package examples
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Type Members
-
class
AllQuantifiedConditionalAxiomHelper
extends AnyRef
Auxiliary structure to deal with axioms of the schema: Forall variables cond1 -> cond2 -> ...
Auxiliary structure to deal with axioms of the schema: Forall variables cond1 -> cond2 -> ... -> condn -> consequence |- ...
- trait ExplicitEqualityTactics extends AnyRef
- trait ProofSequence extends AnyRef
- class Script extends App
-
class
nTape2
extends AnalysisWithCeresOmega
Version 2 of the higher-order n-Tape proof.
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class
nTape3
extends AnalysisWithCeresOmega
Version 3 of the higher-order n-Tape proof.
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class
nTape4
extends AnalysisWithCeresOmega
Version 3 of the higher-order n-Tape proof.
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class
nTape5
extends nTape4
Version 5 of the higher-order n-Tape proof, where if-then-else is directly axiomatized i.e.
Version 5 of the higher-order n-Tape proof, where if-then-else is directly axiomatized i.e. it has 2 additional axioms P -> if code(P) then t else f = t and -P -> if code(P) then t else f = f which were theorems before. In contrast to nTape4 it cuts on instances of the theorem C for specific upper bounds n. Since the instantiated proofs were generated manually, only nTape5(2) to nTape5(4) work.
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class
nTape5Arith
extends nTape4
Version 5 of the higher-order n-Tape proof, where if-then-else is still proved in arithmetic.
Version 5 of the higher-order n-Tape proof, where if-then-else is still proved in arithmetic. In contrast to nTape4 it cuts on instances of the theorem C for specific upper bounds n. Since the instantiated proofs were generated manually, only nTape5Arith(2) works.
Value Members
- val proofSequences: Seq[ProofSequence]
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object
BussTautology
Creates the n-th tautology of a sequence that has only exponential-size cut-free proofs
Creates the n-th tautology of a sequence that has only exponential-size cut-free proofs
This sequence is taken from: S. Buss. "Weak Formal Systems and Connections to Computational Complexity". Lecture Notes for a Topics Course, UC Berkeley, 1988, available from: http://www.math.ucsd.edu/~sbuss/ResearchWeb/index.html
- object CERESExpansionExampleProof
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object
CountingEquivalence
Sequence of valid first-order formulas about equivalent counting methods.
Sequence of valid first-order formulas about equivalent counting methods.
Consider the formula ∀z ∃=1i ∀x ∃y a_i(x,y,z), where ∃=1i is a quantifier that says that there exists exactly one i (in 0..n) such that ∀x ∃y a_i(x,y,z) is true.
This function returns the equivalence between two implementations of the formula: first, using a naive quadratic implementation; and second, using an O(n*log(n)) implementation with threshold formulas.
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object
FactorialFunctionEqualityExampleProof
extends ProofSequence
Proof of f(n) = g(n, 1), where f is the head recursive and g the tail recursive formulation of the factorial function
- object FactorialFunctionEqualityExampleProof2 extends ProofSequence
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object
Formulas
Contains some commonly used formulas.
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object
LinearEqExampleProof
extends TacticsProof with ProofSequence with ExplicitEqualityTactics
Functions to construct cut-free FOL LK proofs of the sequents
Functions to construct cut-free FOL LK proofs of the sequents
Refl, Trans, \ALL x. f(x) = x :- fn(a) = a
where n is an Integer parameter >= 0.
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object
LinearExampleProof
extends ProofSequence
Constructs cut-free FOL LK proofs of the sequents
Constructs cut-free FOL LK proofs of the sequents
P(0), ∀x. P(x) → P(s(x)) :- P(sn(0))
where n is an Integer parameter >= 0.
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object
MonoidCancellation
extends TacticsProof
Monoid cancellation benchmark from Gregory Malecha and Jesper Bengtson: Extensible and Efficient Automation Through Reflective Tactics, ESOP 2016.
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object
PQPairs
Creates the n-th formula of a sequence where distributivity-based algorithm produces only exponential CNFs.
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object
Permutations
Given n >= 2 creates an unsatisfiable first-order clause set based on a statement about the permutations in S_n.
- object Pi2Pigeonhole extends TacticsProof
- object Pi3Pigeonhole extends TacticsProof
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object
PigeonHolePrinciple
Constructs a formula representing the pigeon hole principle.
Constructs a formula representing the pigeon hole principle. More precisely: PigeonHolePrinciple( p, h ) states that if p pigeons are put into h holes then there is a hole which contains two pigeons. PigeonHolePrinciple( p, h ) is a tautology iff p > h.
Since we want to avoid empty disjunctions, we assume > 1 pigeons.
- object ReductionDemo extends Script
- object ReforestDemo extends Script
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object
SquareDiagonalExampleProof
extends ProofSequence
Functions to construct cut-free FOL LK proofs of the sequents
Functions to construct cut-free FOL LK proofs of the sequents
P(0,0), ∀x,y. P(x,y) → P(s(x),y), ∀x,y. P(x,y) → P(x,s(y)) :- P(sn(0),sn(0))
where n is an Integer parameter >= 0.
The proofs constructed here go along the diagonal of P, i.e. one x-step, then one y-step, etc.
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object
SquareEdges2DimExampleProof
extends ProofSequence
Functions to construct cut-free FOL LK proofs of the sequents
Functions to construct cut-free FOL LK proofs of the sequents
P(a,b), ∀x,y. P(x,y) → P(sx(x),y), ∀x,y. P(x,y) → P(x,sy(y)) :- P(sxn(a),syn(b))
where n is an Integer parameter >= 0.
