TSTP Solution File: LCL499+1 by Princess---230619
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%------------------------------------------------------------------------------
% File : Princess---230619
% Problem : LCL499+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 : n009.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 08:11:28 EDT 2023
% Result : Theorem 15.90s 2.96s
% Output : Proof 86.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.05/0.10 % Problem : LCL499+1 : TPTP v8.1.2. Released v3.3.0.
% 0.05/0.10 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s
% 0.10/0.31 % Computer : n009.cluster.edu
% 0.10/0.31 % Model : x86_64 x86_64
% 0.10/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.31 % Memory : 8042.1875MB
% 0.10/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.31 % CPULimit : 300
% 0.10/0.31 % WCLimit : 300
% 0.10/0.31 % DateTime : Thu Aug 24 18:29:36 EDT 2023
% 0.10/0.31 % CPUTime :
% 0.15/0.58 ________ _____
% 0.15/0.58 ___ __ \_________(_)________________________________
% 0.15/0.58 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/
% 0.15/0.58 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ )
% 0.15/0.58 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/
% 0.15/0.58
% 0.15/0.58 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic
% 0.15/0.58 (2023-06-19)
% 0.15/0.58
% 0.15/0.58 (c) Philipp Rümmer, 2009-2023
% 0.15/0.58 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen,
% 0.15/0.58 Amanda Stjerna.
% 0.15/0.58 Free software under BSD-3-Clause.
% 0.15/0.58
% 0.15/0.58 For more information, visit http://www.philipp.ruemmer.org/princess.shtml
% 0.15/0.58
% 0.15/0.58 Loading /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 0.15/0.60 Running up to 7 provers in parallel.
% 0.15/0.62 Prover 0: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1042961893
% 0.15/0.62 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1571432423
% 0.15/0.62 Prover 3: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1922548996
% 0.15/0.62 Prover 2: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMinimalAndEmpty -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1065072994
% 0.15/0.62 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=1868514696
% 0.15/0.62 Prover 6: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximalOutermost -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1399714365
% 0.15/0.62 Prover 5: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=complete -randomSeed=1259561288
% 3.06/1.16 Prover 4: Preprocessing ...
% 3.29/1.18 Prover 1: Preprocessing ...
% 3.29/1.21 Prover 5: Preprocessing ...
% 3.29/1.21 Prover 3: Preprocessing ...
% 3.29/1.21 Prover 6: Preprocessing ...
% 3.29/1.21 Prover 2: Preprocessing ...
% 3.29/1.21 Prover 0: Preprocessing ...
% 9.39/2.05 Prover 5: Proving ...
% 9.39/2.10 Prover 1: Constructing countermodel ...
% 9.39/2.10 Prover 6: Constructing countermodel ...
% 10.06/2.13 Prover 4: Constructing countermodel ...
% 10.64/2.22 Prover 3: Constructing countermodel ...
% 10.64/2.22 Prover 0: Proving ...
% 11.62/2.32 Prover 2: Proving ...
% 15.90/2.95 Prover 0: proved (2345ms)
% 15.90/2.95
% 15.90/2.96 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p
% 15.90/2.96
% 15.90/2.96 Prover 3: stopped
% 15.90/2.96 Prover 6: stopped
% 15.90/2.97 Prover 5: stopped
% 15.90/2.97 Prover 2: stopped
% 15.90/2.97 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089
% 15.90/2.97 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470
% 15.90/2.97 Prover 10: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=919308125
% 15.90/2.97 Prover 11: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1509710984
% 15.90/2.97 Prover 13: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=1138197443
% 16.79/3.03 Prover 8: Preprocessing ...
% 16.79/3.06 Prover 13: Preprocessing ...
% 16.79/3.06 Prover 7: Preprocessing ...
% 16.79/3.06 Prover 11: Preprocessing ...
% 16.79/3.08 Prover 10: Preprocessing ...
% 17.56/3.27 Prover 1: gave up
% 18.47/3.28 Prover 16: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=completeFrugal -randomSeed=-2043353683
% 18.47/3.29 Prover 8: Warning: ignoring some quantifiers
% 18.47/3.32 Prover 16: Preprocessing ...
% 18.47/3.33 Prover 8: Constructing countermodel ...
% 18.47/3.34 Prover 13: Warning: ignoring some quantifiers
% 19.42/3.36 Prover 13: Constructing countermodel ...
% 19.42/3.38 Prover 10: Constructing countermodel ...
