TSTP Solution File: LCL512+1 by Princess---230619
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%------------------------------------------------------------------------------
% File : Princess---230619
% Problem : LCL512+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 : n001.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 08:11:31 EDT 2023
% Result : Theorem 11.97s 2.40s
% Output : Proof 15.45s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL512+1 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.12 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s
% 0.12/0.33 % Computer : n001.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
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Aug 25 05:47:12 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.18/0.59 ________ _____
% 0.18/0.59 ___ __ \_________(_)________________________________
% 0.18/0.59 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/
% 0.18/0.59 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ )
% 0.18/0.59 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/
% 0.18/0.59
% 0.18/0.59 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic
% 0.18/0.59 (2023-06-19)
% 0.18/0.59
% 0.18/0.59 (c) Philipp Rümmer, 2009-2023
% 0.18/0.59 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen,
% 0.18/0.59 Amanda Stjerna.
% 0.18/0.59 Free software under BSD-3-Clause.
% 0.18/0.59
% 0.18/0.59 For more information, visit http://www.philipp.ruemmer.org/princess.shtml
% 0.18/0.59
% 0.18/0.59 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 0.18/0.60 Running up to 7 provers in parallel.
% 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 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 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 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 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
% 2.99/1.12 Prover 1: Preprocessing ...
% 2.99/1.12 Prover 4: Preprocessing ...
% 3.51/1.16 Prover 6: Preprocessing ...
% 3.51/1.16 Prover 2: Preprocessing ...
% 3.51/1.16 Prover 5: Preprocessing ...
% 3.51/1.16 Prover 3: Preprocessing ...
% 3.51/1.16 Prover 0: Preprocessing ...
% 8.02/1.84 Prover 5: Proving ...
% 8.37/1.88 Prover 6: Constructing countermodel ...
% 8.37/1.91 Prover 3: Constructing countermodel ...
% 8.37/1.91 Prover 1: Constructing countermodel ...
% 8.37/1.93 Prover 4: Constructing countermodel ...
% 9.76/2.07 Prover 0: Proving ...
% 9.76/2.09 Prover 2: Proving ...
% 11.97/2.39 Prover 5: proved (1773ms)
% 11.97/2.39
% 11.97/2.40 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% 11.97/2.40
% 11.97/2.40 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470
% 11.97/2.41 Prover 3: stopped
% 11.97/2.41 Prover 2: stopped
% 11.97/2.42 Prover 6: stopped
% 11.97/2.43 Prover 0: stopped
% 11.97/2.44 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089
% 11.97/2.44 Prover 10: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=919308125
% 11.97/2.44 Prover 11: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1509710984
% 11.97/2.44 Prover 13: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=1138197443
% 11.97/2.45 Prover 7: Preprocessing ...
% 11.97/2.45 Prover 8: Preprocessing ...
% 12.64/2.48 Prover 10: Preprocessing ...
% 12.64/2.49 Prover 11: Preprocessing ...
% 12.93/2.52 Prover 13: Preprocessing ...
% 13.59/2.64 Prover 7: Constructing countermodel ...
% 13.59/2.65 Prover 8: Warning: ignoring some quantifiers
% 13.59/2.67 Prover 8: Constructing countermodel ...
% 14.23/2.70 Prover 13: Warning: ignoring some quantifiers
% 14.23/2.71 Prover 13: Constructing countermodel ...
% 14.23/2.72 Prover 10: Constructing countermodel ...
% 14.23/2.74 Prover 11: Constructing countermodel ...
% 15.24/2.85 Prover 4: Found proof (size 61)
% 15.24/2.85 Prover 4: proved (2242ms)
% 15.24/2.85 Prover 8: stopped
% 15.24/2.86 Prover 11: stopped
% 15.24/2.86 Prover 7: stopped
% 15.24/2.86 Prover 13: stopped
% 15.24/2.86 Prover 1: stopped
% 15.24/2.86 Prover 10: stopped
% 15.24/2.86
% 15.24/2.86 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% 15.24/2.86
% 15.24/2.87 % SZS output start Proof for theBenchmark
% 15.24/2.88 Assumptions after simplification:
% 15.24/2.88 ---------------------------------
% 15.24/2.88
% 15.24/2.88 (equivalence_1)
% 15.24/2.91 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ? [v4: $i] : ? [v5:
% 15.24/2.91 int] : ($i(v1) & $i(v0) & (( ~ (v5 = 0) & equiv(v0, v1) = v2 & implies(v2,
% 15.24/2.91 v3) = v4 & implies(v0, v1) = v3 & is_a_theorem(v4) = v5 & $i(v4) &
% 15.24/2.91 $i(v3) & $i(v2) & ~ equivalence_1) | (equivalence_1 & ! [v6: $i] : !
