TSTP Solution File: LCL459+1 by Princess---230619
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%------------------------------------------------------------------------------
% File : Princess---230619
% Problem : LCL459+1 : TPTP v8.1.2. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 08:11:21 EDT 2023
% Result : Theorem 11.38s 2.32s
% Output : Proof 21.69s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : LCL459+1 : TPTP v8.1.2. Released v3.3.0.
% 0.10/0.12 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s
% 0.12/0.33 % Computer : n005.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Aug 24 17:47:53 EDT 2023
% 0.12/0.33 % CPUTime :
% 0.18/0.60 ________ _____
% 0.18/0.60 ___ __ \_________(_)________________________________
% 0.18/0.60 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/
% 0.18/0.60 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ )
% 0.18/0.60 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/
% 0.18/0.60
% 0.18/0.60 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic
% 0.18/0.60 (2023-06-19)
% 0.18/0.60
% 0.18/0.60 (c) Philipp Rümmer, 2009-2023
% 0.18/0.60 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen,
% 0.18/0.60 Amanda Stjerna.
% 0.18/0.60 Free software under BSD-3-Clause.
% 0.18/0.60
% 0.18/0.60 For more information, visit http://www.philipp.ruemmer.org/princess.shtml
% 0.18/0.60
% 0.18/0.60 Loading /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 0.18/0.61 Running up to 7 provers in parallel.
% 0.18/0.62 Prover 0: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1042961893
% 0.18/0.62 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1571432423
% 0.18/0.62 Prover 2: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMinimalAndEmpty -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1065072994
% 0.18/0.62 Prover 3: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1922548996
% 0.18/0.62 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=1868514696
% 0.18/0.62 Prover 5: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=complete -randomSeed=1259561288
% 0.18/0.62 Prover 6: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximalOutermost -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1399714365
% 3.05/1.11 Prover 1: Preprocessing ...
% 3.05/1.11 Prover 4: Preprocessing ...
% 3.37/1.15 Prover 5: Preprocessing ...
% 3.37/1.15 Prover 2: Preprocessing ...
% 3.37/1.15 Prover 6: Preprocessing ...
% 3.37/1.15 Prover 3: Preprocessing ...
% 3.37/1.15 Prover 0: Preprocessing ...
% 7.77/1.78 Prover 5: Proving ...
% 7.77/1.79 Prover 6: Constructing countermodel ...
% 7.77/1.80 Prover 1: Constructing countermodel ...
% 7.77/1.81 Prover 3: Constructing countermodel ...
% 7.77/1.81 Prover 4: Constructing countermodel ...
% 8.31/1.85 Prover 0: Proving ...
% 8.64/1.98 Prover 2: Proving ...
% 11.38/2.32 Prover 0: proved (1697ms)
% 11.38/2.32
% 11.38/2.32 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p
% 11.38/2.32
% 11.38/2.32 Prover 3: stopped
% 11.38/2.33 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470
% 11.38/2.33 Prover 2: stopped
% 11.38/2.34 Prover 5: stopped
% 12.13/2.35 Prover 6: stopped
% 12.13/2.35 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089
% 12.13/2.35 Prover 10: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=919308125
% 12.13/2.36 Prover 11: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1509710984
% 12.13/2.36 Prover 13: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=1138197443
% 12.50/2.39 Prover 7: Preprocessing ...
% 12.50/2.39 Prover 8: Preprocessing ...
% 12.50/2.41 Prover 10: Preprocessing ...
% 12.50/2.42 Prover 13: Preprocessing ...
% 12.50/2.42 Prover 11: Preprocessing ...
% 12.97/2.51 Prover 8: Warning: ignoring some quantifiers
% 12.97/2.52 Prover 8: Constructing countermodel ...
% 12.97/2.55 Prover 1: gave up
% 12.97/2.56 Prover 16: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=completeFrugal -randomSeed=-2043353683
% 13.66/2.59 Prover 13: Warning: ignoring some quantifiers
% 13.66/2.60 Prover 13: Constructing countermodel ...
% 13.66/2.61 Prover 16: Preprocessing ...
% 13.66/2.62 Prover 7: Constructing countermodel ...
