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at.logic.gapt

examples

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package examples

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package.scala
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  1. class AllQuantifiedConditionalAxiomHelper extends AnyRef

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    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 |- ...

  2. trait ExplicitEqualityTactics extends AnyRef

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  3. trait ProofSequence extends AnyRef

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  4. class Script extends App

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  5. class nTape2 extends AnalysisWithCeresOmega

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    Version 2 of the higher-order n-Tape proof.

  6. class nTape3 extends AnalysisWithCeresOmega

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    Version 3 of the higher-order n-Tape proof.

  7. class nTape4 extends AnalysisWithCeresOmega

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    Version 3 of the higher-order n-Tape proof.

  8. class nTape5 extends nTape4

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    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.

  9. class nTape5Arith extends nTape4

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    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

  1. object BussTautology

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    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

  2. object CountingEquivalence

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    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.

  3. object FactorialFunctionEqualityExampleProof extends ProofSequence

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    Proof of f(n) = g(n, 1), where f is the head recursive and g the tail recursive formulation of the factorial function

  4. object FactorialFunctionEqualityExampleProof2 extends ProofSequence

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  5. object Formulas

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    Contains some commonly used formulas.

  6. object LinearEqExampleProof extends TacticsProof with ProofSequence with ExplicitEqualityTactics

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    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.

  7. object LinearExampleProof extends ProofSequence

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    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.

  8. object PQPairs

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    Creates the n-th formula of a sequence where distributivity-based algorithm produces only exponential CNFs.

  9. object Permutations

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    Given n >= 2 creates an unsatisfiable first-order clause set based on a statement about the permutations in S_n.

  10. object Pi2Pigeonhole

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  11. object PigeonHolePrinciple

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    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.

  12. object ReductionDemo extends Script

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  13. object ReforestDemo extends Script

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  14. object SquareDiagonalExampleProof extends ProofSequence

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    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.

  15. object SquareEdges2DimExampleProof extends ProofSequence

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    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.

  16. object SquareEdgesExampleProof extends ProofSequence

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    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

  17. object SumExampleProof extends ProofSequence

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    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.

  18. object SumOfOnesExampleProof extends ProofSequence

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    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.

  19. object SumOfOnesF2ExampleProof extends TacticsProof with ProofSequence with ExplicitEqualityTactics

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  20. object SumOfOnesFExampleProof extends TacticsProof with ProofSequence with ExplicitEqualityTactics

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  21. object UniformAssociativity3ExampleProof extends ProofSequence

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  22. object complex

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  23. object drinker

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  24. object epsilon extends Script

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  25. object fol1 extends TacticsProof

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  26. package hoare

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  27. package induction

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  28. object instprover extends Script

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  29. object lattice extends TacticsProof

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  30. object lkTests

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  31. object meta

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  32. object nTape2 extends nTape2

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  33. object nTape3 extends nTape3

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  34. object nTape4

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  35. object nTape5

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    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.

  36. object nTape5Arith

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    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.

  37. object nTape6

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    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.

  38. object nTapeInstances

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  39. object philsci

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  40. package poset

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  41. package prime

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  42. object primediv extends TacticsProof

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  43. val proofSequences: Seq[ProofSequence]

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  44. object propositional

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  45. package recschem

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  46. object tape extends TacticsProof

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  47. object tapeUrban extends TacticsProof

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    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.

  48. object tbillc

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    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

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