% 15.24/2.91 [v7: $i] : ! [v8: $i] : ( ~ (equiv(v6, v7) = v8) | ~ $i(v7) | ~
% 15.24/2.91 $i(v6) | ? [v9: $i] : ? [v10: $i] : (implies(v8, v9) = v10 &
% 15.24/2.91 implies(v6, v7) = v9 & is_a_theorem(v10) = 0 & $i(v10) & $i(v9))) &
% 15.24/2.91 ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (implies(v6, v7) = v8) | ~
% 15.24/2.91 $i(v7) | ~ $i(v6) | ? [v9: $i] : ? [v10: $i] : (equiv(v6, v7) = v9
% 15.24/2.91 & implies(v9, v8) = v10 & is_a_theorem(v10) = 0 & $i(v10) &
% 15.24/2.91 $i(v9))))))
% 15.24/2.91
% 15.24/2.91 (hilbert_equivalence_1)
% 15.24/2.91 ~ equivalence_1
% 15.24/2.91
% 15.24/2.91 (hilbert_op_equiv)
% 15.24/2.91 op_equiv
% 15.24/2.91
% 15.24/2.91 (kn2)
% 15.24/2.91 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ? [v4: int] : ($i(v1)
% 15.24/2.91 & $i(v0) & (( ~ (v4 = 0) & and(v0, v1) = v2 & implies(v2, v0) = v3 &
% 15.24/2.91 is_a_theorem(v3) = v4 & $i(v3) & $i(v2) & ~ kn2) | (kn2 & ! [v5: $i] :
% 15.24/2.91 ! [v6: $i] : ! [v7: $i] : ( ~ (and(v5, v6) = v7) | ~ $i(v6) | ~
% 15.24/2.91 $i(v5) | ? [v8: $i] : (implies(v7, v5) = v8 & is_a_theorem(v8) = 0 &
% 15.24/2.91 $i(v8))))))
% 15.24/2.91
% 15.24/2.91 (modus_ponens)
% 15.24/2.92 ? [v0: $i] : ? [v1: $i] : ? [v2: int] : ? [v3: $i] : ? [v4: int] : ?
% 15.24/2.92 [v5: int] : ($i(v1) & $i(v0) & ((v4 = 0 & v2 = 0 & ~ (v5 = 0) & implies(v0,
% 15.24/2.92 v1) = v3 & is_a_theorem(v3) = 0 & is_a_theorem(v1) = v5 &
% 15.24/2.92 is_a_theorem(v0) = 0 & $i(v3) & ~ modus_ponens) | (modus_ponens & !
% 15.24/2.92 [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (implies(v6, v7) = v8) | ~
% 15.24/2.92 $i(v7) | ~ $i(v6) | ? [v9: any] : ? [v10: any] : ? [v11: any] :
% 15.24/2.92 (is_a_theorem(v8) = v10 & is_a_theorem(v7) = v11 & is_a_theorem(v6) =
% 15.24/2.92 v9 & ( ~ (v10 = 0) | ~ (v9 = 0) | v11 = 0))))))
% 15.24/2.92
% 15.24/2.92 (op_equiv)
% 15.24/2.92 ~ op_equiv | ( ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (equiv(v0, v1) =
% 15.24/2.92 v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : (and(v3, v4) =
% 15.24/2.92 v2 & implies(v1, v0) = v4 & implies(v0, v1) = v3 & $i(v4) & $i(v3) &
% 15.24/2.92 $i(v2))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v1,
% 15.24/2.92 v0) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] :
% 15.24/2.92 (and(v4, v2) = v3 & equiv(v0, v1) = v3 & implies(v0, v1) = v4 & $i(v4) &
% 15.24/2.92 $i(v3))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0,
% 15.24/2.92 v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] :
% 15.24/2.92 (and(v2, v4) = v3 & equiv(v0, v1) = v3 & implies(v1, v0) = v4 & $i(v4) &
% 15.24/2.92 $i(v3))))
% 15.24/2.92
% 15.24/2.92 (rosser_kn2)
% 15.24/2.92 kn2
% 15.24/2.92
% 15.24/2.92 (rosser_modus_ponens)
% 15.24/2.92 modus_ponens
% 15.24/2.92
% 15.24/2.92 (rosser_op_equiv)
% 15.24/2.92 op_equiv
% 15.24/2.92
% 15.24/2.92 (function-axioms)
% 15.45/2.92 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (or(v3,
% 15.45/2.92 v2) = v1) | ~ (or(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2:
% 15.45/2.92 $i] : ! [v3: $i] : (v1 = v0 | ~ (and(v3, v2) = v1) | ~ (and(v3, v2) =
% 15.45/2.92 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 |
% 15.45/2.92 ~ (equiv(v3, v2) = v1) | ~ (equiv(v3, v2) = v0)) & ! [v0: $i] : ! [v1:
% 15.45/2.92 $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (implies(v3, v2) = v1) | ~
% 15.45/2.92 (implies(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0
% 15.45/2.92 | ~ (not(v2) = v1) | ~ (not(v2) = v0)) & ! [v0: MultipleValueBool] : !
