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package examples

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  1. 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 |- ...

  2. trait ExplicitEqualityTactics extends AnyRef
  3. trait ProofSequence extends AnyRef
  4. class Script extends App
  5. class nTape2 extends AnalysisWithCeresOmega

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

  6. class nTape3 extends AnalysisWithCeresOmega

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

  7. class nTape4 extends AnalysisWithCeresOmega

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

  8. 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.

  9. 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

  1. val proofSequences: Seq[ProofSequence]
  2. 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

  3. object CASCData
  4. object CASCEvaluation
  5. object CERESExpansionExampleProof
  6. 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.

  7. object ECSJumpSchema extends TacticsProof
  8. object EventuallyConstantSchema extends TacticsProof
  9. object EventuallyConstantSchemaInductionRefutation extends TacticsProof
  10. object EventuallyConstantSchemaRefutation extends TacticsProof
  11. object ExponentialCompression extends TacticsProof
  12. 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

  13. object FactorialFunctionEqualityExampleProof2 extends ProofSequence
  14. object FirstSchema0 extends TacticsProof
  15. object FirstSchema1 extends TacticsProof
  16. object FirstSchema2 extends TacticsProof
  17. object FirstSchema3 extends TacticsProof
  18. object FirstSchema4 extends TacticsProof
  19. object FirstSchema5 extends TacticsProof
  20. object FirstSchema6 extends TacticsProof
  21. object FirstSchema7 extends TacticsProof
  22. object FirstSchema8 extends TacticsProof
  23. object FirstSchema9 extends TacticsProof
  24. object Formulas

    Contains some commonly used formulas.

  25. object FourStrictMonotoneSchema extends TacticsProof
  26. object FunctionIterationRefutation extends TacticsProof
  27. object FunctionIterationRefutationPos extends TacticsProof
  28. object FunctionIterationSchema extends TacticsProof
  29. object GradedStrictMonotoneSchema extends TacticsProof
  30. object GradedStrictMonotoneSequenceRefutation extends TacticsProof
  31. object GradedStrictMonotoneSequenceSchema extends TacticsProof
  32. object LinearCutExampleProof extends ProofSequence

    Constructs short FOL LK proofs of the sequents

    Constructs short FOL LK proofs of the sequents

    P(0), ∀x. P(x) → P(s(x)) :- P(s2 n (0))

    where n is an Integer parameter >= 0.

  33. 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.

  34. 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.

  35. object MonoidCancellation extends TacticsProof

    Monoid cancellation benchmark from Gregory Malecha and Jesper Bengtson: Extensible and Efficient Automation Through Reflective Tactics, ESOP 2016.

  36. object NdiffSchema extends TacticsProof
  37. object NiaSchema extends TacticsProof
  38. object NiaSchemaRefutation extends TacticsProof
  39. object OneStrictMonotoneRefutation extends TacticsProof
  40. object OneStrictMonotoneSchema extends TacticsProof
  41. object OneStrictMonotoneSequenceRefutation extends TacticsProof
  42. object OneStrictMonotoneSequenceSchema extends TacticsProof
  43. object PQPairs

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

  44. object Permutations

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

  45. object Pi2Pigeonhole extends TacticsProof
  46. object Pi3Pigeonhole extends TacticsProof
  47. 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.

  48. object ReductionDemo extends Script
  49. object ReforestDemo extends Script
  50. object SimpleMutualInductionSchema extends TacticsProof
  51. 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.

  52. 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.

  53. 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

  54. object StrictMonotoneSchema extends TacticsProof
  55. object StrongStrictMonotoneSchema extends TacticsProof
  56. 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.

  57. 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.

  58. object SumOfOnesF2ExampleProof extends TacticsProof with ProofSequence with ExplicitEqualityTactics
  59. object SumOfOnesFExampleProof extends TacticsProof with ProofSequence with ExplicitEqualityTactics
  60. object ThreeStrictMonotoneRefutation extends TacticsProof
  61. object ThreeStrictMonotoneSchema extends TacticsProof
  62. object TwoStrictMonotoneRefutation extends TacticsProof
  63. object TwoStrictMonotoneSchema extends TacticsProof
  64. object UniformAssociativity3ExampleProof extends ProofSequence
  65. object VeryWeakLexicoPHPSchema extends TacticsProof
  66. object VeryWeakLexicoPHPSchemaVariant extends TacticsProof
  67. object VeryWeakLexicoPHPSchemaVariantRefutation extends TacticsProof
  68. object VeryWeakPHPSeqSchema extends TacticsProof
  69. object VeryWeakPHPSeqTwoSchema extends TacticsProof
  70. object VeryWeakPHPSequenceVariantSchema extends TacticsProof
  71. object divisionByTwo extends TacticsProof
  72. object drinker extends TacticsProof
  73. object elemAtIndex extends TacticsProof
  74. object epsilon extends Script
  75. object fol1 extends TacticsProof
  76. 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.

  77. object gapticExamples
  78. object gniaSchema extends TacticsProof
  79. object instprover extends Script
  80. object lattice extends TacticsProof
  81. object nTape2 extends nTape2
  82. object nTape3 extends nTape3
  83. object nTape4
  84. 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.

  85. 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.

  86. 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.

  87. object nTapeInstances
  88. object philsci
  89. object primediv extends TacticsProof
  90. object successor extends TacticsProof
  91. object tape extends TacticsProof
  92. 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.

  93. object tautSchema extends TacticsProof
  94. object tbillc extends TacticsProof

    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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