% 13.66/2.62 Prover 10: Constructing countermodel ...
% 14.16/2.64 Prover 11: Constructing countermodel ...
% 14.93/2.74 Prover 16: Warning: ignoring some quantifiers
% 14.93/2.75 Prover 16: Constructing countermodel ...
% 16.48/2.94 Prover 13: gave up
% 16.48/2.95 Prover 10: gave up
% 16.48/2.96 Prover 19: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=-1780594085
% 16.48/2.99 Prover 19: Preprocessing ...
% 17.25/3.08 Prover 19: Warning: ignoring some quantifiers
% 17.80/3.12 Prover 19: Constructing countermodel ...
% 18.30/3.17 Prover 8: gave up
% 20.01/3.43 Prover 19: gave up
% 21.69/3.66 Prover 7: Found proof (size 50)
% 21.69/3.66 Prover 7: proved (1343ms)
% 21.69/3.66 Prover 4: stopped
% 21.69/3.66 Prover 16: stopped
% 21.69/3.67 Prover 11: stopped
% 21.69/3.67
% 21.69/3.67 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p
% 21.69/3.67
% 21.69/3.67 % SZS output start Proof for theBenchmark
% 21.69/3.68 Assumptions after simplification:
% 21.69/3.68 ---------------------------------
% 21.69/3.68
% 21.69/3.68 (and_3)
% 21.69/3.70 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ? [v4: $i] : ($i(v1)
% 21.69/3.70 & $i(v0) & ((and(v0, v1) = v2 & implies(v1, v2) = v3 & implies(v0, v3) = v4
% 21.69/3.70 & $i(v4) & $i(v3) & $i(v2) & ~ and_3 & ~ is_a_theorem(v4)) | (and_3 &
% 21.69/3.70 ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ( ~ (and(v5, v6) = v7) | ~
% 21.69/3.70 $i(v6) | ~ $i(v5) | ? [v8: $i] : ? [v9: $i] : (implies(v6, v7) = v8
% 21.69/3.70 & implies(v5, v8) = v9 & $i(v9) & $i(v8) & is_a_theorem(v9))))))
% 21.69/3.70
% 21.69/3.70 (hilbert_and_3)
% 21.69/3.70 and_3
% 21.69/3.70
% 21.69/3.70 (hilbert_implies_2)
% 21.69/3.70 implies_2
% 21.69/3.70
% 21.69/3.70 (hilbert_modus_ponens)
% 21.69/3.70 modus_ponens
% 21.69/3.70
% 21.69/3.70 (implies_2)
% 21.69/3.71 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ? [v4: $i] : ($i(v1)
% 21.69/3.71 & $i(v0) & ((implies(v3, v2) = v4 & implies(v0, v2) = v3 & implies(v0, v1) =
% 21.69/3.71 v2 & $i(v4) & $i(v3) & $i(v2) & ~ implies_2 & ~ is_a_theorem(v4)) |
% 21.69/3.71 (implies_2 & ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ( ~ (implies(v5,
% 21.69/3.71 v6) = v7) | ~ $i(v6) | ~ $i(v5) | ? [v8: $i] : ? [v9: $i] :
% 21.69/3.71 (implies(v8, v7) = v9 & implies(v5, v7) = v8 & $i(v9) & $i(v8) &
% 21.69/3.71 is_a_theorem(v9))))))
% 21.69/3.71
% 21.69/3.71 (kn1)
% 21.69/3.71 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ($i(v0) & ((and(v0, v0) = v1 &
% 21.69/3.71 implies(v0, v1) = v2 & $i(v2) & $i(v1) & ~ kn1 & ~ is_a_theorem(v2)) |
% 21.69/3.71 (kn1 & ! [v3: $i] : ! [v4: $i] : ( ~ (and(v3, v3) = v4) | ~ $i(v3) | ?
% 21.69/3.71 [v5: $i] : (implies(v3, v4) = v5 & $i(v5) & is_a_theorem(v5))))))
% 21.69/3.71
% 21.69/3.71 (modus_ponens)
% 21.69/3.71 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ($i(v1) & $i(v0) & ((implies(v0, v1)
% 21.69/3.71 = v2 & $i(v2) & is_a_theorem(v2) & is_a_theorem(v0) & ~
% 21.69/3.71 is_a_theorem(v1) & ~ modus_ponens) | (modus_ponens & ! [v3: $i] : !
