="q"/>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <variable symbol="p"/>
                                                                  </function>
                                                                  <function symbol="f">
                                                                   <variable symbol="q"/>
                                                                  </function>
                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                               </quantifiedformula>
                                                              </quantifiedformula>
                                                              <quantifiedformula type="exists">
                                                               <variable symbol="p"/>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="q"/>
                                                                <conjunctiveformula type="and">
                                                                 <conjunctiveformula type="neg">
                                                                  <constantatomformula symbol="=">
                                                                   <variable symbol="p"/>
                                                                   <variable symbol="q"/>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <variable symbol="p"/>
                                                                  </function>
                                                                  <function symbol="f">
                                                                   <variable symbol="q"/>
                                                                  </function>
                                                                 </constantatomformula>
                                                                </conjunctiveformula>
                                                               </quantifiedformula>
                                                              </quantifiedformula>
                                                             </formulalist>
                                                            </sequent>
                                                            <rule type="weakl" symbol="w:l">
                                                             <sequent>
                                                              <formulalist>
                                                               <constantatomformula symbol="A" definition="A"/>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <variable symbol="x_{109}"/>
                                                                  <function symbol="+">
                                                                   <constant symbol="1"/>
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{110}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <variable symbol="x_{111}"/>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </constantatomformula>
                                                               <constantatomformula symbol="=">
                                                                <function symbol="f">
                                                                 <function symbol="+">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{109}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </function>
                                                                  <variable symbol="x_{112}"/>
                                                                 </function>
                                                                </function>
                                                                <constant symbol="1"/>
                                                               </constantatomformula>
                                                              </formulalist>
                                                              <formulalist>
                                                               <quantifiedformula type="exists">
                                                                <variable symbol="p"/>
                                                                <quantifiedformula type="exists">
                                                                 <variable symbol="q"/>
                                                                 <conjunctiveformula type="and">
                                                                  <conjunctiveformula type="neg">
                                                                   <constantatomformula symbol="=">
                                                                    <variable symbol="p"/>
                                                                    <variable symbol="q"/>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <variable symbol="p"/>
                                                                   </function>
                                                                   <function symbol="f">
                                                                    <variable symbol="q"/>
                                                                   </function>
                                                                  </constantatomformula>
                                                                 </conjunctiveformula>
                                                                </quantifiedformula>
                                                               </quantifiedformula>
                                                              </formulalist>
                                                             </sequent>
                                                             <rule type="existsr" symbol="\ex:r">
                                                              <sequent>
                                                               <formulalist>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <function symbol="+">
                                                                   <variable symbol="x_{109}"/>
                                                                   <function symbol="+">
                                                                    <constant symbol="1"/>
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{110}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <variable symbol="x_{111}"/>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </constantatomformula>
                                                                <constantatomformula symbol="=">
                                                                 <function symbol="f">
                                                                  <function symbol="+">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{109}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </function>
                                                                   <variable symbol="x_{112}"/>
                                                                  </function>
                                                                 </function>
                                                                 <constant symbol="1"/>
                                                                </constantatomformula>
                                                               </formulalist>
                                                               <formulalist>
                                                                <quantifiedformula type="exists">
                                                                 <variable symbol="p"/>
                                                                 <quantifiedformula type="exists">
                                                                  <variable symbol="q"/>
                                                                  <conjunctiveformula type="and">
                                                                   <conjunctiveformula type="neg">
                                                                    <constantatomformula symbol="=">
                                                                     <variable symbol="p"/>
                                                                     <variable symbol="q"/>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <variable symbol="p"/>
                                                                    </function>
                                                                    <function symbol="f">
                                                                     <variable symbol="q"/>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                 </quantifiedformula>
                                                                </quantifiedformula>
                                                               </formulalist>
                                                              </sequent>
                                                              <rule type="existsr" symbol="\ex:r">
                                                               <sequent>
                                                                <formulalist>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <function symbol="+">
                                                                    <variable symbol="x_{109}"/>
                                                                    <function symbol="+">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{110}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{111}"/>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
                                                                 </constantatomformula>
                                                                 <constantatomformula symbol="=">
                                                                  <function symbol="f">
                                                                   <function symbol="+">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{109}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </function>
                                                                    <variable symbol="x_{112}"/>