% 15.45/2.92 [v1: MultipleValueBool] : ! [v2: $i] : (v1 = v0 | ~ (is_a_theorem(v2) = v1)
% 15.45/2.92 | ~ (is_a_theorem(v2) = v0))
% 15.45/2.92
% 15.45/2.92 Further assumptions not needed in the proof:
% 15.45/2.92 --------------------------------------------
% 15.45/2.93 and_1, and_2, and_3, cn1, cn2, cn3, equivalence_2, equivalence_3,
% 15.45/2.93 hilbert_op_implies_and, hilbert_op_or, implies_1, implies_2, implies_3, kn1,
% 15.45/2.93 kn3, modus_tollens, op_and, op_implies_and, op_implies_or, op_or, or_1, or_2,
% 15.45/2.93 or_3, r1, r2, r3, r4, r5, rosser_kn1, rosser_kn3, rosser_op_implies_and,
% 15.45/2.93 rosser_op_or, substitution_of_equivalents
% 15.45/2.93
% 15.45/2.93 Those formulas are unsatisfiable:
% 15.45/2.93 ---------------------------------
% 15.45/2.93
% 15.45/2.93 Begin of proof
% 15.45/2.93 |
% 15.45/2.93 | ALPHA: (function-axioms) implies:
% 15.45/2.93 | (1) ! [v0: MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: $i] :
% 15.45/2.93 | (v1 = v0 | ~ (is_a_theorem(v2) = v1) | ~ (is_a_theorem(v2) = v0))
% 15.45/2.93 | (2) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~
% 15.45/2.93 | (implies(v3, v2) = v1) | ~ (implies(v3, v2) = v0))
% 15.45/2.93 | (3) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~
% 15.45/2.93 | (equiv(v3, v2) = v1) | ~ (equiv(v3, v2) = v0))
% 15.45/2.93 |
% 15.45/2.93 | DELTA: instantiating (kn2) with fresh symbols all_14_0, all_14_1, all_14_2,
% 15.45/2.93 | all_14_3, all_14_4 gives:
% 15.45/2.93 | (4) $i(all_14_3) & $i(all_14_4) & (( ~ (all_14_0 = 0) & and(all_14_4,
% 15.45/2.93 | all_14_3) = all_14_2 & implies(all_14_2, all_14_4) = all_14_1 &
% 15.45/2.93 | is_a_theorem(all_14_1) = all_14_0 & $i(all_14_1) & $i(all_14_2) &
% 15.45/2.93 | ~ kn2) | (kn2 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 15.45/2.93 | (and(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] :
% 15.45/2.93 | (implies(v2, v0) = v3 & is_a_theorem(v3) = 0 & $i(v3)))))
% 15.45/2.93 |
% 15.45/2.93 | ALPHA: (4) implies:
% 15.45/2.93 | (5) ( ~ (all_14_0 = 0) & and(all_14_4, all_14_3) = all_14_2 &
% 15.45/2.93 | implies(all_14_2, all_14_4) = all_14_1 & is_a_theorem(all_14_1) =
% 15.45/2.93 | all_14_0 & $i(all_14_1) & $i(all_14_2) & ~ kn2) | (kn2 & ! [v0: $i]
% 15.45/2.93 | : ! [v1: $i] : ! [v2: $i] : ( ~ (and(v0, v1) = v2) | ~ $i(v1) | ~
% 15.45/2.93 | $i(v0) | ? [v3: $i] : (implies(v2, v0) = v3 & is_a_theorem(v3) = 0
% 15.45/2.93 | & $i(v3))))
% 15.45/2.93 |
% 15.45/2.93 | DELTA: instantiating (modus_ponens) with fresh symbols all_30_0, all_30_1,
% 15.45/2.93 | all_30_2, all_30_3, all_30_4, all_30_5 gives:
% 15.45/2.93 | (6) $i(all_30_4) & $i(all_30_5) & ((all_30_1 = 0 & all_30_3 = 0 & ~
% 15.45/2.93 | (all_30_0 = 0) & implies(all_30_5, all_30_4) = all_30_2 &
% 15.45/2.93 | is_a_theorem(all_30_2) = 0 & is_a_theorem(all_30_4) = all_30_0 &
% 15.45/2.93 | is_a_theorem(all_30_5) = 0 & $i(all_30_2) & ~ modus_ponens) |
% 15.45/2.93 | (modus_ponens & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 15.45/2.93 | (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: any] :
% 15.45/2.93 | ? [v4: any] : ? [v5: any] : (is_a_theorem(v2) = v4 &
% 15.45/2.93 | is_a_theorem(v1) = v5 & is_a_theorem(v0) = v3 & ( ~ (v4 = 0) |
% 15.45/2.94 | ~ (v3 = 0) | v5 = 0)))))
% 15.45/2.94 |
% 15.45/2.94 | ALPHA: (6) implies:
% 15.45/2.94 | (7) (all_30_1 = 0 & all_30_3 = 0 & ~ (all_30_0 = 0) & implies(all_30_5,
% 15.45/2.94 | all_30_4) = all_30_2 & is_a_theorem(all_30_2) = 0 &
% 15.45/2.94 | is_a_theorem(all_30_4) = all_30_0 & is_a_theorem(all_30_5) = 0 &
% 15.45/2.94 | $i(all_30_2) & ~ modus_ponens) | (modus_ponens & ! [v0: $i] : !