% 21.69/3.71 [v4: $i] : ! [v5: $i] : ( ~ (implies(v3, v4) = v5) | ~ $i(v4) | ~
% 21.69/3.71 $i(v3) | ~ is_a_theorem(v5) | ~ is_a_theorem(v3) |
% 21.69/3.71 is_a_theorem(v4)))))
% 21.69/3.71
% 21.69/3.71 (rosser_kn1)
% 21.69/3.71 ~ kn1
% 21.69/3.71
% 21.69/3.71 (function-axioms)
% 21.69/3.71 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (or(v3,
% 21.69/3.71 v2) = v1) | ~ (or(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2:
% 21.69/3.71 $i] : ! [v3: $i] : (v1 = v0 | ~ (and(v3, v2) = v1) | ~ (and(v3, v2) =
% 21.69/3.71 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 |
% 21.69/3.71 ~ (equiv(v3, v2) = v1) | ~ (equiv(v3, v2) = v0)) & ! [v0: $i] : ! [v1:
% 21.69/3.71 $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (implies(v3, v2) = v1) | ~
% 21.69/3.71 (implies(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0
% 21.69/3.71 | ~ (not(v2) = v1) | ~ (not(v2) = v0))
% 21.69/3.71
% 21.69/3.71 Further assumptions not needed in the proof:
% 21.69/3.71 --------------------------------------------
% 21.69/3.71 and_1, and_2, cn1, cn2, cn3, equivalence_1, equivalence_2, equivalence_3,
% 21.69/3.71 hilbert_and_1, hilbert_and_2, hilbert_equivalence_1, hilbert_equivalence_2,
% 21.69/3.71 hilbert_equivalence_3, hilbert_implies_1, hilbert_implies_3,
% 21.69/3.71 hilbert_modus_tollens, hilbert_op_equiv, hilbert_op_implies_and, hilbert_op_or,
% 21.69/3.71 hilbert_or_1, hilbert_or_2, hilbert_or_3, implies_1, implies_3, kn2, kn3,
% 21.69/3.71 modus_tollens, op_and, op_equiv, op_implies_and, op_implies_or, op_or, or_1,
% 21.69/3.71 or_2, or_3, r1, r2, r3, r4, r5, rosser_op_equiv, rosser_op_implies_and,
% 21.69/3.71 rosser_op_or, substitution_of_equivalents
% 21.69/3.71
% 21.69/3.71 Those formulas are unsatisfiable:
% 21.69/3.71 ---------------------------------
% 21.69/3.71
% 21.69/3.71 Begin of proof
% 21.69/3.71 |
% 21.69/3.71 | ALPHA: (function-axioms) implies:
% 21.69/3.72 | (1) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~
% 21.69/3.72 | (implies(v3, v2) = v1) | ~ (implies(v3, v2) = v0))
% 21.69/3.72 |
% 21.69/3.72 | DELTA: instantiating (kn1) with fresh symbols all_4_0, all_4_1, all_4_2 gives:
% 21.69/3.72 | (2) $i(all_4_2) & ((and(all_4_2, all_4_2) = all_4_1 & implies(all_4_2,
% 21.69/3.72 | all_4_1) = all_4_0 & $i(all_4_0) & $i(all_4_1) & ~ kn1 & ~
% 21.69/3.72 | is_a_theorem(all_4_0)) | (kn1 & ! [v0: $i] : ! [v1: $i] : ( ~
% 21.69/3.72 | (and(v0, v0) = v1) | ~ $i(v0) | ? [v2: $i] : (implies(v0, v1) =
% 21.69/3.72 | v2 & $i(v2) & is_a_theorem(v2)))))
% 21.69/3.72 |
% 21.69/3.72 | ALPHA: (2) implies:
% 21.69/3.72 | (3) $i(all_4_2)
% 21.69/3.72 | (4) (and(all_4_2, all_4_2) = all_4_1 & implies(all_4_2, all_4_1) = all_4_0
% 21.69/3.72 | & $i(all_4_0) & $i(all_4_1) & ~ kn1 & ~ is_a_theorem(all_4_0)) |
% 21.69/3.72 | (kn1 & ! [v0: $i] : ! [v1: $i] : ( ~ (and(v0, v0) = v1) | ~ $i(v0) |
% 21.69/3.72 | ? [v2: $i] : (implies(v0, v1) = v2 & $i(v2) & is_a_theorem(v2))))
% 21.69/3.72 |
% 21.69/3.72 | DELTA: instantiating (modus_ponens) with fresh symbols all_10_0, all_10_1,
% 21.69/3.72 | all_10_2 gives:
% 21.69/3.72 | (5) $i(all_10_1) & $i(all_10_2) & ((implies(all_10_2, all_10_1) = all_10_0
% 21.69/3.72 | & $i(all_10_0) & is_a_theorem(all_10_0) & is_a_theorem(all_10_2) &
% 21.69/3.72 | ~ is_a_theorem(all_10_1) & ~ modus_ponens) | (modus_ponens & !
