<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE proofdatabase
SYSTEM "http://www.logic.at/ceres/xml/4.0/proofdatabase.dtd">
<proofdatabase>
 <definitionlist/>
 <axiomset/>
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                 <variable symbol="k"/>
                </constantatomformula>
               </formulalist>
              </sequent>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="p_{sk}" calculus="LK">
  <rule type="cut" symbol="cut">
   <sequent>
    <formulalist>
     <quantifiedformula type="all">
      <variable symbol="m"/>
      <quantifiedformula type="all">
       <variable symbol="n"/>
       <conjunctiveformula type="impl">
        <constantatomformula symbol="P">
         <variable symbol="m"/>
         <variable symbol="n"/>
        </constantatomformula>
        <constantatomformula symbol="Q">
         <variable symbol="m"/>
         <variable symbol="n"/>
        </constantatomformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
    </formulalist>
    <formulalist>
     <quantifiedformula type="exists">
      <variable symbol="m"/>
      <quantifiedformula type="exists">
       <variable symbol="n"/>
       <conjunctiveformula type="impl">
        <conjunctiveformula type="neg">
         <constantatomformula symbol="Q">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
        <conjunctiveformula type="neg">
         <constantatomformula symbol="P">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
    </formulalist>
   </sequent>
   <rule type="defr" symbol="d:r" param="();[()];()">
    <sequent>
     <formulalist>
      <quantifiedformula type="all">
       <variable symbol="m"/>
       <quantifiedformula type="all">
        <variable symbol="n"/>
        <conjunctiveformula type="impl">
         <constantatomformula symbol="P">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
         <constantatomformula symbol="Q">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="φ_{cutf}" definition="φ_{cutf}"/>
     </formulalist>
    </sequent>
    <rule type="forallr" symbol="\all:r">
     <sequent>
      <formulalist>
       <quantifiedformula type="all">
        <variable symbol="m"/>
        <quantifiedformula type="all">
         <variable symbol="n"/>
         <conjunctiveformula type="impl">
          <constantatomformula symbol="P">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
        </quantifiedformula>
       </quantifiedformula>
      </formulalist>
      <formulalist>
       <quantifiedformula type="all">
        <variable symbol="m"/>
        <quantifiedformula type="exists">
         <variable symbol="n"/>
         <conjunctiveformula type="or">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="P">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
        </quantifiedformula>
       </quantifiedformula>
      </formulalist>
     </sequent>
     <rule type="existsr" symbol="\ex:r">
      <sequent>
       <formulalist>
        <quantifiedformula type="all">
         <variable symbol="m"/>
         <quantifiedformula type="all">
          <variable symbol="n"/>
          <conjunctiveformula type="impl">
           <constantatomformula symbol="P">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
           <constantatomformula symbol="Q">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
         </quantifiedformula>
        </quantifiedformula>
       </formulalist>
       <formulalist>
        <quantifiedformula type="exists">
         <variable symbol="n"/>
         <conjunctiveformula type="or">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="P">
            <variable symbol="k"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
          <constantatomformula symbol="Q">
           <variable symbol="k"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
        </quantifiedformula>
       </formulalist>
      </sequent>
      <rule type="foralll" symbol="\all:l">
       <sequent>
        <formulalist>
         <quantifiedformula type="all">
          <variable symbol="m"/>
          <quantifiedformula type="all">
           <variable symbol="n"/>
           <conjunctiveformula type="impl">
            <constantatomformula symbol="P">
             <variable symbol="m"/>
             <variable symbol="n"/>
            </constantatomformula>
            <constantatomformula symbol="Q">
             <variable symbol="m"/>
             <variable symbol="n"/>
            </constantatomformula>
           </conjunctiveformula>
          </quantifiedformula>
         </quantifiedformula>
        </formulalist>
        <formulalist>
         <conjunctiveformula type="or">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="P">
            <variable symbol="k"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
          <constantatomformula symbol="Q">
           <variable symbol="k"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
        </formulalist>
       </sequent>
       <rule type="foralll" symbol="\all:l">
        <sequent>
         <formulalist>
          <quantifiedformula type="all">
           <variable symbol="n"/>
           <conjunctiveformula type="impl">
            <constantatomformula symbol="P">
             <variable symbol="k"/>
             <variable symbol="n"/>
            </constantatomformula>
            <constantatomformula symbol="Q">
             <variable symbol="k"/>
             <variable symbol="n"/>
            </constantatomformula>
           </conjunctiveformula>
          </quantifiedformula>
         </formulalist>
         <formulalist>
          <conjunctiveformula type="or">
           <conjunctiveformula type="neg">
            <constantatomformula symbol="P">
             <variable symbol="k"/>
             <variable symbol="n"/>
            </constantatomformula>
           </conjunctiveformula>
           <constantatomformula symbol="Q">
            <variable symbol="k"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
         </formulalist>
        </sequent>
        <rule type="contrr" symbol="c:r" param="2">
         <sequent>
          <formulalist>
           <conjunctiveformula type="impl">
            <constantatomformula symbol="P">
             <variable symbol="k"/>
             <variable symbol="n"/>
            </constantatomformula>
            <constantatomformula symbol="Q">
             <variable symbol="k"/>
             <variable symbol="n"/>
            </constantatomformula>
           </conjunctiveformula>
          </formulalist>
          <formulalist>
           <conjunctiveformula type="or">
            <conjunctiveformula type="neg">
             <constantatomformula symbol="P">
              <variable symbol="k"/>
              <variable symbol="n"/>
             </constantatomformula>
            </conjunctiveformula>
            <constantatomformula symbol="Q">
             <variable symbol="k"/>
             <variable symbol="n"/>
            </constantatomformula>
           </conjunctiveformula>
          </formulalist>
         </sequent>
         <rule type="orr1" symbol="\or:r1">
          <sequent>
           <formulalist>
            <conjunctiveformula type="impl">
             <constantatomformula symbol="P">
              <variable symbol="k"/>
              <variable symbol="n"/>
             </constantatomformula>
             <constantatomformula symbol="Q">
              <variable symbol="k"/>
              <variable symbol="n"/>
             </constantatomformula>
            </conjunctiveformula>
           </formulalist>
           <formulalist>
            <conjunctiveformula type="or">
             <conjunctiveformula type="neg">
              <constantatomformula symbol="P">
               <variable symbol="k"/>
               <variable symbol="n"/>
              </constantatomformula>
             </conjunctiveformula>
             <constantatomformula symbol="Q">
              <variable symbol="k"/>
              <variable symbol="n"/>
             </constantatomformula>
            </conjunctiveformula>
            <conjunctiveformula type="or">
             <conjunctiveformula type="neg">
              <constantatomformula symbol="P">
               <variable symbol="k"/>
               <variable symbol="n"/>
              </constantatomformula>
             </conjunctiveformula>
             <constantatomformula symbol="Q">
              <variable symbol="k"/>
              <variable symbol="n"/>