% 15.45/2.94 | [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~
% 15.45/2.94 | $i(v0) | ? [v3: any] : ? [v4: any] : ? [v5: any] :
% 15.45/2.94 | (is_a_theorem(v2) = v4 & is_a_theorem(v1) = v5 & is_a_theorem(v0) =
% 15.45/2.94 | v3 & ( ~ (v4 = 0) | ~ (v3 = 0) | v5 = 0))))
% 15.45/2.94 |
% 15.45/2.94 | DELTA: instantiating (equivalence_1) with fresh symbols all_43_0, all_43_1,
% 15.45/2.94 | all_43_2, all_43_3, all_43_4, all_43_5 gives:
% 15.45/2.94 | (8) $i(all_43_4) & $i(all_43_5) & (( ~ (all_43_0 = 0) & equiv(all_43_5,
% 15.45/2.94 | all_43_4) = all_43_3 & implies(all_43_3, all_43_2) = all_43_1 &
% 15.45/2.94 | implies(all_43_5, all_43_4) = all_43_2 & is_a_theorem(all_43_1) =
% 15.45/2.94 | all_43_0 & $i(all_43_1) & $i(all_43_2) & $i(all_43_3) & ~
% 15.45/2.94 | equivalence_1) | (equivalence_1 & ! [v0: $i] : ! [v1: $i] : !
% 15.45/2.94 | [v2: $i] : ( ~ (equiv(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ?
% 15.45/2.94 | [v3: $i] : ? [v4: $i] : (implies(v2, v3) = v4 & implies(v0, v1)
% 15.45/2.94 | = v3 & is_a_theorem(v4) = 0 & $i(v4) & $i(v3))) & ! [v0: $i] :
% 15.45/2.94 | ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~ $i(v1)
% 15.45/2.94 | | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : (equiv(v0, v1) = v3 &
% 15.45/2.94 | implies(v3, v2) = v4 & is_a_theorem(v4) = 0 & $i(v4) &
% 15.45/2.94 | $i(v3)))))
% 15.45/2.94 |
% 15.45/2.94 | ALPHA: (8) implies:
% 15.45/2.94 | (9) $i(all_43_5)
% 15.45/2.94 | (10) $i(all_43_4)
% 15.45/2.94 | (11) ( ~ (all_43_0 = 0) & equiv(all_43_5, all_43_4) = all_43_3 &
% 15.45/2.94 | implies(all_43_3, all_43_2) = all_43_1 & implies(all_43_5, all_43_4)
% 15.45/2.94 | = all_43_2 & is_a_theorem(all_43_1) = all_43_0 & $i(all_43_1) &
% 15.45/2.94 | $i(all_43_2) & $i(all_43_3) & ~ equivalence_1) | (equivalence_1 &
% 15.45/2.94 | ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (equiv(v0, v1) = v2) |
% 15.45/2.94 | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : (implies(v2,
% 15.45/2.94 | v3) = v4 & implies(v0, v1) = v3 & is_a_theorem(v4) = 0 &
% 15.45/2.94 | $i(v4) & $i(v3))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (
% 15.45/2.94 | ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] :
% 15.45/2.94 | ? [v4: $i] : (equiv(v0, v1) = v3 & implies(v3, v2) = v4 &
% 15.45/2.94 | is_a_theorem(v4) = 0 & $i(v4) & $i(v3))))
% 15.45/2.94 |
% 15.45/2.94 | BETA: splitting (11) gives:
% 15.45/2.94 |
% 15.45/2.94 | Case 1:
% 15.45/2.94 | |
% 15.45/2.94 | | (12) ~ (all_43_0 = 0) & equiv(all_43_5, all_43_4) = all_43_3 &
% 15.45/2.94 | | implies(all_43_3, all_43_2) = all_43_1 & implies(all_43_5, all_43_4)
% 15.45/2.94 | | = all_43_2 & is_a_theorem(all_43_1) = all_43_0 & $i(all_43_1) &
% 15.45/2.94 | | $i(all_43_2) & $i(all_43_3) & ~ equivalence_1
% 15.45/2.94 | |
% 15.45/2.94 | | ALPHA: (12) implies:
% 15.45/2.94 | | (13) ~ (all_43_0 = 0)
% 15.45/2.94 | | (14) $i(all_43_3)
% 15.45/2.94 | | (15) $i(all_43_2)
% 15.45/2.94 | | (16) is_a_theorem(all_43_1) = all_43_0
% 15.45/2.94 | | (17) implies(all_43_5, all_43_4) = all_43_2
% 15.45/2.94 | | (18) implies(all_43_3, all_43_2) = all_43_1
% 15.45/2.94 | | (19) equiv(all_43_5, all_43_4) = all_43_3
% 15.45/2.94 | |
% 15.45/2.94 | | BETA: splitting (op_equiv) gives:
% 15.45/2.94 | |
% 15.45/2.94 | | Case 1:
% 15.45/2.94 | | |
% 15.45/2.94 | | | (20) ~ op_equiv
% 15.45/2.94 | | |
% 15.45/2.94 | | | PRED_UNIFY: (20), (rosser_op_equiv) imply:
% 15.45/2.94 | | | (21) $false
% 15.45/2.95 | | |
% 15.45/2.95 | | | CLOSE: (21) is inconsistent.