% 21.69/3.72 | [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) |
% 21.69/3.72 | ~ $i(v1) | ~ $i(v0) | ~ is_a_theorem(v2) | ~ is_a_theorem(v0)
% 21.69/3.72 | | is_a_theorem(v1))))
% 21.69/3.72 |
% 21.69/3.72 | ALPHA: (5) implies:
% 21.69/3.72 | (6) (implies(all_10_2, all_10_1) = all_10_0 & $i(all_10_0) &
% 21.69/3.72 | is_a_theorem(all_10_0) & is_a_theorem(all_10_2) & ~
% 21.69/3.72 | is_a_theorem(all_10_1) & ~ modus_ponens) | (modus_ponens & ! [v0:
% 21.69/3.72 | $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~
% 21.69/3.72 | $i(v1) | ~ $i(v0) | ~ is_a_theorem(v2) | ~ is_a_theorem(v0) |
% 21.69/3.72 | is_a_theorem(v1)))
% 21.69/3.72 |
% 21.69/3.72 | DELTA: instantiating (implies_2) with fresh symbols all_31_0, all_31_1,
% 21.69/3.72 | all_31_2, all_31_3, all_31_4 gives:
% 21.69/3.72 | (7) $i(all_31_3) & $i(all_31_4) & ((implies(all_31_1, all_31_2) = all_31_0
% 21.69/3.72 | & implies(all_31_4, all_31_2) = all_31_1 & implies(all_31_4,
% 21.69/3.72 | all_31_3) = all_31_2 & $i(all_31_0) & $i(all_31_1) & $i(all_31_2)
% 21.69/3.72 | & ~ implies_2 & ~ is_a_theorem(all_31_0)) | (implies_2 & ! [v0:
% 21.69/3.72 | $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~
% 21.69/3.72 | $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : (implies(v3, v2)
% 21.69/3.72 | = v4 & implies(v0, v2) = v3 & $i(v4) & $i(v3) &
% 21.69/3.72 | is_a_theorem(v4)))))
% 21.69/3.72 |
% 21.69/3.72 | ALPHA: (7) implies:
% 21.69/3.73 | (8) (implies(all_31_1, all_31_2) = all_31_0 & implies(all_31_4, all_31_2) =
% 21.69/3.73 | all_31_1 & implies(all_31_4, all_31_3) = all_31_2 & $i(all_31_0) &
% 21.69/3.73 | $i(all_31_1) & $i(all_31_2) & ~ implies_2 & ~
% 21.69/3.73 | is_a_theorem(all_31_0)) | (implies_2 & ! [v0: $i] : ! [v1: $i] : !
% 21.69/3.73 | [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ?