% 19.42/3.39 Prover 7: Constructing countermodel ...
% 20.75/3.55 Prover 11: Constructing countermodel ...
% 20.75/3.55 Prover 16: Warning: ignoring some quantifiers
% 20.75/3.59 Prover 16: Constructing countermodel ...
% 23.24/3.94 Prover 8: gave up
% 23.24/3.95 Prover 19: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=-1780594085
% 24.13/4.04 Prover 19: Preprocessing ...
% 24.13/4.04 Prover 13: gave up
% 25.00/4.12 Prover 10: gave up
% 25.27/4.21 Prover 19: Warning: ignoring some quantifiers
% 25.27/4.22 Prover 19: Constructing countermodel ...
% 30.58/4.84 Prover 19: gave up
% 36.94/5.90 Prover 16: gave up
% 84.93/13.78 Prover 11: Found proof (size 162)
% 84.93/13.78 Prover 11: proved (10803ms)
% 84.93/13.78 Prover 7: stopped
% 84.93/13.78 Prover 4: stopped
% 84.93/13.78
% 84.93/13.78 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p
% 84.93/13.78
% 84.93/13.81 % SZS output start Proof for theBenchmark
% 84.93/13.81 Assumptions after simplification:
% 84.93/13.81 ---------------------------------
% 84.93/13.81
% 84.93/13.81 (kn1)
% 85.43/13.85 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: int] : ($i(v0) & (( ~ (v3 =
% 85.43/13.85 0) & and(v0, v0) = v1 & implies(v0, v1) = v2 & is_a_theorem(v2) = v3 &
% 85.43/13.85 $i(v2) & $i(v1) & ~ kn1) | (kn1 & ! [v4: $i] : ! [v5: $i] : ( ~
% 85.43/13.85 (and(v4, v4) = v5) | ~ $i(v4) | ? [v6: $i] : (implies(v4, v5) = v6 &
% 85.43/13.85 is_a_theorem(v6) = 0 & $i(v6))))))
% 85.43/13.85
% 85.43/13.85 (modus_ponens)
% 85.43/13.85 ? [v0: $i] : ? [v1: $i] : ? [v2: int] : ? [v3: $i] : ? [v4: int] : ?
% 85.43/13.85 [v5: int] : ($i(v1) & $i(v0) & ((v4 = 0 & v2 = 0 & ~ (v5 = 0) & implies(v0,
% 85.43/13.85 v1) = v3 & is_a_theorem(v3) = 0 & is_a_theorem(v1) = v5 &
% 85.43/13.85 is_a_theorem(v0) = 0 & $i(v3) & ~ modus_ponens) | (modus_ponens & !
% 85.43/13.85 [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (implies(v6, v7) = v8) | ~
% 85.43/13.85 $i(v7) | ~ $i(v6) | ? [v9: int] : ? [v10: int] : ? [v11: int] :
% 85.43/13.85 ((v11 = 0 & is_a_theorem(v7) = 0) | ( ~ (v10 = 0) & is_a_theorem(v8) =
% 85.43/13.85 v10) | ( ~ (v9 = 0) & is_a_theorem(v6) = v9))))))
% 85.43/13.85
% 85.43/13.85 (op_and)
% 85.43/13.86 ~ op_and | ( ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4:
% 85.43/13.86 $i] : ( ~ (or(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = v2) | ~
% 85.43/13.86 $i(v1) | ~ $i(v0) | ? [v5: $i] : (and(v0, v1) = v5 & not(v4) = v5 &
% 85.43/13.86 $i(v5))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (and(v0, v1) =
% 85.43/13.86 v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : ? [v5: $i] :
% 85.43/13.86 (or(v3, v4) = v5 & not(v5) = v2 & not(v1) = v4 & not(v0) = v3 & $i(v5) &
% 85.43/13.86 $i(v4) & $i(v3) & $i(v2))))
% 85.43/13.86
% 85.43/13.86 (op_equiv)
% 85.43/13.86 ~ op_equiv | ( ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (equiv(v0, v1) =
% 85.43/13.86 v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : (and(v3, v4) =
% 85.43/13.86 v2 & implies(v1, v0) = v4 & implies(v0, v1) = v3 & $i(v4) & $i(v3) &
% 85.43/13.86 $i(v2))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v1,
% 85.43/13.86 v0) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] :
% 85.43/13.86 (and(v4, v2) = v3 & equiv(v0, v1) = v3 & implies(v0, v1) = v4 & $i(v4) &
% 85.43/13.86 $i(v3))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0,
% 85.43/13.86 v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] :
% 85.43/13.86 (and(v2, v4) = v3 & equiv(v0, v1) = v3 & implies(v1, v0) = v4 & $i(v4) &
% 85.43/13.86 $i(v3))))
% 85.43/13.86
% 85.43/13.86 (op_implies_and)
% 85.43/13.87 ~ op_implies_and | ( ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (
% 85.43/13.87 ~ (and(v0, v2) = v3) | ~ (not(v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v4:
% 85.43/13.87 $i] : (not(v3) = v4 & implies(v0, v1) = v4 & $i(v4))) & ! [v0: $i] : !