% 15.45/2.95 | | |
% 15.45/2.95 | | Case 2:
% 15.45/2.95 | | |
% 15.45/2.95 | | | (22) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (equiv(v0, v1) = v2)
% 15.45/2.95 | | | | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : (and(v3,
% 15.45/2.95 | | | v4) = v2 & implies(v1, v0) = v4 & implies(v0, v1) = v3 &
% 15.45/2.95 | | | $i(v4) & $i(v3) & $i(v2))) & ! [v0: $i] : ! [v1: $i] : !
% 15.45/2.95 | | | [v2: $i] : ( ~ (implies(v1, v0) = v2) | ~ $i(v1) | ~ $i(v0) | ?
% 15.45/2.95 | | | [v3: $i] : ? [v4: $i] : (and(v4, v2) = v3 & equiv(v0, v1) = v3
% 15.45/2.95 | | | & implies(v0, v1) = v4 & $i(v4) & $i(v3))) & ! [v0: $i] : !
% 15.45/2.95 | | | [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) |
% 15.45/2.95 | | | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : (and(v2, v4) = v3 &
% 15.45/2.95 | | | equiv(v0, v1) = v3 & implies(v1, v0) = v4 & $i(v4) & $i(v3)))
% 15.45/2.95 | | |
% 15.45/2.95 | | | ALPHA: (22) implies:
% 15.45/2.95 | | | (23) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) =
% 15.45/2.95 | | | v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] :
% 15.45/2.95 | | | (and(v2, v4) = v3 & equiv(v0, v1) = v3 & implies(v1, v0) = v4 &
% 15.45/2.95 | | | $i(v4) & $i(v3)))
% 15.45/2.95 | | | (24) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v1, v0) =
% 15.45/2.95 | | | v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] :
% 15.45/2.95 | | | (and(v4, v2) = v3 & equiv(v0, v1) = v3 & implies(v0, v1) = v4 &
% 15.45/2.95 | | | $i(v4) & $i(v3)))
% 15.45/2.95 | | | (25) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (equiv(v0, v1) = v2)
% 15.45/2.95 | | | | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : (and(v3,
% 15.45/2.95 | | | v4) = v2 & implies(v1, v0) = v4 & implies(v0, v1) = v3 &
% 15.45/2.95 | | | $i(v4) & $i(v3) & $i(v2)))
% 15.45/2.95 | | |
% 15.45/2.95 | | | BETA: splitting (5) gives:
% 15.45/2.95 | | |
% 15.45/2.95 | | | Case 1:
% 15.45/2.95 | | | |
% 15.45/2.95 | | | | (26) ~ (all_14_0 = 0) & and(all_14_4, all_14_3) = all_14_2 &
% 15.45/2.95 | | | | implies(all_14_2, all_14_4) = all_14_1 & is_a_theorem(all_14_1)
% 15.45/2.95 | | | | = all_14_0 & $i(all_14_1) & $i(all_14_2) & ~ kn2
% 15.45/2.95 | | | |
% 15.45/2.95 | | | | ALPHA: (26) implies:
% 15.45/2.95 | | | | (27) ~ kn2
% 15.45/2.95 | | | |
% 15.45/2.95 | | | | PRED_UNIFY: (27), (rosser_kn2) imply:
% 15.45/2.95 | | | | (28) $false
% 15.45/2.95 | | | |
% 15.45/2.95 | | | | CLOSE: (28) is inconsistent.