% 21.69/3.73 | [v3: $i] : ? [v4: $i] : (implies(v3, v2) = v4 & implies(v0, v2) =
% 21.69/3.73 | v3 & $i(v4) & $i(v3) & is_a_theorem(v4))))
% 21.69/3.73 |
% 21.69/3.73 | DELTA: instantiating (and_3) with fresh symbols all_33_0, all_33_1, all_33_2,
% 21.69/3.73 | all_33_3, all_33_4 gives:
% 21.69/3.73 | (9) $i(all_33_3) & $i(all_33_4) & ((and(all_33_4, all_33_3) = all_33_2 &
% 21.69/3.73 | implies(all_33_3, all_33_2) = all_33_1 & implies(all_33_4,
% 21.69/3.73 | all_33_1) = all_33_0 & $i(all_33_0) & $i(all_33_1) & $i(all_33_2)
% 21.69/3.73 | & ~ and_3 & ~ is_a_theorem(all_33_0)) | (and_3 & ! [v0: $i] : !
% 21.69/3.73 | [v1: $i] : ! [v2: $i] : ( ~ (and(v0, v1) = v2) | ~ $i(v1) | ~
% 21.69/3.73 | $i(v0) | ? [v3: $i] : ? [v4: $i] : (implies(v1, v2) = v3 &
% 21.69/3.73 | implies(v0, v3) = v4 & $i(v4) & $i(v3) & is_a_theorem(v4)))))
% 21.69/3.73 |
% 21.69/3.73 | ALPHA: (9) implies:
% 21.69/3.73 | (10) (and(all_33_4, all_33_3) = all_33_2 & implies(all_33_3, all_33_2) =
% 21.69/3.73 | all_33_1 & implies(all_33_4, all_33_1) = all_33_0 & $i(all_33_0) &
% 21.69/3.73 | $i(all_33_1) & $i(all_33_2) & ~ and_3 & ~ is_a_theorem(all_33_0))
% 21.69/3.73 | | (and_3 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (and(v0, v1)
% 21.69/3.73 | = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] :
% 21.69/3.73 | (implies(v1, v2) = v3 & implies(v0, v3) = v4 & $i(v4) & $i(v3) &
% 21.69/3.73 | is_a_theorem(v4))))
% 21.69/3.73 |
% 21.69/3.73 | BETA: splitting (4) gives:
% 21.69/3.73 |
% 21.69/3.73 | Case 1:
% 21.69/3.73 | |
% 21.69/3.73 | | (11) and(all_4_2, all_4_2) = all_4_1 & implies(all_4_2, all_4_1) =
% 21.69/3.73 | | all_4_0 & $i(all_4_0) & $i(all_4_1) & ~ kn1 & ~
% 21.69/3.73 | | is_a_theorem(all_4_0)
% 21.69/3.73 | |
% 21.69/3.73 | | ALPHA: (11) implies:
% 21.69/3.73 | | (12) ~ is_a_theorem(all_4_0)
% 21.69/3.73 | | (13) $i(all_4_1)
% 21.69/3.73 | | (14) implies(all_4_2, all_4_1) = all_4_0
% 21.69/3.73 | | (15) and(all_4_2, all_4_2) = all_4_1
% 21.69/3.73 | |
% 21.69/3.73 | | BETA: splitting (8) gives:
% 21.69/3.73 | |
% 21.69/3.73 | | Case 1:
% 21.69/3.73 | | |
% 21.69/3.73 | | | (16) implies(all_31_1, all_31_2) = all_31_0 & implies(all_31_4,
% 21.69/3.73 | | | all_31_2) = all_31_1 & implies(all_31_4, all_31_3) = all_31_2 &
% 21.69/3.73 | | | $i(all_31_0) & $i(all_31_1) & $i(all_31_2) & ~ implies_2 & ~
% 21.69/3.73 | | | is_a_theorem(all_31_0)
% 21.69/3.73 | | |
% 21.69/3.73 | | | ALPHA: (16) implies:
% 21.69/3.73 | | | (17) ~ implies_2
% 21.69/3.73 | | |
% 21.69/3.73 | | | PRED_UNIFY: (17), (hilbert_implies_2) imply:
% 21.69/3.73 | | | (18) $false
% 21.69/3.73 | | |
% 21.69/3.73 | | | CLOSE: (18) is inconsistent.