% 85.43/13.87 [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0)
% 85.43/13.87 | ? [v3: $i] : ? [v4: $i] : (and(v0, v3) = v4 & not(v4) = v2 & not(v1) =
% 85.43/13.87 v3 & $i(v4) & $i(v3) & $i(v2))))
% 85.43/13.87
% 85.43/13.87 (op_implies_or)
% 85.43/13.87 ~ op_implies_or | ( ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (
% 85.43/13.87 ~ (or(v2, v1) = v3) | ~ (not(v0) = v2) | ~ $i(v1) | ~ $i(v0) |
% 85.43/13.87 (implies(v0, v1) = v3 & $i(v3))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i]
% 85.43/13.87 : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : (or(v3,
% 85.43/13.87 v1) = v2 & not(v0) = v3 & $i(v3) & $i(v2))))
% 85.43/13.87
% 85.43/13.87 (op_or)
% 85.43/13.88 ~ op_or | ( ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4:
% 85.43/13.88 $i] : ( ~ (and(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = v2) | ~
% 85.43/13.88 $i(v1) | ~ $i(v0) | ? [v5: $i] : (or(v0, v1) = v5 & not(v4) = v5 &
% 85.43/13.88 $i(v5))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (or(v0, v1) =
% 85.43/13.88 v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : ? [v5: $i] :
% 85.43/13.88 (and(v3, v4) = v5 & not(v5) = v2 & not(v1) = v4 & not(v0) = v3 & $i(v5) &
% 85.43/13.88 $i(v4) & $i(v3) & $i(v2))))
% 85.43/13.88
% 85.43/13.88 (principia_modus_ponens)
% 85.43/13.88 modus_ponens
% 85.43/13.88
% 85.43/13.88 (principia_op_and)
% 85.43/13.88 op_and
% 85.43/13.88
% 85.43/13.88 (principia_op_equiv)
% 85.43/13.88 op_equiv
% 85.43/13.88
% 85.43/13.88 (principia_op_implies_or)
% 85.43/13.88 op_implies_or
% 85.43/13.88
% 85.43/13.88 (principia_r1)
% 85.43/13.88 r1
% 85.43/13.88
% 85.43/13.88 (principia_r2)
% 85.43/13.88 r2
% 85.43/13.88
% 85.43/13.88 (principia_r3)
% 85.43/13.88 r3
% 85.43/13.88
% 85.43/13.88 (principia_r4)
% 85.43/13.88 r4
% 85.43/13.88
% 85.43/13.88 (r1)
% 85.43/13.88 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: int] : ($i(v0) & (( ~ (v3 =
% 85.43/13.88 0) & or(v0, v0) = v1 & implies(v1, v0) = v2 & is_a_theorem(v2) = v3 &
% 85.43/13.88 $i(v2) & $i(v1) & ~ r1) | (r1 & ! [v4: $i] : ! [v5: $i] : ( ~ (or(v4,
% 85.43/13.88 v4) = v5) | ~ $i(v4) | ? [v6: $i] : (implies(v5, v4) = v6 &
% 85.43/13.88 is_a_theorem(v6) = 0 & $i(v6))))))
% 85.43/13.88
% 85.43/13.88 (r2)
% 85.43/13.89 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ? [v4: int] : ($i(v1)
% 85.43/13.89 & $i(v0) & (( ~ (v4 = 0) & or(v0, v1) = v2 & implies(v1, v2) = v3 &
% 85.43/13.89 is_a_theorem(v3) = v4 & $i(v3) & $i(v2) & ~ r2) | (r2 & ! [v5: $i] :
% 85.43/13.89 ! [v6: $i] : ! [v7: $i] : ( ~ (or(v5, v6) = v7) | ~ $i(v6) | ~ $i(v5)
% 85.43/13.89 | ? [v8: $i] : (implies(v6, v7) = v8 & is_a_theorem(v8) = 0 &
% 85.43/13.89 $i(v8))))))
% 85.43/13.89
% 85.43/13.89 (r3)
% 85.43/13.89 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ? [v4: $i] : ? [v5:
% 85.43/13.89 int] : ($i(v1) & $i(v0) & (( ~ (v5 = 0) & or(v1, v0) = v3 & or(v0, v1) = v2