             </constantatomformula>
            </conjunctiveformula>
           </formulalist>
          </sequent>
          <rule type="permr" symbol="\pi:r" param="(1 2)">
           <sequent>
            <formulalist>
             <conjunctiveformula type="impl">
              <constantatomformula symbol="P">
               <variable symbol="k"/>
               <variable symbol="n"/>
              </constantatomformula>
              <constantatomformula symbol="Q">
               <variable symbol="k"/>
               <variable symbol="n"/>
              </constantatomformula>
             </conjunctiveformula>
            </formulalist>
            <formulalist>
             <conjunctiveformula type="or">
              <conjunctiveformula type="neg">
               <constantatomformula symbol="P">
                <variable symbol="k"/>
                <variable symbol="n"/>
               </constantatomformula>
              </conjunctiveformula>
              <constantatomformula symbol="Q">
               <variable symbol="k"/>
               <variable symbol="n"/>
              </constantatomformula>
             </conjunctiveformula>
             <conjunctiveformula type="neg">
              <constantatomformula symbol="P">
               <variable symbol="k"/>
               <variable symbol="n"/>
              </constantatomformula>
             </conjunctiveformula>
            </formulalist>
           </sequent>
           <rule type="orr2" symbol="\or:r2">
            <sequent>
             <formulalist>
              <conjunctiveformula type="impl">
               <constantatomformula symbol="P">
                <variable symbol="k"/>
                <variable symbol="n"/>
               </constantatomformula>
               <constantatomformula symbol="Q">
                <variable symbol="k"/>
                <variable symbol="n"/>
               </constantatomformula>
              </conjunctiveformula>
             </formulalist>
             <formulalist>
              <conjunctiveformula type="neg">
               <constantatomformula symbol="P">
                <variable symbol="k"/>
                <variable symbol="n"/>
               </constantatomformula>
              </conjunctiveformula>
              <conjunctiveformula type="or">
               <conjunctiveformula type="neg">
                <constantatomformula symbol="P">
                 <variable symbol="k"/>
                 <variable symbol="n"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="Q">
                <variable symbol="k"/>
                <variable symbol="n"/>
               </constantatomformula>
              </conjunctiveformula>
             </formulalist>
            </sequent>
            <rule type="impll" symbol="\impl:l">
             <sequent>
              <formulalist>
               <conjunctiveformula type="impl">
                <constantatomformula symbol="P">
                 <variable symbol="k"/>
                 <variable symbol="n"/>
                </constantatomformula>
                <constantatomformula symbol="Q">
                 <variable symbol="k"/>
                 <variable symbol="n"/>
                </constantatomformula>
               </conjunctiveformula>
              </formulalist>
              <formulalist>
               <conjunctiveformula type="neg">
                <constantatomformula symbol="P">
                 <variable symbol="k"/>
                 <variable symbol="n"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="Q">
                <variable symbol="k"/>
                <variable symbol="n"/>
               </constantatomformula>
              </formulalist>
             </sequent>
             <rule type="permr" symbol="\pi:r" param="(1 2)">
              <sequent>
               <formulalist/>
               <formulalist>
                <conjunctiveformula type="neg">
                 <constantatomformula symbol="P">
                  <variable symbol="k"/>
                  <variable symbol="n"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="P">
                 <variable symbol="k"/>
                 <variable symbol="n"/>
                </constantatomformula>
               </formulalist>
              </sequent>
              <rule type="negr" symbol="\neg:r">
               <sequent>
                <formulalist/>
                <formulalist>
                 <constantatomformula symbol="P">
                  <variable symbol="k"/>
                  <variable symbol="n"/>
                 </constantatomformula>
                 <conjunctiveformula type="neg">
                  <constantatomformula symbol="P">
                   <variable symbol="k"/>
                   <variable symbol="n"/>
                  </constantatomformula>
                 </conjunctiveformula>
                </formulalist>
               </sequent>
               <rule type="axiom" symbol="axiom">
                <sequent>
                 <formulalist>
                  <constantatomformula symbol="P">
                   <variable symbol="k"/>
                   <variable symbol="n"/>
                  </constantatomformula>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="P">
                   <variable symbol="k"/>
                   <variable symbol="n"/>
                  </constantatomformula>
                 </formulalist>
                </sequent>
               </rule>
              </rule>
             </rule>
             <rule type="axiom" symbol="axiom">
              <sequent>
               <formulalist>
                <constantatomformula symbol="Q">
                 <variable symbol="k"/>
                 <variable symbol="n"/>
                </constantatomformula>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="Q">
                 <variable symbol="k"/>
                 <variable symbol="n"/>
                </constantatomformula>
               </formulalist>
              </sequent>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
   <rule type="defl" symbol="d:l" param="();[()];()">
    <sequent>
     <formulalist>
      <constantatomformula symbol="φ_{cutf}" definition="φ_{cutf}"/>
     </formulalist>
     <formulalist>
      <quantifiedformula type="exists">
       <variable symbol="m"/>
       <quantifiedformula type="exists">
        <variable symbol="n"/>
        <conjunctiveformula type="impl">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
         <conjunctiveformula type="neg">
          <constantatomformula symbol="P">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
     </formulalist>
    </sequent>
    <rule type="foralll" symbol="\all:l">
     <sequent>
      <formulalist>
       <quantifiedformula type="all">
        <variable symbol="m"/>
        <quantifiedformula type="exists">
         <variable symbol="n"/>
         <conjunctiveformula type="or">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="P">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
        </quantifiedformula>
       </quantifiedformula>
      </formulalist>
      <formulalist>
       <quantifiedformula type="exists">
        <variable symbol="m"/>
        <quantifiedformula type="exists">
         <variable symbol="n"/>
         <conjunctiveformula type="impl">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="Q">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
          <conjunctiveformula type="neg">
           <constantatomformula symbol="P">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
         </conjunctiveformula>
        </quantifiedformula>
       </quantifiedformula>
      </formulalist>
     </sequent>
     <rule type="existsl" symbol="\ex:l">
      <sequent>
       <formulalist>
        <quantifiedformula type="exists">
         <variable symbol="n"/>
         <conjunctiveformula type="or">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="P">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
        </quantifiedformula>
       </formulalist>
       <formulalist>
        <quantifiedformula type="exists">
         <variable symbol="m"/>
         <quantifiedformula type="exists">
          <variable symbol="n"/>
          <conjunctiveformula type="impl">
           <conjunctiveformula type="neg">
            <constantatomformula symbol="Q">
             <variable symbol="m"/>
             <variable symbol="n"/>
            </constantatomformula>
           </conjunctiveformula>