% 21.69/3.73 | | |
% 21.69/3.73 | | Case 2:
% 21.69/3.73 | | |
% 21.69/3.74 | | | (19) implies_2 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 21.69/3.74 | | | (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] :
% 21.69/3.74 | | | ? [v4: $i] : (implies(v3, v2) = v4 & implies(v0, v2) = v3 &
% 21.69/3.74 | | | $i(v4) & $i(v3) & is_a_theorem(v4)))
% 21.69/3.74 | | |
% 21.69/3.74 | | | ALPHA: (19) implies:
% 21.69/3.74 | | | (20) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) =
% 21.69/3.74 | | | v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] :
% 21.69/3.74 | | | (implies(v3, v2) = v4 & implies(v0, v2) = v3 & $i(v4) & $i(v3) &
% 21.69/3.74 | | | is_a_theorem(v4)))
% 21.69/3.74 | | |
% 21.69/3.74 | | | BETA: splitting (10) gives:
% 21.69/3.74 | | |
% 21.69/3.74 | | | Case 1:
% 21.69/3.74 | | | |
% 21.69/3.74 | | | | (21) and(all_33_4, all_33_3) = all_33_2 & implies(all_33_3, all_33_2)
% 21.69/3.74 | | | | = all_33_1 & implies(all_33_4, all_33_1) = all_33_0 &
% 21.69/3.74 | | | | $i(all_33_0) & $i(all_33_1) & $i(all_33_2) & ~ and_3 & ~
% 21.69/3.74 | | | | is_a_theorem(all_33_0)
% 21.69/3.74 | | | |
% 21.69/3.74 | | | | ALPHA: (21) implies:
% 21.69/3.74 | | | | (22) ~ and_3
% 21.69/3.74 | | | |
% 21.69/3.74 | | | | PRED_UNIFY: (22), (hilbert_and_3) imply:
% 21.69/3.74 | | | | (23) $false
% 21.69/3.74 | | | |
% 21.69/3.74 | | | | CLOSE: (23) is inconsistent.
% 21.69/3.74 | | | |
% 21.69/3.74 | | | Case 2:
% 21.69/3.74 | | | |
% 21.69/3.74 | | | | (24) and_3 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (and(v0,
% 21.69/3.74 | | | | v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4:
% 21.69/3.74 | | | | $i] : (implies(v1, v2) = v3 & implies(v0, v3) = v4 & $i(v4)
% 21.69/3.74 | | | | & $i(v3) & is_a_theorem(v4)))
% 21.69/3.74 | | | |
% 21.69/3.74 | | | | ALPHA: (24) implies:
% 21.69/3.74 | | | | (25) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (and(v0, v1) = v2)
% 21.69/3.74 | | | | | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] :
% 21.69/3.74 | | | | (implies(v1, v2) = v3 & implies(v0, v3) = v4 & $i(v4) & $i(v3)
% 21.69/3.74 | | | | & is_a_theorem(v4)))
% 21.69/3.74 | | | |
% 21.69/3.74 | | | | BETA: splitting (6) gives:
% 21.69/3.74 | | | |
% 21.69/3.74 | | | | Case 1:
% 21.69/3.74 | | | | |
% 21.69/3.74 | | | | | (26) implies(all_10_2, all_10_1) = all_10_0 & $i(all_10_0) &
% 21.69/3.74 | | | | | is_a_theorem(all_10_0) & is_a_theorem(all_10_2) & ~
% 21.69/3.74 | | | | | is_a_theorem(all_10_1) & ~ modus_ponens
% 21.69/3.74 | | | | |
% 21.69/3.74 | | | | | ALPHA: (26) implies:
% 21.69/3.74 | | | | | (27) ~ modus_ponens
% 21.69/3.74 | | | | |
% 21.69/3.74 | | | | | PRED_UNIFY: (27), (hilbert_modus_ponens) imply:
% 21.69/3.74 | | | | | (28) $false
% 21.69/3.74 | | | | |
% 21.69/3.74 | | | | | CLOSE: (28) is inconsistent.