% 15.45/2.95 | | | |
% 15.45/2.95 | | | Case 2:
% 15.45/2.95 | | | |
% 15.45/2.95 | | | | (29) kn2 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (and(v0, v1)
% 15.45/2.95 | | | | = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : (implies(v2,
% 15.45/2.95 | | | | v0) = v3 & is_a_theorem(v3) = 0 & $i(v3)))
% 15.45/2.95 | | | |
% 15.45/2.95 | | | | ALPHA: (29) implies:
% 15.45/2.95 | | | | (30) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (and(v0, v1) = v2)
% 15.45/2.95 | | | | | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : (implies(v2, v0) = v3
% 15.45/2.95 | | | | & is_a_theorem(v3) = 0 & $i(v3)))
% 15.45/2.95 | | | |
% 15.45/2.95 | | | | BETA: splitting (7) gives:
% 15.45/2.95 | | | |
% 15.45/2.95 | | | | Case 1:
% 15.45/2.95 | | | | |
% 15.45/2.95 | | | | | (31) all_30_1 = 0 & all_30_3 = 0 & ~ (all_30_0 = 0) &
% 15.45/2.95 | | | | | implies(all_30_5, all_30_4) = all_30_2 &
% 15.45/2.95 | | | | | is_a_theorem(all_30_2) = 0 & is_a_theorem(all_30_4) = all_30_0
% 15.45/2.95 | | | | | & is_a_theorem(all_30_5) = 0 & $i(all_30_2) & ~ modus_ponens
% 15.45/2.95 | | | | |
% 15.45/2.95 | | | | | ALPHA: (31) implies:
% 15.45/2.95 | | | | | (32) ~ modus_ponens
% 15.45/2.95 | | | | |
% 15.45/2.95 | | | | | PRED_UNIFY: (32), (rosser_modus_ponens) imply:
% 15.45/2.95 | | | | | (33) $false
% 15.45/2.95 | | | | |
% 15.45/2.95 | | | | | CLOSE: (33) is inconsistent.
% 15.45/2.95 | | | | |
% 15.45/2.95 | | | | Case 2:
% 15.45/2.95 | | | | |
% 15.45/2.95 | | | | | (34) modus_ponens & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 15.45/2.95 | | | | | (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3:
% 15.45/2.95 | | | | | any] : ? [v4: any] : ? [v5: any] : (is_a_theorem(v2) =
% 15.45/2.95 | | | | | v4 & is_a_theorem(v1) = v5 & is_a_theorem(v0) = v3 & ( ~
% 15.45/2.95 | | | | | (v4 = 0) | ~ (v3 = 0) | v5 = 0)))
% 15.45/2.95 | | | | |
% 15.45/2.95 | | | | | ALPHA: (34) implies:
% 15.45/2.96 | | | | | (35) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1)
% 15.45/2.96 | | | | | = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: any] : ? [v4:
% 15.45/2.96 | | | | | any] : ? [v5: any] : (is_a_theorem(v2) = v4 &
% 15.45/2.96 | | | | | is_a_theorem(v1) = v5 & is_a_theorem(v0) = v3 & ( ~ (v4 =
% 15.45/2.96 | | | | | 0) | ~ (v3 = 0) | v5 = 0)))
% 15.45/2.96 | | | | |
% 15.45/2.96 | | | | | GROUND_INST: instantiating (24) with all_43_4, all_43_5, all_43_2,
% 15.45/2.96 | | | | | simplifying with (9), (10), (17) gives:
% 15.45/2.96 | | | | | (36) ? [v0: $i] : ? [v1: $i] : (and(v1, all_43_2) = v0 &
% 15.45/2.96 | | | | | equiv(all_43_4, all_43_5) = v0 & implies(all_43_4, all_43_5)
% 15.45/2.96 | | | | | = v1 & $i(v1) & $i(v0))
% 15.45/2.96 | | | | |
% 15.45/2.96 | | | | | GROUND_INST: instantiating (23) with all_43_5, all_43_4, all_43_2,
% 15.45/2.96 | | | | | simplifying with (9), (10), (17) gives:
% 15.45/2.96 | | | | | (37) ? [v0: $i] : ? [v1: $i] : (and(all_43_2, v1) = v0 &
% 15.45/2.96 | | | | | equiv(all_43_5, all_43_4) = v0 & implies(all_43_4, all_43_5)
% 15.45/2.96 | | | | | = v1 & $i(v1) & $i(v0))
% 15.45/2.96 | | | | |
% 15.45/2.96 | | | | | GROUND_INST: instantiating (35) with all_43_3, all_43_2, all_43_1,
% 15.45/2.96 | | | | | simplifying with (14), (15), (18) gives:
% 15.45/2.96 | | | | | (38) ? [v0: any] : ? [v1: any] : ? [v2: any] :
% 15.45/2.96 | | | | | (is_a_theorem(all_43_1) = v1 & is_a_theorem(all_43_2) = v2 &
% 15.45/2.96 | | | | | is_a_theorem(all_43_3) = v0 & ( ~ (v1 = 0) | ~ (v0 = 0) |
% 15.45/2.96 | | | | | v2 = 0))
% 15.45/2.96 | | | | |