                                                                   </function>
                                                                  </function>
                                                                  <constant symbol="1"/>
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                                                                </formulalist>
                                                                <formulalist>
                                                                 <quantifiedformula type="exists">
                                                                  <variable symbol="q"/>
                                                                  <conjunctiveformula type="and">
                                                                   <conjunctiveformula type="neg">
                                                                    <constantatomformula symbol="=">
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                                                                      <variable symbol="x_{109}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <variable symbol="q"/>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{109}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="f">
                                                                     <variable symbol="q"/>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                 </quantifiedformula>
                                                                </formulalist>
                                                               </sequent>
                                                               <rule type="andr" symbol="\and:r">
                                                                <sequent>
                                                                 <formulalist>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <function symbol="+">
                                                                     <variable symbol="x_{109}"/>
                                                                     <function symbol="+">
                                                                      <constant symbol="1"/>
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{110}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <variable symbol="x_{111}"/>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </constantatomformula>
                                                                  <constantatomformula symbol="=">
                                                                   <function symbol="f">
                                                                    <function symbol="+">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </function>
                                                                     <variable symbol="x_{112}"/>
                                                                    </function>
                                                                   </function>
                                                                   <constant symbol="1"/>
                                                                  </constantatomformula>
                                                                 </formulalist>
                                                                 <formulalist>
                                                                  <conjunctiveformula type="and">
                                                                   <conjunctiveformula type="neg">
                                                                    <constantatomformula symbol="=">
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                                                                      <variable symbol="x_{109}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <variable symbol="x_{112}"/>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
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                                                                      <variable symbol="x_{109}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <variable symbol="x_{112}"/>
                                                                     </function>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </conjunctiveformula>
                                                                 </formulalist>
                                                                </sequent>
                                                                <rule type="negr" symbol="\neg:r">
                                                                 <sequent>
                                                                  <formulalist/>
                                                                  <formulalist>
                                                                   <conjunctiveformula type="neg">
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{109}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <variable symbol="x_{112}"/>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </conjunctiveformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                 <rule type="eql1" symbol="=:l1" param="(2)">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{109}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <variable symbol="x_{112}"/>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                   <formulalist/>
                                                                  </sequent>
                                                                  <rule type="axiom" symbol="axiom">
                                                                   <sequent>
                                                                    <formulalist/>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{112}"/>
                                                                       </function>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{109}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <variable symbol="x_{112}"/>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                  </rule>
                                                                  <rule type="axiom" symbol="axiom">
                                                                   <sequent>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <variable symbol="x_{112}"/>
                                                                       </function>
                                                                      </function>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                    <formulalist/>
                                                                   </sequent>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                                <rule type="eqr2" symbol="=:r2" param="(1)">
                                                                 <sequent>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{109}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </constantatomformula>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <variable symbol="x_{112}"/>
                                                                     </function>
                                                                    </function>
                                                                    <constant symbol="1"/>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                  <formulalist>
                                                                   <constantatomformula symbol="=">
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <variable symbol="x_{109}"/>
                                                                      <function symbol="+">
                                                                       <constant symbol="1"/>
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{110}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <variable symbol="x_{111}"/>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                    </function>
                                                                    <function symbol="f">
                                                                     <function symbol="+">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <variable symbol="x_{109}"/>
                                                                        <function symbol="+">
                                                                         <constant symbol="1"/>
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{110}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <variable symbol="x_{111}"/>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                       <constant symbol="1"/>
                                                                      </function>
                                                                      <variable symbol="x_{112}"/>
                                                                     </function>
                                                                    </function>
                                                                   </constantatomformula>
                                                                  </formulalist>
                                                                 </sequent>
                                                                 <rule type="axiom" symbol="axiom">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <variable symbol="x_{109}"/>
                                                                       <function symbol="+">
                                                                        <constant symbol="1"/>
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{110}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <variable symbol="x_{111}"/>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                       </function>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                 </rule>
                                                                 <rule type="eqr2" symbol="=:r2" param="(2)">
                                                                  <sequent>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{109}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <variable symbol="x_{112}"/>