% 21.69/3.74 | | | | |
% 21.69/3.74 | | | | Case 2:
% 21.69/3.74 | | | | |
% 21.69/3.74 | | | | | (29) modus_ponens & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 21.69/3.74 | | | | | (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ~
% 21.69/3.74 | | | | | is_a_theorem(v2) | ~ is_a_theorem(v0) | is_a_theorem(v1))
% 21.69/3.74 | | | | |
% 21.69/3.74 | | | | | ALPHA: (29) implies:
% 21.69/3.74 | | | | | (30) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1)
% 21.69/3.74 | | | | | = v2) | ~ $i(v1) | ~ $i(v0) | ~ is_a_theorem(v2) | ~
% 21.69/3.74 | | | | | is_a_theorem(v0) | is_a_theorem(v1))
% 21.69/3.74 | | | | |
% 21.69/3.74 | | | | | GROUND_INST: instantiating (20) with all_4_2, all_4_1, all_4_0,
% 21.69/3.74 | | | | | simplifying with (3), (13), (14) gives:
% 21.69/3.74 | | | | | (31) ? [v0: $i] : ? [v1: $i] : (implies(v0, all_4_0) = v1 &
% 21.69/3.74 | | | | | implies(all_4_2, all_4_0) = v0 & $i(v1) & $i(v0) &
% 21.69/3.74 | | | | | is_a_theorem(v1))
% 21.69/3.74 | | | | |
% 21.69/3.74 | | | | | GROUND_INST: instantiating (25) with all_4_2, all_4_2, all_4_1,
% 21.69/3.74 | | | | | simplifying with (3), (15) gives:
% 21.69/3.75 | | | | | (32) ? [v0: $i] : ? [v1: $i] : (implies(all_4_2, v0) = v1 &
% 21.69/3.75 | | | | | implies(all_4_2, all_4_1) = v0 & $i(v1) & $i(v0) &
% 21.69/3.75 | | | | | is_a_theorem(v1))
% 21.69/3.75 | | | | |
% 21.69/3.75 | | | | | DELTA: instantiating (31) with fresh symbols all_143_0, all_143_1
% 21.69/3.75 | | | | | gives:
% 21.69/3.75 | | | | | (33) implies(all_143_1, all_4_0) = all_143_0 & implies(all_4_2,
% 21.69/3.75 | | | | | all_4_0) = all_143_1 & $i(all_143_0) & $i(all_143_1) &
% 21.69/3.75 | | | | | is_a_theorem(all_143_0)
% 21.69/3.75 | | | | |
% 21.69/3.75 | | | | | ALPHA: (33) implies:
% 21.69/3.75 | | | | | (34) is_a_theorem(all_143_0)
% 21.69/3.75 | | | | | (35) implies(all_4_2, all_4_0) = all_143_1
% 21.69/3.75 | | | | | (36) implies(all_143_1, all_4_0) = all_143_0
% 21.69/3.75 | | | | |
% 21.69/3.75 | | | | | DELTA: instantiating (32) with fresh symbols all_147_0, all_147_1
% 21.69/3.75 | | | | | gives:
% 21.69/3.75 | | | | | (37) implies(all_4_2, all_147_1) = all_147_0 & implies(all_4_2,
% 21.69/3.75 | | | | | all_4_1) = all_147_1 & $i(all_147_0) & $i(all_147_1) &
% 21.69/3.75 | | | | | is_a_theorem(all_147_0)
% 21.69/3.75 | | | | |
% 21.69/3.75 | | | | | ALPHA: (37) implies:
% 21.69/3.75 | | | | | (38) is_a_theorem(all_147_0)
% 21.69/3.75 | | | | | (39) $i(all_147_1)
% 21.69/3.75 | | | | | (40) $i(all_147_0)
% 21.69/3.75 | | | | | (41) implies(all_4_2, all_4_1) = all_147_1
% 21.69/3.75 | | | | | (42) implies(all_4_2, all_147_1) = all_147_0
% 21.69/3.75 | | | | |
% 21.69/3.75 | | | | | GROUND_INST: instantiating (1) with all_4_0, all_147_1, all_4_1,