% 15.45/2.96 | | | | | GROUND_INST: instantiating (25) with all_43_5, all_43_4, all_43_3,
% 15.45/2.96 | | | | | simplifying with (9), (10), (19) gives:
% 15.45/2.96 | | | | | (39) ? [v0: $i] : ? [v1: $i] : (and(v0, v1) = all_43_3 &
% 15.45/2.96 | | | | | implies(all_43_4, all_43_5) = v1 & implies(all_43_5,
% 15.45/2.96 | | | | | all_43_4) = v0 & $i(v1) & $i(v0) & $i(all_43_3))
% 15.45/2.96 | | | | |
% 15.45/2.96 | | | | | DELTA: instantiating (37) with fresh symbols all_97_0, all_97_1 gives:
% 15.45/2.96 | | | | | (40) and(all_43_2, all_97_0) = all_97_1 & equiv(all_43_5, all_43_4)
% 15.45/2.96 | | | | | = all_97_1 & implies(all_43_4, all_43_5) = all_97_0 &
% 15.45/2.96 | | | | | $i(all_97_0) & $i(all_97_1)
% 15.45/2.96 | | | | |
% 15.45/2.96 | | | | | ALPHA: (40) implies:
% 15.45/2.96 | | | | | (41) implies(all_43_4, all_43_5) = all_97_0
% 15.45/2.96 | | | | | (42) equiv(all_43_5, all_43_4) = all_97_1
% 15.45/2.96 | | | | | (43) and(all_43_2, all_97_0) = all_97_1
% 15.45/2.96 | | | | |
% 15.45/2.96 | | | | | DELTA: instantiating (36) with fresh symbols all_99_0, all_99_1 gives:
% 15.45/2.96 | | | | | (44) and(all_99_0, all_43_2) = all_99_1 & equiv(all_43_4, all_43_5)
% 15.45/2.96 | | | | | = all_99_1 & implies(all_43_4, all_43_5) = all_99_0 &
% 15.45/2.96 | | | | | $i(all_99_0) & $i(all_99_1)
% 15.45/2.96 | | | | |
% 15.45/2.96 | | | | | ALPHA: (44) implies:
% 15.45/2.96 | | | | | (45) $i(all_99_0)
% 15.45/2.96 | | | | | (46) implies(all_43_4, all_43_5) = all_99_0
% 15.45/2.96 | | | | |
% 15.45/2.96 | | | | | DELTA: instantiating (38) with fresh symbols all_105_0, all_105_1,
% 15.45/2.96 | | | | | all_105_2 gives:
% 15.45/2.96 | | | | | (47) is_a_theorem(all_43_1) = all_105_1 & is_a_theorem(all_43_2) =
% 15.45/2.96 | | | | | all_105_0 & is_a_theorem(all_43_3) = all_105_2 & ( ~
% 15.45/2.96 | | | | | (all_105_1 = 0) | ~ (all_105_2 = 0) | all_105_0 = 0)
% 15.45/2.96 | | | | |
% 15.45/2.96 | | | | | ALPHA: (47) implies:
% 15.45/2.96 | | | | | (48) is_a_theorem(all_43_1) = all_105_1
% 15.45/2.96 | | | | |
% 15.45/2.96 | | | | | DELTA: instantiating (39) with fresh symbols all_109_0, all_109_1
% 15.45/2.96 | | | | | gives:
% 15.45/2.96 | | | | | (49) and(all_109_1, all_109_0) = all_43_3 & implies(all_43_4,
% 15.45/2.96 | | | | | all_43_5) = all_109_0 & implies(all_43_5, all_43_4) =
% 15.45/2.96 | | | | | all_109_1 & $i(all_109_0) & $i(all_109_1) & $i(all_43_3)
% 15.45/2.96 | | | | |
% 15.45/2.96 | | | | | ALPHA: (49) implies:
% 15.45/2.96 | | | | | (50) $i(all_109_1)
% 15.45/2.96 | | | | | (51) implies(all_43_5, all_43_4) = all_109_1
% 15.45/2.96 | | | | | (52) implies(all_43_4, all_43_5) = all_109_0
% 15.45/2.96 | | | | |
% 15.45/2.96 | | | | | GROUND_INST: instantiating (1) with all_43_0, all_105_1, all_43_1,
% 15.45/2.96 | | | | | simplifying with (16), (48) gives:
% 15.45/2.96 | | | | | (53) all_105_1 = all_43_0
% 15.45/2.96 | | | | |
% 15.45/2.96 | | | | | GROUND_INST: instantiating (2) with all_43_2, all_109_1, all_43_4,
% 15.45/2.96 | | | | | all_43_5, simplifying with (17), (51) gives:
% 15.45/2.96 | | | | | (54) all_109_1 = all_43_2
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | GROUND_INST: instantiating (2) with all_99_0, all_109_0, all_43_5,
% 15.45/2.97 | | | | | all_43_4, simplifying with (46), (52) gives:
% 15.45/2.97 | | | | | (55) all_109_0 = all_99_0
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | GROUND_INST: instantiating (2) with all_97_0, all_109_0, all_43_5,
% 15.45/2.97 | | | | | all_43_4, simplifying with (41), (52) gives:
% 15.45/2.97 | | | | | (56) all_109_0 = all_97_0