The proofs constructed here go along the edges of P, i.e. first all X-steps are performed, then all Y-steps are performed, but unlike SquareEdgesExampleProof, different functions are used for the X- and the Y-directions.
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object
SquareEdgesExampleProof
extends ProofSequence
Functions to construct cut-free FOL LK proofs of the sequents
Functions to construct cut-free FOL LK proofs of the sequents
P(0,0), ∀x,y. P(x,y) → P(s(x),y), ∀x,y. P(x,y) → P(x,s(y)) :- P(sn(0),sn(0))
where n is an Integer parameter >= 0.
The proofs constructed here go along the edges of P, i.e. first all X-steps are performed, then all Y-steps are performed
-
object
SumExampleProof
extends ProofSequence
Functions to construct the straightforward cut-free FOL LK proofs of the sequents
Functions to construct the straightforward cut-free FOL LK proofs of the sequents
P(sn(0),0), ∀x,y. P(s(x),y) → P(x,s(y)) :- P(0,sn(0))
where n is an Integer parameter >= 0.
This sequent is shown to have no cut-free proof which can be compressed by a single cut with a single quantifier in S. Eberhard, S. Hetzl: On the compressibility of finite languages and formal proofs, submitted, 2015.
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object
SumOfOnesExampleProof
extends ProofSequence
Functions to construct cut-free FOL LK proofs of the sequents
Functions to construct cut-free FOL LK proofs of the sequents
Refl, Trans, CongSuc, ABase, ASuc, :- sum( n ) = sn(0)
where n is an Integer parameter >= 0.
- object SumOfOnesF2ExampleProof extends TacticsProof with ProofSequence with ExplicitEqualityTactics
- object SumOfOnesFExampleProof extends TacticsProof with ProofSequence with ExplicitEqualityTactics
- object UniformAssociativity3ExampleProof extends ProofSequence
- object drinker
- object epsilon extends Script
- object fol1 extends TacticsProof
-
object
fol2
Provides a simple intuitionistic proof of ¬p ∨ p ⊢ ¬¬p ⊃ p.
Provides a simple intuitionistic proof of ¬p ∨ p ⊢ ¬¬p ⊃ p. Applying the CERES method will create a non-intuitionistic proof but reductive cut-elimination will always create an intuitionistic one. Therefore this is an example that CERES produces cut-free proofs which reductive cut-elimination cannot.
- object gapticExamples
- object gniaSchema extends TacticsProof
- object instprover extends Script
- object lattice extends TacticsProof
- object nTape2 extends nTape2
- object nTape3 extends nTape3
- object nTape4
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object
nTape5
Version 5 of the higher-order n-Tape proof.
Version 5 of the higher-order n-Tape proof. In contrast to nTape4 it cuts on instances of the theorem C for specific upper bounds n. Since the instantiated proofs were generated manually, only nTape5(2) to nTape5(4) work.
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object
nTape5Arith
Version 5 of the higher-order n-Tape proof, where if-then-else is still proved in arithmetic.
Version 5 of the higher-order n-Tape proof, where if-then-else is still proved in arithmetic. In contrast to nTape4 it cuts on instances of the theorem C for specific upper bounds n. Since the instantiated proofs were generated manually, only nTape5Arith(2) works.
-
object
nTape6
The object nTape6 generates hard problems for higher order theorem provers containing an axiomatization of if-then-else.
The object nTape6 generates hard problems for higher order theorem provers containing an axiomatization of if-then-else. Formulas: f1,f2 ... if-then-else axiomatizations f3,f4 ... properties of the successor function (0 is no successor and a number is always different from its successor) conclusion0 ... there exists a function h s.t. h(0) = 1, h(1) = 0 conclusion1 ... there exists a function h s.t. h(0) = 1, h(1) = 0, h(2) = 0 conclusion2 ... there exists a function h s.t. h(0) = 1, h(1) = 0, h(2) = 1 w1 ... witness for sc w2 ... witness for sc2
The problems are (in sequent notation):
P0: f1, f2 :- conclusion0 P1: f1, f2, f3, f4 :- conclusion1 P2: f1, f2, f3, f4 :- conclusion2
The generated filenames are "ntape6-${i}-without-witness.tptp" for i = 0 to 2.
To show that there are actual witnesses for the function h, we provide a witness, where the witness w1 can be used for both W0 and W1:
W0: { w1 :- } x P0 W1: { w1 :- } x P1 W2: { w2 :- } x P2
The generated filenames are "ntape6-${i}-with-witness.tptp" for i = 0 to 2.
- object nTapeInstances
- object niaSchema extends TacticsProof
- object philsci
- object primediv extends TacticsProof
- object tape extends TacticsProof
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object
tapeUrban
extends TacticsProof
Formalisation of the tape-proof as described in C.
Formalisation of the tape-proof as described in C. Urban: Classical Logic and Computation, PhD Thesis, Cambridge University, 2000.
- object tautSchema extends TacticsProof
-
object
tbillc
This is an example used in the talk[1] at TbiLLC 2013.
This is an example used in the talk[1] at TbiLLC 2013. It generates a (cut-free) LK proof where the extracted expansion tree has nested quantifiers.
[1] http://www.illc.uva.nl/Tbilisi/Tbilisi2013/uploaded_files/inlineitem/riener.pdf