           <conjunctiveformula type="neg">
            <constantatomformula symbol="P">
             <variable symbol="m"/>
             <variable symbol="n"/>
            </constantatomformula>
           </conjunctiveformula>
          </conjunctiveformula>
         </quantifiedformula>
        </quantifiedformula>
       </formulalist>
      </sequent>
      <rule type="existsr" symbol="\ex:r">
       <sequent>
        <formulalist>
         <conjunctiveformula type="or">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="P">
            <variable symbol="m"/>
            <variable symbol="k"/>
           </constantatomformula>
          </conjunctiveformula>
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="k"/>
          </constantatomformula>
         </conjunctiveformula>
        </formulalist>
        <formulalist>
         <quantifiedformula type="exists">
          <variable symbol="m"/>
          <quantifiedformula type="exists">
           <variable symbol="n"/>
           <conjunctiveformula type="impl">
            <conjunctiveformula type="neg">
             <constantatomformula symbol="Q">
              <variable symbol="m"/>
              <variable symbol="n"/>
             </constantatomformula>
            </conjunctiveformula>
            <conjunctiveformula type="neg">
             <constantatomformula symbol="P">
              <variable symbol="m"/>
              <variable symbol="n"/>
             </constantatomformula>
            </conjunctiveformula>
           </conjunctiveformula>
          </quantifiedformula>
         </quantifiedformula>
        </formulalist>
       </sequent>
       <rule type="existsr" symbol="\ex:r">
        <sequent>
         <formulalist>
          <conjunctiveformula type="or">
           <conjunctiveformula type="neg">
            <constantatomformula symbol="P">
             <variable symbol="m"/>
             <variable symbol="k"/>
            </constantatomformula>
           </conjunctiveformula>
           <constantatomformula symbol="Q">
            <variable symbol="m"/>
            <variable symbol="k"/>
           </constantatomformula>
          </conjunctiveformula>
         </formulalist>
         <formulalist>
          <quantifiedformula type="exists">
           <variable symbol="n"/>
           <conjunctiveformula type="impl">
            <conjunctiveformula type="neg">
             <constantatomformula symbol="Q">
              <variable symbol="m"/>
              <variable symbol="n"/>
             </constantatomformula>
            </conjunctiveformula>
            <conjunctiveformula type="neg">
             <constantatomformula symbol="P">
              <variable symbol="m"/>
              <variable symbol="n"/>
             </constantatomformula>
            </conjunctiveformula>
           </conjunctiveformula>
          </quantifiedformula>
         </formulalist>
        </sequent>
        <rule type="implr" symbol="\impl:r">
         <sequent>
          <formulalist>
           <conjunctiveformula type="or">
            <conjunctiveformula type="neg">
             <constantatomformula symbol="P">
              <variable symbol="m"/>
              <variable symbol="k"/>
             </constantatomformula>
            </conjunctiveformula>
            <constantatomformula symbol="Q">
             <variable symbol="m"/>
             <variable symbol="k"/>
            </constantatomformula>
           </conjunctiveformula>
          </formulalist>
          <formulalist>
           <conjunctiveformula type="impl">
            <conjunctiveformula type="neg">
             <constantatomformula symbol="Q">
              <variable symbol="m"/>
              <variable symbol="k"/>
             </constantatomformula>
            </conjunctiveformula>
            <conjunctiveformula type="neg">
             <constantatomformula symbol="P">
              <variable symbol="m"/>
              <variable symbol="k"/>
             </constantatomformula>
            </conjunctiveformula>
           </conjunctiveformula>
          </formulalist>
         </sequent>
         <rule type="perml" symbol="\pi:l" param="(1 2)">
          <sequent>
           <formulalist>
            <conjunctiveformula type="neg">
             <constantatomformula symbol="Q">
              <variable symbol="m"/>
              <variable symbol="k"/>
             </constantatomformula>
            </conjunctiveformula>
            <conjunctiveformula type="or">
             <conjunctiveformula type="neg">
              <constantatomformula symbol="P">
               <variable symbol="m"/>
               <variable symbol="k"/>
              </constantatomformula>
             </conjunctiveformula>
             <constantatomformula symbol="Q">
              <variable symbol="m"/>
              <variable symbol="k"/>
             </constantatomformula>
            </conjunctiveformula>
           </formulalist>
           <formulalist>
            <conjunctiveformula type="neg">
             <constantatomformula symbol="P">
              <variable symbol="m"/>
              <variable symbol="k"/>
             </constantatomformula>
            </conjunctiveformula>
           </formulalist>
          </sequent>
          <rule type="orl" symbol="\or:l">
           <sequent>
            <formulalist>
             <conjunctiveformula type="or">
              <conjunctiveformula type="neg">
               <constantatomformula symbol="P">
                <variable symbol="m"/>
                <variable symbol="k"/>
               </constantatomformula>
              </conjunctiveformula>
              <constantatomformula symbol="Q">
               <variable symbol="m"/>
               <variable symbol="k"/>
              </constantatomformula>
             </conjunctiveformula>
             <conjunctiveformula type="neg">
              <constantatomformula symbol="Q">
               <variable symbol="m"/>
               <variable symbol="k"/>
              </constantatomformula>
             </conjunctiveformula>
            </formulalist>
            <formulalist>
             <conjunctiveformula type="neg">
              <constantatomformula symbol="P">
               <variable symbol="m"/>
               <variable symbol="k"/>
              </constantatomformula>
             </conjunctiveformula>
            </formulalist>
           </sequent>
           <rule type="negl" symbol="\neg:l">
            <sequent>
             <formulalist>
              <conjunctiveformula type="neg">
               <constantatomformula symbol="P">
                <variable symbol="m"/>
                <variable symbol="k"/>
               </constantatomformula>
              </conjunctiveformula>
             </formulalist>
             <formulalist>
              <conjunctiveformula type="neg">
               <constantatomformula symbol="P">
                <variable symbol="m"/>
                <variable symbol="k"/>
               </constantatomformula>
              </conjunctiveformula>
             </formulalist>
            </sequent>
            <rule type="permr" symbol="\pi:r" param="(1 2)">
             <sequent>
              <formulalist/>
              <formulalist>
               <conjunctiveformula type="neg">
                <constantatomformula symbol="P">
                 <variable symbol="m"/>
                 <variable symbol="k"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="P">
                <variable symbol="m"/>
                <variable symbol="k"/>
               </constantatomformula>
              </formulalist>
             </sequent>
             <rule type="negr" symbol="\neg:r">
              <sequent>
               <formulalist/>
               <formulalist>
                <constantatomformula symbol="P">
                 <variable symbol="m"/>
                 <variable symbol="k"/>
                </constantatomformula>
                <conjunctiveformula type="neg">
                 <constantatomformula symbol="P">
                  <variable symbol="m"/>
                  <variable symbol="k"/>
                 </constantatomformula>
                </conjunctiveformula>
               </formulalist>
              </sequent>
              <rule type="axiom" symbol="axiom">
               <sequent>
                <formulalist>
                 <constantatomformula symbol="P">
                  <variable symbol="m"/>