% 21.69/3.75 | | | | | all_4_2, simplifying with (14), (41) gives:
% 21.69/3.75 | | | | | (43) all_147_1 = all_4_0
% 21.69/3.75 | | | | |
% 21.69/3.75 | | | | | GROUND_INST: instantiating (1) with all_143_1, all_147_0, all_4_0,
% 21.69/3.75 | | | | | all_4_2, simplifying with (35) gives:
% 21.69/3.75 | | | | | (44) all_147_0 = all_143_1 | ~ (implies(all_4_2, all_4_0) =
% 21.69/3.75 | | | | | all_147_0)
% 21.69/3.75 | | | | |
% 21.69/3.75 | | | | | REDUCE: (42), (43) imply:
% 21.69/3.75 | | | | | (45) implies(all_4_2, all_4_0) = all_147_0
% 21.69/3.75 | | | | |
% 21.69/3.75 | | | | | REDUCE: (39), (43) imply:
% 21.69/3.75 | | | | | (46) $i(all_4_0)
% 21.69/3.75 | | | | |
% 21.69/3.75 | | | | | BETA: splitting (44) gives:
% 21.69/3.75 | | | | |
% 21.69/3.75 | | | | | Case 1:
% 21.69/3.75 | | | | | |
% 21.69/3.75 | | | | | | (47) ~ (implies(all_4_2, all_4_0) = all_147_0)
% 21.69/3.75 | | | | | |
% 21.69/3.75 | | | | | | PRED_UNIFY: (45), (47) imply:
% 21.69/3.75 | | | | | | (48) $false
% 21.69/3.75 | | | | | |
% 21.69/3.75 | | | | | | CLOSE: (48) is inconsistent.
% 21.69/3.75 | | | | | |
% 21.69/3.75 | | | | | Case 2:
% 21.69/3.75 | | | | | |
% 21.69/3.75 | | | | | | (49) all_147_0 = all_143_1
% 21.69/3.75 | | | | | |
% 21.69/3.75 | | | | | | REDUCE: (40), (49) imply:
% 21.69/3.75 | | | | | | (50) $i(all_143_1)
% 21.69/3.75 | | | | | |
% 21.69/3.75 | | | | | | REDUCE: (38), (49) imply:
% 21.69/3.75 | | | | | | (51) is_a_theorem(all_143_1)
% 21.69/3.75 | | | | | |
% 21.69/3.75 | | | | | | GROUND_INST: instantiating (30) with all_143_1, all_4_0, all_143_0,
% 21.69/3.75 | | | | | | simplifying with (12), (34), (36), (46), (50), (51)
% 21.69/3.75 | | | | | | gives:
% 21.69/3.75 | | | | | | (52) $false
% 21.69/3.75 | | | | | |
% 21.69/3.75 | | | | | | CLOSE: (52) is inconsistent.
% 21.69/3.75 | | | | | |
% 21.69/3.75 | | | | | End of split
% 21.69/3.75 | | | | |
% 21.69/3.75 | | | | End of split
% 21.69/3.75 | | | |
% 21.69/3.75 | | | End of split
% 21.69/3.75 | | |
% 21.69/3.75 | | End of split
% 21.69/3.75 | |
% 21.69/3.75 | Case 2:
% 21.69/3.75 | |
% 21.69/3.75 | | (53) kn1 & ! [v0: $i] : ! [v1: $i] : ( ~ (and(v0, v0) = v1) | ~ $i(v0)
% 21.69/3.75 | | | ? [v2: $i] : (implies(v0, v1) = v2 & $i(v2) &
% 21.69/3.75 | | is_a_theorem(v2)))
% 21.69/3.75 | |
% 21.69/3.75 | | ALPHA: (53) implies:
% 21.69/3.75 | | (54) kn1
% 21.69/3.75 | |
% 21.69/3.75 | | PRED_UNIFY: (54), (rosser_kn1) imply:
% 21.69/3.75 | | (55) $false
% 21.69/3.75 | |
% 21.69/3.75 | | CLOSE: (55) is inconsistent.
% 21.69/3.75 | |
% 21.69/3.75 | End of split
% 21.69/3.75 |
% 21.69/3.75 End of proof
% 21.69/3.75 % SZS output end Proof for theBenchmark
% 21.69/3.75
% 21.69/3.75 3153ms
%------------------------------------------------------------------------------