% 85.43/13.89 & implies(v2, v3) = v4 & is_a_theorem(v4) = v5 & $i(v4) & $i(v3) &
% 85.43/13.89 $i(v2) & ~ r3) | (r3 & ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~
% 85.43/13.89 (or(v7, v6) = v8) | ~ $i(v7) | ~ $i(v6) | ? [v9: $i] : ? [v10: $i]
% 85.43/13.89 : (or(v6, v7) = v9 & implies(v9, v8) = v10 & is_a_theorem(v10) = 0 &
% 85.43/13.89 $i(v10) & $i(v9))) & ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~
% 85.43/13.89 (or(v6, v7) = v8) | ~ $i(v7) | ~ $i(v6) | ? [v9: $i] : ? [v10: $i]
% 85.43/13.89 : (or(v7, v6) = v9 & implies(v8, v9) = v10 & is_a_theorem(v10) = 0 &
% 85.43/13.89 $i(v10) & $i(v9))))))
% 85.43/13.89
% 85.43/13.89 (r4)
% 85.70/13.90 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ? [v4: $i] : ? [v5:
% 85.70/13.90 $i] : ? [v6: $i] : ? [v7: $i] : ? [v8: int] : ($i(v2) & $i(v1) & $i(v0) &
% 85.70/13.90 (( ~ (v8 = 0) & or(v1, v5) = v6 & or(v1, v2) = v3 & or(v0, v3) = v4 & or(v0,
% 85.70/13.90 v2) = v5 & implies(v4, v6) = v7 & is_a_theorem(v7) = v8 & $i(v7) &
% 85.70/13.90 $i(v6) & $i(v5) & $i(v4) & $i(v3) & ~ r4) | (r4 & ! [v9: $i] : !
% 85.70/13.90 [v10: $i] : ! [v11: $i] : ! [v12: $i] : ! [v13: $i] : ( ~ (or(v10,
% 85.70/13.90 v12) = v13) | ~ (or(v9, v11) = v12) | ~ $i(v11) | ~ $i(v10) |
% 85.70/13.90 ~ $i(v9) | ? [v14: $i] : ? [v15: $i] : ? [v16: $i] : (or(v10, v11)
% 85.70/13.90 = v14 & or(v9, v14) = v15 & implies(v15, v13) = v16 &
% 85.70/13.90 is_a_theorem(v16) = 0 & $i(v16) & $i(v15) & $i(v14))) & ! [v9: $i]
% 85.70/13.90 : ! [v10: $i] : ! [v11: $i] : ! [v12: $i] : ! [v13: $i] : ( ~
% 85.70/13.90 (or(v10, v11) = v12) | ~ (or(v9, v12) = v13) | ~ $i(v11) | ~
% 85.70/13.90 $i(v10) | ~ $i(v9) | ? [v14: $i] : ? [v15: $i] : ? [v16: $i] :
% 85.70/13.90 (or(v10, v14) = v15 & or(v9, v11) = v14 & implies(v13, v15) = v16 &
% 85.70/13.90 is_a_theorem(v16) = 0 & $i(v16) & $i(v15) & $i(v14))))))
% 85.70/13.90
% 85.70/13.90 (rosser_kn1)
% 85.70/13.90 ~ kn1
% 85.70/13.90
% 85.70/13.90 (rosser_op_equiv)
% 85.70/13.90 op_equiv
% 85.70/13.90
% 85.70/13.90 (rosser_op_implies_and)
% 85.70/13.90 op_implies_and
% 85.70/13.90
% 85.70/13.90 (rosser_op_or)
% 85.70/13.90 op_or
% 85.70/13.90
% 85.70/13.90 (function-axioms)
% 85.70/13.90 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (or(v3,
% 85.70/13.90 v2) = v1) | ~ (or(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2:
% 85.70/13.90 $i] : ! [v3: $i] : (v1 = v0 | ~ (and(v3, v2) = v1) | ~ (and(v3, v2) =
% 85.70/13.90 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 |
% 85.70/13.90 ~ (equiv(v3, v2) = v1) | ~ (equiv(v3, v2) = v0)) & ! [v0: $i] : ! [v1:
% 85.70/13.90 $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (implies(v3, v2) = v1) | ~
% 85.70/13.90 (implies(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0
% 85.70/13.90 | ~ (not(v2) = v1) | ~ (not(v2) = v0)) & ! [v0: MultipleValueBool] : !