                  <variable symbol="k"/>
                 </constantatomformula>
                </formulalist>
                <formulalist>
                 <constantatomformula symbol="P">
                  <variable symbol="m"/>
                  <variable symbol="k"/>
                 </constantatomformula>
                </formulalist>
               </sequent>
              </rule>
             </rule>
            </rule>
           </rule>
           <rule type="perml" symbol="\pi:l" param="(1 2)">
            <sequent>
             <formulalist>
              <constantatomformula symbol="Q">
               <variable symbol="m"/>
               <variable symbol="k"/>
              </constantatomformula>
              <conjunctiveformula type="neg">
               <constantatomformula symbol="Q">
                <variable symbol="m"/>
                <variable symbol="k"/>
               </constantatomformula>
              </conjunctiveformula>
             </formulalist>
             <formulalist/>
            </sequent>
            <rule type="negl" symbol="\neg:l">
             <sequent>
              <formulalist>
               <conjunctiveformula type="neg">
                <constantatomformula symbol="Q">
                 <variable symbol="m"/>
                 <variable symbol="k"/>
                </constantatomformula>
               </conjunctiveformula>
               <constantatomformula symbol="Q">
                <variable symbol="m"/>
                <variable symbol="k"/>
               </constantatomformula>
              </formulalist>
              <formulalist/>
             </sequent>
             <rule type="axiom" symbol="axiom">
              <sequent>
               <formulalist>
                <constantatomformula symbol="Q">
                 <variable symbol="m"/>
                 <variable symbol="k"/>
                </constantatomformula>
               </formulalist>
               <formulalist>
                <constantatomformula symbol="Q">
                 <variable symbol="m"/>
                 <variable symbol="k"/>
                </constantatomformula>
               </formulalist>
              </sequent>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{1}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <quantifiedformula type="all">
      <variable symbol="m"/>
      <quantifiedformula type="all">
       <variable symbol="n"/>
       <conjunctiveformula type="impl">
        <constantatomformula symbol="P">
         <variable symbol="m"/>
         <variable symbol="n"/>
        </constantatomformula>
        <constantatomformula symbol="Q">
         <variable symbol="m"/>
         <variable symbol="n"/>
        </constantatomformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
     <constantatomformula symbol="Q">
      <variable symbol="m"/>
      <variable symbol="k"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <quantifiedformula type="exists">
      <variable symbol="m"/>
      <quantifiedformula type="exists">
       <variable symbol="n"/>
       <conjunctiveformula type="impl">
        <conjunctiveformula type="neg">
         <constantatomformula symbol="Q">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
        <conjunctiveformula type="neg">
         <constantatomformula symbol="P">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
    </formulalist>
   </sequent>
   <rule type="existsr" symbol="\ex:r">
    <sequent>
     <formulalist>
      <constantatomformula symbol="Q">
       <variable symbol="m"/>
       <variable symbol="k"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <quantifiedformula type="exists">
       <variable symbol="m"/>
       <quantifiedformula type="exists">
        <variable symbol="n"/>
        <conjunctiveformula type="impl">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
         <conjunctiveformula type="neg">
          <constantatomformula symbol="P">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
     </formulalist>
    </sequent>
    <rule type="existsr" symbol="\ex:r">
     <sequent>
      <formulalist>
       <constantatomformula symbol="Q">
        <variable symbol="m"/>
        <variable symbol="k"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <quantifiedformula type="exists">
        <variable symbol="n"/>
        <conjunctiveformula type="impl">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
         <conjunctiveformula type="neg">
          <constantatomformula symbol="P">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
        </conjunctiveformula>
       </quantifiedformula>
      </formulalist>
     </sequent>
     <rule type="implr" symbol="\impl:r">
      <sequent>
       <formulalist>
        <constantatomformula symbol="Q">
         <variable symbol="m"/>
         <variable symbol="k"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <conjunctiveformula type="impl">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="k"/>
          </constantatomformula>
         </conjunctiveformula>
         <conjunctiveformula type="neg">
          <constantatomformula symbol="P">
           <variable symbol="m"/>
           <variable symbol="k"/>
          </constantatomformula>
         </conjunctiveformula>
        </conjunctiveformula>
       </formulalist>
      </sequent>
      <rule type="weakr" symbol="w:r">
       <sequent>
        <formulalist>
         <conjunctiveformula type="neg">
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="k"/>
          </constantatomformula>
         </conjunctiveformula>
         <constantatomformula symbol="Q">
          <variable symbol="m"/>
          <variable symbol="k"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <conjunctiveformula type="neg">
          <constantatomformula symbol="P">
           <variable symbol="m"/>
           <variable symbol="k"/>
          </constantatomformula>
         </conjunctiveformula>
        </formulalist>
       </sequent>
       <rule type="negl" symbol="\neg:l">
        <sequent>
         <formulalist>
          <conjunctiveformula type="neg">
           <constantatomformula symbol="Q">
            <variable symbol="m"/>
            <variable symbol="k"/>
           </constantatomformula>
          </conjunctiveformula>
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="k"/>
          </constantatomformula>
         </formulalist>
         <formulalist/>
        </sequent>
        <rule type="axiom" symbol="axiom">
         <sequent>
          <formulalist>
           <constantatomformula symbol="Q">
            <variable symbol="m"/>
            <variable symbol="k"/>
           </constantatomformula>
          </formulalist>
          <formulalist>
           <constantatomformula symbol="Q">
            <variable symbol="m"/>
            <variable symbol="k"/>
           </constantatomformula>
          </formulalist>
         </sequent>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{2}" calculus="LK">
  <rule type="weakl" symbol="w:l">
   <sequent>
    <formulalist>
     <quantifiedformula type="all">
      <variable symbol="m"/>
      <quantifiedformula type="all">
       <variable symbol="n"/>
       <conjunctiveformula type="impl">
        <constantatomformula symbol="P">
         <variable symbol="m"/>
         <variable symbol="n"/>
        </constantatomformula>
        <constantatomformula symbol="Q">
         <variable symbol="m"/>
         <variable symbol="n"/>
        </constantatomformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="P">
      <variable symbol="m"/>
      <variable symbol="k"/>
     </constantatomformula>
     <quantifiedformula type="exists">
      <variable symbol="m"/>
      <quantifiedformula type="exists">
       <variable symbol="n"/>
       <conjunctiveformula type="impl">
        <conjunctiveformula type="neg">
         <constantatomformula symbol="Q">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
        <conjunctiveformula type="neg">
         <constantatomformula symbol="P">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
    </formulalist>
   </sequent>
   <rule type="existsr" symbol="\ex:r">
    <sequent>
     <formulalist/>
     <formulalist>
      <constantatomformula symbol="P">
       <variable symbol="m"/>
       <variable symbol="k"/>
      </constantatomformula>
      <quantifiedformula type="exists">
       <variable symbol="m"/>
       <quantifiedformula type="exists">
        <variable symbol="n"/>