                                                                      </function>
                                                                     </function>
                                                                     <constant symbol="1"/>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                   <formulalist>
                                                                    <constantatomformula symbol="=">
                                                                     <constant symbol="1"/>
                                                                     <function symbol="f">
                                                                      <function symbol="+">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <variable symbol="x_{109}"/>
                                                                         <function symbol="+">
                                                                          <constant symbol="1"/>
                                                                          <function symbol="+">
                                                                           <variable symbol="x_{110}"/>
                                                                           <function symbol="+">
                                                                            <constant symbol="1"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <variable symbol="x_{111}"/>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                        </function>
                                                                        <constant symbol="1"/>
                                                                       </function>
                                                                       <variable symbol="x_{112}"/>
                                                                      </function>
                                                                     </function>
                                                                    </constantatomformula>
                                                                   </formulalist>
                                                                  </sequent>
                                                                  <rule type="axiom" symbol="axiom">
                                                                   <sequent>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                        <variable symbol="x_{112}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <function symbol="f">
                                                                       <function symbol="+">
                                                                        <function symbol="+">
                                                                         <function symbol="+">
                                                                          <variable symbol="x_{109}"/>
                                                                          <function symbol="+">
                                                                           <constant symbol="1"/>
                                                                           <function symbol="+">
                                                                            <variable symbol="x_{110}"/>
                                                                            <function symbol="+">
                                                                             <constant symbol="1"/>
                                                                             <function symbol="+">
                                                                              <constant symbol="1"/>
                                                                              <variable symbol="x_{111}"/>
                                                                             </function>
                                                                            </function>
                                                                           </function>
                                                                          </function>
                                                                         </function>
                                                                         <constant symbol="1"/>
                                                                        </function>
                                                                        <variable symbol="x_{112}"/>
                                                                       </function>
                                                                      </function>
                                                                      <constant symbol="1"/>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                  </rule>
                                                                  <rule type="axiom" symbol="axiom">
                                                                   <sequent>
                                                                    <formulalist/>
                                                                    <formulalist>
                                                                     <constantatomformula symbol="=">
                                                                      <constant symbol="1"/>
                                                                      <constant symbol="1"/>
                                                                     </constantatomformula>
                                                                    </formulalist>
                                                                   </sequent>
                                                                  </rule>
                                                                 </rule>
                                                                </rule>
                                                               </rule>
                                                              </rule>
                                                             </rule>
                                                            </rule>
                                                           </rule>
                                                          </rule>
                                                         </rule>
                                                        </rule>
                                                       </rule>
                                                      </rule>
                                                     </rule>
                                                    </rule>
                                                   </rule>
                                                  </rule>
                                                 </rule>
                                                </rule>
                                               </rule>
                                              </rule>
                                             </rule>
                                            </rule>
                                           </rule>
                                          </rule>
                                         </rule>
                                        </rule>
                                       </rule>
                                      </rule>
                                     </rule>
                                    </rule>
                                   </rule>
                                  </rule>
                                 </rule>
                                </rule>
                               </rule>
                              </rule>
                             </rule>
                            </rule>
                           </rule>
                          </rule>
                         </rule>
                        </rule>
                       </rule>
                      </rule>
                     </rule>
                    </rule>
                   </rule>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <sequentlist symbol="clause set">
  <sequent>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="f">
      <function symbol="+">
       <variable symbol="n_0"/>
       <variable symbol="k_0"/>
      </function>
     </function>
     <constant symbol="1"/>
    </constantatomformula>
    <constantatomformula symbol="=">
     <function symbol="f">
      <function symbol="+">
       <function symbol="+">
        <function symbol="+">
         <variable symbol="n_0"/>
         <variable symbol="k_0"/>
        </function>
        <constant symbol="1"/>
       </function>
       <variable symbol="k_1"/>
      </function>
     </function>
     <constant symbol="1"/>
    </constantatomformula>
   </formulalist>
   <formulalist/>
  </sequent>
  <sequent>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="f">
      <function symbol="+">
       <variable symbol="n_0"/>
       <variable symbol="k_0"/>
      </function>
     </function>
     <constant symbol="0"/>
    </constantatomformula>
    <constantatomformula symbol="=">
     <function symbol="f">
      <function symbol="+">
       <function symbol="+">
        <function symbol="+">
         <variable symbol="n_0"/>
         <variable symbol="k_0"/>
        </function>
        <constant symbol="1"/>
       </function>
       <variable symbol="k_1"/>
      </function>
     </function>
     <constant symbol="0"/>
    </constantatomformula>
   </formulalist>
   <formulalist/>
  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="=">
     <function symbol="f">
      <function symbol="+">
       <variable symbol="n_0"/>
       <variable symbol="n_1"/>
      </function>
     </function>
     <constant symbol="0"/>
    </constantatomformula>
    <constantatomformula symbol="=">
     <function symbol="f">
      <function symbol="+">
       <variable symbol="n_1"/>
       <variable symbol="n_0"/>
      </function>
     </function>
     <constant symbol="1"/>
    </constantatomformula>
   </formulalist>
  </sequent>
 </sequentlist>
 <variabledefinitions/>
</proofdatabase>