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | GROUND_INST: instantiating (3) with all_43_3, all_97_1, all_43_4,
% 15.45/2.97 | | | | | all_43_5, simplifying with (19), (42) gives:
% 15.45/2.97 | | | | | (57) all_97_1 = all_43_3
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | COMBINE_EQS: (55), (56) imply:
% 15.45/2.97 | | | | | (58) all_99_0 = all_97_0
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | SIMP: (58) implies:
% 15.45/2.97 | | | | | (59) all_99_0 = all_97_0
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | REDUCE: (43), (57) imply:
% 15.45/2.97 | | | | | (60) and(all_43_2, all_97_0) = all_43_3
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | REDUCE: (45), (59) imply:
% 15.45/2.97 | | | | | (61) $i(all_97_0)
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | GROUND_INST: instantiating (30) with all_43_2, all_97_0, all_43_3,
% 15.45/2.97 | | | | | simplifying with (15), (60), (61) gives:
% 15.45/2.97 | | | | | (62) ? [v0: $i] : (implies(all_43_3, all_43_2) = v0 &
% 15.45/2.97 | | | | | is_a_theorem(v0) = 0 & $i(v0))
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | DELTA: instantiating (62) with fresh symbol all_148_0 gives:
% 15.45/2.97 | | | | | (63) implies(all_43_3, all_43_2) = all_148_0 &
% 15.45/2.97 | | | | | is_a_theorem(all_148_0) = 0 & $i(all_148_0)
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | ALPHA: (63) implies:
% 15.45/2.97 | | | | | (64) is_a_theorem(all_148_0) = 0
% 15.45/2.97 | | | | | (65) implies(all_43_3, all_43_2) = all_148_0
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | GROUND_INST: instantiating (2) with all_43_1, all_148_0, all_43_2,
% 15.45/2.97 | | | | | all_43_3, simplifying with (18), (65) gives:
% 15.45/2.97 | | | | | (66) all_148_0 = all_43_1
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | REDUCE: (64), (66) imply:
% 15.45/2.97 | | | | | (67) is_a_theorem(all_43_1) = 0
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | GROUND_INST: instantiating (1) with all_43_0, 0, all_43_1, simplifying
% 15.45/2.97 | | | | | with (16), (67) gives:
% 15.45/2.97 | | | | | (68) all_43_0 = 0
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | REDUCE: (13), (68) imply:
% 15.45/2.97 | | | | | (69) $false
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | | CLOSE: (69) is inconsistent.
% 15.45/2.97 | | | | |
% 15.45/2.97 | | | | End of split
% 15.45/2.97 | | | |
% 15.45/2.97 | | | End of split
% 15.45/2.97 | | |
% 15.45/2.97 | | End of split
% 15.45/2.97 | |
% 15.45/2.97 | Case 2:
% 15.45/2.97 | |
% 15.45/2.97 | | (70) equivalence_1 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 15.45/2.97 | | (equiv(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ?
% 15.45/2.97 | | [v4: $i] : (implies(v2, v3) = v4 & implies(v0, v1) = v3 &
% 15.45/2.97 | | is_a_theorem(v4) = 0 & $i(v4) & $i(v3))) & ! [v0: $i] : ! [v1:
% 15.45/2.97 | | $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~
% 15.45/2.97 | | $i(v0) | ? [v3: $i] : ? [v4: $i] : (equiv(v0, v1) = v3 &
% 15.45/2.97 | | implies(v3, v2) = v4 & is_a_theorem(v4) = 0 & $i(v4) & $i(v3)))
% 15.45/2.97 | |
% 15.45/2.97 | | ALPHA: (70) implies:
% 15.45/2.97 | | (71) equivalence_1
% 15.45/2.97 | |
% 15.45/2.97 | | PRED_UNIFY: (71), (hilbert_equivalence_1) imply:
% 15.45/2.97 | | (72) $false
% 15.45/2.97 | |
% 15.45/2.97 | | CLOSE: (72) is inconsistent.
% 15.45/2.97 | |
% 15.45/2.97 | End of split
% 15.45/2.97 |
% 15.45/2.97 End of proof
% 15.45/2.97 % SZS output end Proof for theBenchmark
% 15.45/2.97
% 15.45/2.97 2384ms
%------------------------------------------------------------------------------