        <conjunctiveformula type="impl">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
         <conjunctiveformula type="neg">
          <constantatomformula symbol="P">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
     </formulalist>
    </sequent>
    <rule type="existsr" symbol="\ex:r">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="P">
        <variable symbol="m"/>
        <variable symbol="k"/>
       </constantatomformula>
       <quantifiedformula type="exists">
        <variable symbol="n"/>
        <conjunctiveformula type="impl">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
         <conjunctiveformula type="neg">
          <constantatomformula symbol="P">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
        </conjunctiveformula>
       </quantifiedformula>
      </formulalist>
     </sequent>
     <rule type="implr" symbol="\impl:r">
      <sequent>
       <formulalist/>
       <formulalist>
        <constantatomformula symbol="P">
         <variable symbol="m"/>
         <variable symbol="k"/>
        </constantatomformula>
        <conjunctiveformula type="impl">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="k"/>
          </constantatomformula>
         </conjunctiveformula>
         <conjunctiveformula type="neg">
          <constantatomformula symbol="P">
           <variable symbol="m"/>
           <variable symbol="k"/>
          </constantatomformula>
         </conjunctiveformula>
        </conjunctiveformula>
       </formulalist>
      </sequent>
      <rule type="weakl" symbol="w:l">
       <sequent>
        <formulalist>
         <conjunctiveformula type="neg">
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="k"/>
          </constantatomformula>
         </conjunctiveformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="P">
          <variable symbol="m"/>
          <variable symbol="k"/>
         </constantatomformula>
         <conjunctiveformula type="neg">
          <constantatomformula symbol="P">
           <variable symbol="m"/>
           <variable symbol="k"/>
          </constantatomformula>
         </conjunctiveformula>
        </formulalist>
       </sequent>
       <rule type="negr" symbol="\neg:r">
        <sequent>
         <formulalist/>
         <formulalist>
          <constantatomformula symbol="P">
           <variable symbol="m"/>
           <variable symbol="k"/>
          </constantatomformula>
          <conjunctiveformula type="neg">
           <constantatomformula symbol="P">
            <variable symbol="m"/>
            <variable symbol="k"/>
           </constantatomformula>
          </conjunctiveformula>
         </formulalist>
        </sequent>
        <rule type="axiom" symbol="axiom">
         <sequent>
          <formulalist>
           <constantatomformula symbol="P">
            <variable symbol="m"/>
            <variable symbol="k"/>
           </constantatomformula>
          </formulalist>
          <formulalist>
           <constantatomformula symbol="P">
            <variable symbol="m"/>
            <variable symbol="k"/>
           </constantatomformula>
          </formulalist>
         </sequent>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="\psi_{3}" calculus="LK">
  <rule type="weakr" symbol="w:r">
   <sequent>
    <formulalist>
     <quantifiedformula type="all">
      <variable symbol="m"/>
      <quantifiedformula type="all">
       <variable symbol="n"/>
       <conjunctiveformula type="impl">
        <constantatomformula symbol="P">
         <variable symbol="m"/>
         <variable symbol="n"/>
        </constantatomformula>
        <constantatomformula symbol="Q">
         <variable symbol="m"/>
         <variable symbol="n"/>
        </constantatomformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
     <constantatomformula symbol="P">
      <variable symbol="k"/>
      <variable symbol="n"/>
     </constantatomformula>
    </formulalist>
    <formulalist>
     <constantatomformula symbol="Q">
      <variable symbol="k"/>
      <variable symbol="n"/>
     </constantatomformula>
     <quantifiedformula type="exists">
      <variable symbol="m"/>
      <quantifiedformula type="exists">
       <variable symbol="n"/>
       <conjunctiveformula type="impl">
        <conjunctiveformula type="neg">
         <constantatomformula symbol="Q">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
        <conjunctiveformula type="neg">
         <constantatomformula symbol="P">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
    </formulalist>
   </sequent>
   <rule type="foralll" symbol="\all:l">
    <sequent>
     <formulalist>
      <quantifiedformula type="all">
       <variable symbol="m"/>
       <quantifiedformula type="all">
        <variable symbol="n"/>
        <conjunctiveformula type="impl">
         <constantatomformula symbol="P">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
         <constantatomformula symbol="Q">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
      <constantatomformula symbol="P">
       <variable symbol="k"/>
       <variable symbol="n"/>
      </constantatomformula>
     </formulalist>
     <formulalist>
      <constantatomformula symbol="Q">
       <variable symbol="k"/>
       <variable symbol="n"/>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="foralll" symbol="\all:l">
     <sequent>
      <formulalist>
       <quantifiedformula type="all">
        <variable symbol="n"/>
        <conjunctiveformula type="impl">
         <constantatomformula symbol="P">
          <variable symbol="k"/>
          <variable symbol="n"/>
         </constantatomformula>
         <constantatomformula symbol="Q">
          <variable symbol="k"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
       </quantifiedformula>
       <constantatomformula symbol="P">
        <variable symbol="k"/>
        <variable symbol="n"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="Q">
        <variable symbol="k"/>
        <variable symbol="n"/>
       </constantatomformula>
      </formulalist>
     </sequent>
     <rule type="impll" symbol="\impl:l">
      <sequent>
       <formulalist>
        <conjunctiveformula type="impl">
         <constantatomformula symbol="P">
          <variable symbol="k"/>
          <variable symbol="n"/>
         </constantatomformula>
         <constantatomformula symbol="Q">
          <variable symbol="k"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
        <constantatomformula symbol="P">
         <variable symbol="k"/>
         <variable symbol="n"/>
        </constantatomformula>
       </formulalist>
       <formulalist>
        <constantatomformula symbol="Q">
         <variable symbol="k"/>
         <variable symbol="n"/>
        </constantatomformula>
       </formulalist>
      </sequent>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="P">
          <variable symbol="k"/>
          <variable symbol="n"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="P">
          <variable symbol="k"/>
          <variable symbol="n"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
      <rule type="axiom" symbol="axiom">
       <sequent>
        <formulalist>
         <constantatomformula symbol="Q">
          <variable symbol="k"/>
          <variable symbol="n"/>
         </constantatomformula>
        </formulalist>
        <formulalist>
         <constantatomformula symbol="Q">
          <variable symbol="k"/>
          <variable symbol="n"/>
         </constantatomformula>
        </formulalist>
       </sequent>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <proof symbol="refutation" calculus="RES">
  <rule type="res" symbol="r">
   <sequent>
    <formulalist/>
    <formulalist/>
   </sequent>
   <rule type="res" symbol="r">
    <sequent>
     <formulalist/>
     <formulalist>
      <constantatomformula symbol="Q">
       <variable symbol="x_{0}"/>
       <variable symbol="x_{1}"/>
      </constantatomformula>
     </formulalist>
    </sequent>
    <rule type="init" symbol="">
     <sequent>
      <formulalist/>
      <formulalist>
       <constantatomformula symbol="P">
        <variable symbol="x_{0}"/>
        <variable symbol="x_{1}"/>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
    <rule type="init" symbol="">
     <sequent>
      <formulalist>
       <constantatomformula symbol="P">
        <variable symbol="x_{0}"/>
        <variable symbol="x_{1}"/>