% 85.70/13.90 [v1: MultipleValueBool] : ! [v2: $i] : (v1 = v0 | ~ (is_a_theorem(v2) = v1)
% 85.70/13.90 | ~ (is_a_theorem(v2) = v0))
% 85.70/13.90
% 85.70/13.90 Further assumptions not needed in the proof:
% 85.70/13.90 --------------------------------------------
% 85.70/13.90 and_1, and_2, and_3, cn1, cn2, cn3, equivalence_1, equivalence_2, equivalence_3,
% 85.70/13.90 implies_1, implies_2, implies_3, kn2, kn3, modus_tollens, or_1, or_2, or_3,
% 85.70/13.90 principia_r5, r5, substitution_of_equivalents
% 85.70/13.90
% 85.70/13.90 Those formulas are unsatisfiable:
% 85.70/13.90 ---------------------------------
% 85.70/13.90
% 85.70/13.90 Begin of proof
% 85.70/13.90 |
% 85.70/13.90 | ALPHA: (function-axioms) implies:
% 85.70/13.90 | (1) ! [v0: MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: $i] :
% 85.70/13.90 | (v1 = v0 | ~ (is_a_theorem(v2) = v1) | ~ (is_a_theorem(v2) = v0))
% 85.70/13.90 | (2) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (not(v2) = v1)
% 85.70/13.90 | | ~ (not(v2) = v0))
% 85.70/13.91 | (3) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~
% 85.70/13.91 | (implies(v3, v2) = v1) | ~ (implies(v3, v2) = v0))
% 85.70/13.91 | (4) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~
% 85.70/13.91 | (or(v3, v2) = v1) | ~ (or(v3, v2) = v0))
% 85.70/13.91 |
% 85.70/13.91 | DELTA: instantiating (kn1) with fresh symbols all_4_0, all_4_1, all_4_2,
% 85.70/13.91 | all_4_3 gives:
% 85.70/13.91 | (5) $i(all_4_3) & (( ~ (all_4_0 = 0) & and(all_4_3, all_4_3) = all_4_2 &
% 85.70/13.91 | implies(all_4_3, all_4_2) = all_4_1 & is_a_theorem(all_4_1) =
% 85.70/13.91 | all_4_0 & $i(all_4_1) & $i(all_4_2) & ~ kn1) | (kn1 & ! [v0: $i]
% 85.70/13.91 | : ! [v1: $i] : ( ~ (and(v0, v0) = v1) | ~ $i(v0) | ? [v2: $i] :
% 85.70/13.91 | (implies(v0, v1) = v2 & is_a_theorem(v2) = 0 & $i(v2)))))
% 85.70/13.91 |
% 85.70/13.91 | ALPHA: (5) implies:
% 85.70/13.91 | (6) $i(all_4_3)
% 85.70/13.91 | (7) ( ~ (all_4_0 = 0) & and(all_4_3, all_4_3) = all_4_2 & implies(all_4_3,
% 85.70/13.91 | all_4_2) = all_4_1 & is_a_theorem(all_4_1) = all_4_0 & $i(all_4_1)
% 85.70/13.91 | & $i(all_4_2) & ~ kn1) | (kn1 & ! [v0: $i] : ! [v1: $i] : ( ~
% 85.70/13.91 | (and(v0, v0) = v1) | ~ $i(v0) | ? [v2: $i] : (implies(v0, v1) =
% 85.70/13.91 | v2 & is_a_theorem(v2) = 0 & $i(v2))))
% 85.70/13.91 |
% 85.70/13.91 | DELTA: instantiating (r1) with fresh symbols all_6_0, all_6_1, all_6_2,
% 85.70/13.91 | all_6_3 gives:
% 85.70/13.91 | (8) $i(all_6_3) & (( ~ (all_6_0 = 0) & or(all_6_3, all_6_3) = all_6_2 &
% 85.70/13.91 | implies(all_6_2, all_6_3) = all_6_1 & is_a_theorem(all_6_1) =
% 85.70/13.91 | all_6_0 & $i(all_6_1) & $i(all_6_2) & ~ r1) | (r1 & ! [v0: $i] :