       </constantatomformula>
      </formulalist>
      <formulalist>
       <constantatomformula symbol="Q">
        <variable symbol="x_{0}"/>
        <variable symbol="x_{1}"/>
       </constantatomformula>
      </formulalist>
     </sequent>
    </rule>
   </rule>
   <rule type="init" symbol="">
    <sequent>
     <formulalist>
      <constantatomformula symbol="Q">
       <variable symbol="x_{0}"/>
       <variable symbol="x_{1}"/>
      </constantatomformula>
     </formulalist>
     <formulalist/>
    </sequent>
   </rule>
  </rule>
 </proof>
 <proof symbol="acnf" calculus="LK">
  <rule type="contrr" symbol="c:r" param="2">
   <sequent>
    <formulalist>
     <quantifiedformula type="all">
      <variable symbol="m"/>
      <quantifiedformula type="all">
       <variable symbol="n"/>
       <conjunctiveformula type="impl">
        <constantatomformula symbol="P">
         <variable symbol="m"/>
         <variable symbol="n"/>
        </constantatomformula>
        <constantatomformula symbol="Q">
         <variable symbol="m"/>
         <variable symbol="n"/>
        </constantatomformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
    </formulalist>
    <formulalist>
     <quantifiedformula type="exists">
      <variable symbol="m"/>
      <quantifiedformula type="exists">
       <variable symbol="n"/>
       <conjunctiveformula type="impl">
        <conjunctiveformula type="neg">
         <constantatomformula symbol="Q">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
        <conjunctiveformula type="neg">
         <constantatomformula symbol="P">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
       </conjunctiveformula>
      </quantifiedformula>
     </quantifiedformula>
    </formulalist>
   </sequent>
   <rule type="permr" symbol="\pi:r" param="">
    <sequent>
     <formulalist>
      <quantifiedformula type="all">
       <variable symbol="m"/>
       <quantifiedformula type="all">
        <variable symbol="n"/>
        <conjunctiveformula type="impl">
         <constantatomformula symbol="P">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
         <constantatomformula symbol="Q">
          <variable symbol="m"/>
          <variable symbol="n"/>
         </constantatomformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
     </formulalist>
     <formulalist>
      <quantifiedformula type="exists">
       <variable symbol="m"/>
       <quantifiedformula type="exists">
        <variable symbol="n"/>
        <conjunctiveformula type="impl">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
         <conjunctiveformula type="neg">
          <constantatomformula symbol="P">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
      <quantifiedformula type="exists">
       <variable symbol="m"/>
       <quantifiedformula type="exists">
        <variable symbol="n"/>
        <conjunctiveformula type="impl">
         <conjunctiveformula type="neg">
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
         <conjunctiveformula type="neg">
          <constantatomformula symbol="P">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
        </conjunctiveformula>
       </quantifiedformula>
      </quantifiedformula>
     </formulalist>
    </sequent>
    <rule type="contrl" symbol="c:l" param="2">
     <sequent>
      <formulalist>
       <quantifiedformula type="all">
        <variable symbol="m"/>
        <quantifiedformula type="all">
         <variable symbol="n"/>
         <conjunctiveformula type="impl">
          <constantatomformula symbol="P">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
          <constantatomformula symbol="Q">
           <variable symbol="m"/>
           <variable symbol="n"/>
          </constantatomformula>
         </conjunctiveformula>
        </quantifiedformula>
       </quantifiedformula>
      </formulalist>
      <formulalist>
       <quantifiedformula type="exists">
        <variable symbol="m"/>
        <quantifiedformula type="exists">
         <variable symbol="n"/>
         <conjunctiveformula type="impl">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="Q">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
          <conjunctiveformula type="neg">
           <constantatomformula symbol="P">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
         </conjunctiveformula>
        </quantifiedformula>
       </quantifiedformula>
       <quantifiedformula type="exists">
        <variable symbol="m"/>
        <quantifiedformula type="exists">
         <variable symbol="n"/>
         <conjunctiveformula type="impl">
          <conjunctiveformula type="neg">
           <constantatomformula symbol="Q">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
          <conjunctiveformula type="neg">
           <constantatomformula symbol="P">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
         </conjunctiveformula>
        </quantifiedformula>
       </quantifiedformula>
      </formulalist>
     </sequent>
     <rule type="perml" symbol="\pi:l" param="">
      <sequent>
       <formulalist>
        <quantifiedformula type="all">
         <variable symbol="m"/>
         <quantifiedformula type="all">
          <variable symbol="n"/>
          <conjunctiveformula type="impl">
           <constantatomformula symbol="P">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
           <constantatomformula symbol="Q">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
         </quantifiedformula>
        </quantifiedformula>
        <quantifiedformula type="all">
         <variable symbol="m"/>
         <quantifiedformula type="all">
          <variable symbol="n"/>
          <conjunctiveformula type="impl">
           <constantatomformula symbol="P">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
           <constantatomformula symbol="Q">
            <variable symbol="m"/>
            <variable symbol="n"/>
           </constantatomformula>
          </conjunctiveformula>
         </quantifiedformula>
        </quantifiedformula>
       </formulalist>
       <formulalist>
        <quantifiedformula type="exists">
         <variable symbol="m"/>
         <quantifiedformula type="exists">
          <variable symbol="n"/>
          <conjunctiveformula type="impl">
           <conjunctiveformula type="neg">
            <constantatomformula symbol="Q">
             <variable symbol="m"/>
             <variable symbol="n"/>
            </constantatomformula>
           </conjunctiveformula>
           <conjunctiveformula type="neg">
            <constantatomformula symbol="P">
             <variable symbol="m"/>
             <variable symbol="n"/>
            </constantatomformula>
           </conjunctiveformula>
          </conjunctiveformula>
         </quantifiedformula>
        </quantifiedformula>
        <quantifiedformula type="exists">
         <variable symbol="m"/>
         <quantifiedformula type="exists">
          <variable symbol="n"/>
          <conjunctiveformula type="impl">
           <conjunctiveformula type="neg">
            <constantatomformula symbol="Q">
             <variable symbol="m"/>
             <variable symbol="n"/>
            </constantatomformula>
           </conjunctiveformula>
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            <constantatomformula symbol="P">
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                <formulalist>
                 <constantatomformula symbol="Q">
                  <variable symbol="x_{0}"/>
                  <variable symbol="x_{1}"/>
                 </constantatomformula>
                 <quantifiedformula type="exists">
                  <variable symbol="m"/>
                  <quantifiedformula type="exists">
                   <variable symbol="n"/>
                   <conjunctiveformula type="impl">
                    <conjunctiveformula type="neg">
                     <constantatomformula symbol="Q">
                      <variable symbol="m"/>
                      <variable symbol="n"/>
                     </constantatomformula>
                    </conjunctiveformula>
                    <conjunctiveformula type="neg">
                     <constantatomformula symbol="P">
                      <variable symbol="m"/>
                      <variable symbol="n"/>
                     </constantatomformula>
                    </conjunctiveformula>
                   </conjunctiveformula>
                  </quantifiedformula>
                 </quantifiedformula>
                </formulalist>
               </sequent>
               <rule type="foralll" symbol="\all:l">
                <sequent>
                 <formulalist>
                  <quantifiedformula type="all">
                   <variable symbol="m"/>
                   <quantifiedformula type="all">
                    <variable symbol="n"/>
                    <conjunctiveformula type="impl">
                     <constantatomformula symbol="P">
                      <variable symbol="m"/>
                      <variable symbol="n"/>
                     </constantatomformula>
                     <constantatomformula symbol="Q">
                      <variable symbol="m"/>
                      <variable symbol="n"/>
                     </constantatomformula>
                    </conjunctiveformula>
                   </quantifiedformula>
                  </quantifiedformula>
                  <constantatomformula symbol="P">
                   <variable symbol="x_{0}"/>
                   <variable symbol="x_{1}"/>
                  </constantatomformula>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="Q">
                   <variable symbol="x_{0}"/>
                   <variable symbol="x_{1}"/>
                  </constantatomformula>
                 </formulalist>
                </sequent>
                <rule type="foralll" symbol="\all:l">
                 <sequent>
                  <formulalist>
                   <quantifiedformula type="all">
                    <variable symbol="n"/>
                    <conjunctiveformula type="impl">
                     <constantatomformula symbol="P">
                      <variable symbol="x_{0}"/>
                      <variable symbol="n"/>
                     </constantatomformula>
                     <constantatomformula symbol="Q">
                      <variable symbol="x_{0}"/>
                      <variable symbol="n"/>
                     </constantatomformula>
                    </conjunctiveformula>
                   </quantifiedformula>
                   <constantatomformula symbol="P">
                    <variable symbol="x_{0}"/>
                    <variable symbol="x_{1}"/>
                   </constantatomformula>
                  </formulalist>
                  <formulalist>
                   <constantatomformula symbol="Q">
                    <variable symbol="x_{0}"/>
                    <variable symbol="x_{1}"/>
                   </constantatomformula>
                  </formulalist>
                 </sequent>
                 <rule type="impll" symbol="\impl:l">
                  <sequent>
                   <formulalist>
                    <conjunctiveformula type="impl">
                     <constantatomformula symbol="P">
                      <variable symbol="x_{0}"/>
                      <variable symbol="x_{1}"/>
                     </constantatomformula>
                     <constantatomformula symbol="Q">
                      <variable symbol="x_{0}"/>
                      <variable symbol="x_{1}"/>
                     </constantatomformula>
                    </conjunctiveformula>
                    <constantatomformula symbol="P">
                     <variable symbol="x_{0}"/>
                     <variable symbol="x_{1}"/>
                    </constantatomformula>
                   </formulalist>
                   <formulalist>
                    <constantatomformula symbol="Q">
                     <variable symbol="x_{0}"/>
                     <variable symbol="x_{1}"/>
                    </constantatomformula>
                   </formulalist>
                  </sequent>
                  <rule type="axiom" symbol="axiom">
                   <sequent>
                    <formulalist>
                     <constantatomformula symbol="P">
                      <variable symbol="x_{0}"/>
                      <variable symbol="x_{1}"/>
                     </constantatomformula>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="P">
                      <variable symbol="x_{0}"/>
                      <variable symbol="x_{1}"/>
                     </constantatomformula>
                    </formulalist>
                   </sequent>
                  </rule>
                  <rule type="axiom" symbol="axiom">
                   <sequent>
                    <formulalist>
                     <constantatomformula symbol="Q">
                      <variable symbol="x_{0}"/>
                      <variable symbol="x_{1}"/>
                     </constantatomformula>
                    </formulalist>
                    <formulalist>
                     <constantatomformula symbol="Q">
                      <variable symbol="x_{0}"/>
                      <variable symbol="x_{1}"/>
                     </constantatomformula>
                    </formulalist>
                   </sequent>
                  </rule>
                 </rule>
                </rule>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
       <rule type="permr" symbol="\pi:r" param="">
        <sequent>
         <formulalist>
          <constantatomformula symbol="Q">
           <variable symbol="x_{0}"/>
           <variable symbol="x_{1}"/>
          </constantatomformula>
          <quantifiedformula type="all">
           <variable symbol="m"/>
           <quantifiedformula type="all">
            <variable symbol="n"/>
            <conjunctiveformula type="impl">
             <constantatomformula symbol="P">
              <variable symbol="m"/>
              <variable symbol="n"/>
             </constantatomformula>
             <constantatomformula symbol="Q">
              <variable symbol="m"/>
              <variable symbol="n"/>
             </constantatomformula>
            </conjunctiveformula>
           </quantifiedformula>
          </quantifiedformula>
         </formulalist>
         <formulalist>
          <quantifiedformula type="exists">
           <variable symbol="m"/>
           <quantifiedformula type="exists">
            <variable symbol="n"/>
            <conjunctiveformula type="impl">
             <conjunctiveformula type="neg">
              <constantatomformula symbol="Q">
               <variable symbol="m"/>
               <variable symbol="n"/>
              </constantatomformula>
             </conjunctiveformula>
             <conjunctiveformula type="neg">
              <constantatomformula symbol="P">
               <variable symbol="m"/>
               <variable symbol="n"/>
              </constantatomformula>
             </conjunctiveformula>
            </conjunctiveformula>
           </quantifiedformula>
          </quantifiedformula>
         </formulalist>
        </sequent>
        <rule type="perml" symbol="\pi:l" param="(1 2)">
         <sequent>
          <formulalist>
           <constantatomformula symbol="Q">
            <variable symbol="x_{0}"/>
            <variable symbol="x_{1}"/>
           </constantatomformula>
           <quantifiedformula type="all">
            <variable symbol="m"/>
            <quantifiedformula type="all">
             <variable symbol="n"/>
             <conjunctiveformula type="impl">
              <constantatomformula symbol="P">
               <variable symbol="m"/>
               <variable symbol="n"/>
              </constantatomformula>
              <constantatomformula symbol="Q">
               <variable symbol="m"/>
               <variable symbol="n"/>
              </constantatomformula>
             </conjunctiveformula>
            </quantifiedformula>
           </quantifiedformula>
          </formulalist>
          <formulalist>
           <quantifiedformula type="exists">
            <variable symbol="m"/>
            <quantifiedformula type="exists">
             <variable symbol="n"/>
             <conjunctiveformula type="impl">
              <conjunctiveformula type="neg">
               <constantatomformula symbol="Q">
                <variable symbol="m"/>
                <variable symbol="n"/>
               </constantatomformula>
              </conjunctiveformula>
              <conjunctiveformula type="neg">
               <constantatomformula symbol="P">
                <variable symbol="m"/>
                <variable symbol="n"/>
               </constantatomformula>
              </conjunctiveformula>
             </conjunctiveformula>
            </quantifiedformula>
           </quantifiedformula>
          </formulalist>
         </sequent>
         <rule type="weakl" symbol="w:l">
          <sequent>
           <formulalist>
            <quantifiedformula type="all">
             <variable symbol="m"/>
             <quantifiedformula type="all">
              <variable symbol="n"/>
              <conjunctiveformula type="impl">
               <constantatomformula symbol="P">
                <variable symbol="m"/>
                <variable symbol="n"/>
               </constantatomformula>
               <constantatomformula symbol="Q">
                <variable symbol="m"/>
                <variable symbol="n"/>
               </constantatomformula>
              </conjunctiveformula>
             </quantifiedformula>
            </quantifiedformula>
            <constantatomformula symbol="Q">
             <variable symbol="x_{0}"/>
             <variable symbol="x_{1}"/>
            </constantatomformula>
           </formulalist>
           <formulalist>
            <quantifiedformula type="exists">
             <variable symbol="m"/>
             <quantifiedformula type="exists">
              <variable symbol="n"/>
              <conjunctiveformula type="impl">
               <conjunctiveformula type="neg">
                <constantatomformula symbol="Q">
                 <variable symbol="m"/>
                 <variable symbol="n"/>
                </constantatomformula>
               </conjunctiveformula>
               <conjunctiveformula type="neg">
                <constantatomformula symbol="P">
                 <variable symbol="m"/>
                 <variable symbol="n"/>
                </constantatomformula>
               </conjunctiveformula>
              </conjunctiveformula>
             </quantifiedformula>
            </quantifiedformula>
           </formulalist>
          </sequent>
          <rule type="existsr" symbol="\ex:r">
           <sequent>
            <formulalist>
             <constantatomformula symbol="Q">
              <variable symbol="x_{0}"/>
              <variable symbol="x_{1}"/>
             </constantatomformula>
            </formulalist>
            <formulalist>
             <quantifiedformula type="exists">
              <variable symbol="m"/>
              <quantifiedformula type="exists">
               <variable symbol="n"/>
               <conjunctiveformula type="impl">
                <conjunctiveformula type="neg">
                 <constantatomformula symbol="Q">
                  <variable symbol="m"/>
                  <variable symbol="n"/>
                 </constantatomformula>
                </conjunctiveformula>
                <conjunctiveformula type="neg">
                 <constantatomformula symbol="P">
                  <variable symbol="m"/>
                  <variable symbol="n"/>
                 </constantatomformula>
                </conjunctiveformula>
               </conjunctiveformula>
              </quantifiedformula>
             </quantifiedformula>
            </formulalist>
           </sequent>
           <rule type="existsr" symbol="\ex:r">
            <sequent>
             <formulalist>
              <constantatomformula symbol="Q">
               <variable symbol="x_{0}"/>
               <variable symbol="x_{1}"/>
              </constantatomformula>
             </formulalist>
             <formulalist>
              <quantifiedformula type="exists">
               <variable symbol="n"/>
               <conjunctiveformula type="impl">
                <conjunctiveformula type="neg">
                 <constantatomformula symbol="Q">
                  <variable symbol="x_{0}"/>
                  <variable symbol="n"/>
                 </constantatomformula>
                </conjunctiveformula>
                <conjunctiveformula type="neg">
                 <constantatomformula symbol="P">
                  <variable symbol="x_{0}"/>
                  <variable symbol="n"/>
                 </constantatomformula>
                </conjunctiveformula>
               </conjunctiveformula>
              </quantifiedformula>
             </formulalist>
            </sequent>
            <rule type="implr" symbol="\impl:r">
             <sequent>
              <formulalist>
               <constantatomformula symbol="Q">
                <variable symbol="x_{0}"/>
                <variable symbol="x_{1}"/>
               </constantatomformula>
              </formulalist>
              <formulalist>
               <conjunctiveformula type="impl">
                <conjunctiveformula type="neg">
                 <constantatomformula symbol="Q">
                  <variable symbol="x_{0}"/>
                  <variable symbol="x_{1}"/>
                 </constantatomformula>
                </conjunctiveformula>
                <conjunctiveformula type="neg">
                 <constantatomformula symbol="P">
                  <variable symbol="x_{0}"/>
                  <variable symbol="x_{1}"/>
                 </constantatomformula>
                </conjunctiveformula>
               </conjunctiveformula>
              </formulalist>
             </sequent>
             <rule type="weakr" symbol="w:r">
              <sequent>
               <formulalist>
                <conjunctiveformula type="neg">
                 <constantatomformula symbol="Q">
                  <variable symbol="x_{0}"/>
                  <variable symbol="x_{1}"/>
                 </constantatomformula>
                </conjunctiveformula>
                <constantatomformula symbol="Q">
                 <variable symbol="x_{0}"/>
                 <variable symbol="x_{1}"/>
                </constantatomformula>
               </formulalist>
               <formulalist>
                <conjunctiveformula type="neg">
                 <constantatomformula symbol="P">
                  <variable symbol="x_{0}"/>
                  <variable symbol="x_{1}"/>
                 </constantatomformula>
                </conjunctiveformula>
               </formulalist>
              </sequent>
              <rule type="negl" symbol="\neg:l">
               <sequent>
                <formulalist>
                 <conjunctiveformula type="neg">
                  <constantatomformula symbol="Q">
                   <variable symbol="x_{0}"/>
                   <variable symbol="x_{1}"/>
                  </constantatomformula>
                 </conjunctiveformula>
                 <constantatomformula symbol="Q">
                  <variable symbol="x_{0}"/>
                  <variable symbol="x_{1}"/>
                 </constantatomformula>
                </formulalist>
                <formulalist/>
               </sequent>
               <rule type="axiom" symbol="axiom">
                <sequent>
                 <formulalist>
                  <constantatomformula symbol="Q">
                   <variable symbol="x_{0}"/>
                   <variable symbol="x_{1}"/>
                  </constantatomformula>
                 </formulalist>
                 <formulalist>
                  <constantatomformula symbol="Q">
                   <variable symbol="x_{0}"/>
                   <variable symbol="x_{1}"/>
                  </constantatomformula>
                 </formulalist>
                </sequent>
               </rule>
              </rule>
             </rule>
            </rule>
           </rule>
          </rule>
         </rule>
        </rule>
       </rule>
      </rule>
     </rule>
    </rule>
   </rule>
  </rule>
 </proof>
 <sequentlist symbol="clause set">
  <sequent>
   <formulalist>
    <constantatomformula symbol="Q">
     <variable symbol="m"/>
     <variable symbol="k"/>
    </constantatomformula>
   </formulalist>
   <formulalist/>
  </sequent>
  <sequent>
   <formulalist/>
   <formulalist>
    <constantatomformula symbol="P">
     <variable symbol="m"/>
     <variable symbol="k"/>
    </constantatomformula>
   </formulalist>
  </sequent>
  <sequent>
   <formulalist>
    <constantatomformula symbol="P">
     <variable symbol="k"/>
     <variable symbol="n"/>
    </constantatomformula>
   </formulalist>
   <formulalist>
    <constantatomformula symbol="Q">
     <variable symbol="k"/>
     <variable symbol="n"/>
    </constantatomformula>
   </formulalist>
  </sequent>
 </sequentlist>
 <variabledefinitions/>
</proofdatabase>