% 85.70/13.91 | ! [v1: $i] : ( ~ (or(v0, v0) = v1) | ~ $i(v0) | ? [v2: $i] :
% 85.70/13.91 | (implies(v1, v0) = v2 & is_a_theorem(v2) = 0 & $i(v2)))))
% 85.70/13.91 |
% 85.70/13.91 | ALPHA: (8) implies:
% 85.70/13.91 | (9) ( ~ (all_6_0 = 0) & or(all_6_3, all_6_3) = all_6_2 & implies(all_6_2,
% 85.70/13.91 | all_6_3) = all_6_1 & is_a_theorem(all_6_1) = all_6_0 & $i(all_6_1)
% 85.70/13.91 | & $i(all_6_2) & ~ r1) | (r1 & ! [v0: $i] : ! [v1: $i] : ( ~
% 85.70/13.91 | (or(v0, v0) = v1) | ~ $i(v0) | ? [v2: $i] : (implies(v1, v0) = v2
% 85.70/13.91 | & is_a_theorem(v2) = 0 & $i(v2))))
% 85.70/13.91 |
% 85.70/13.91 | DELTA: instantiating (r2) with fresh symbols all_20_0, all_20_1, all_20_2,
% 85.70/13.91 | all_20_3, all_20_4 gives:
% 85.70/13.92 | (10) $i(all_20_3) & $i(all_20_4) & (( ~ (all_20_0 = 0) & or(all_20_4,
% 85.70/13.92 | all_20_3) = all_20_2 & implies(all_20_3, all_20_2) = all_20_1 &
% 85.70/13.92 | is_a_theorem(all_20_1) = all_20_0 & $i(all_20_1) & $i(all_20_2) &
% 85.70/13.92 | ~ r2) | (r2 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 85.70/13.92 | (or(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] :
% 85.70/13.92 | (implies(v1, v2) = v3 & is_a_theorem(v3) = 0 & $i(v3)))))
% 85.70/13.92 |
% 85.70/13.92 | ALPHA: (10) implies:
% 85.70/13.92 | (11) ( ~ (all_20_0 = 0) & or(all_20_4, all_20_3) = all_20_2 &
% 85.70/13.92 | implies(all_20_3, all_20_2) = all_20_1 & is_a_theorem(all_20_1) =
% 85.70/13.92 | all_20_0 & $i(all_20_1) & $i(all_20_2) & ~ r2) | (r2 & ! [v0: $i]
% 85.70/13.92 | : ! [v1: $i] : ! [v2: $i] : ( ~ (or(v0, v1) = v2) | ~ $i(v1) | ~
% 85.70/13.92 | $i(v0) | ? [v3: $i] : (implies(v1, v2) = v3 & is_a_theorem(v3) =
% 85.70/13.92 | 0 & $i(v3))))
% 85.70/13.92 |
% 85.70/13.92 | DELTA: instantiating (modus_ponens) with fresh symbols all_30_0, all_30_1,
% 85.70/13.92 | all_30_2, all_30_3, all_30_4, all_30_5 gives:
% 85.70/13.92 | (12) $i(all_30_4) & $i(all_30_5) & ((all_30_1 = 0 & all_30_3 = 0 & ~
% 85.70/13.92 | (all_30_0 = 0) & implies(all_30_5, all_30_4) = all_30_2 &
% 85.70/13.92 | is_a_theorem(all_30_2) = 0 & is_a_theorem(all_30_4) = all_30_0 &
% 85.70/13.92 | is_a_theorem(all_30_5) = 0 & $i(all_30_2) & ~ modus_ponens) |
% 85.70/13.92 | (modus_ponens & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 85.70/13.92 | (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: int] :
% 85.70/13.92 | ? [v4: int] : ? [v5: int] : ((v5 = 0 & is_a_theorem(v1) = 0) |
% 85.70/13.92 | ( ~ (v4 = 0) & is_a_theorem(v2) = v4) | ( ~ (v3 = 0) &
% 85.70/13.92 | is_a_theorem(v0) = v3)))))
% 85.70/13.92 |
% 85.70/13.92 | ALPHA: (12) implies:
% 85.70/13.92 | (13) (all_30_1 = 0 & all_30_3 = 0 & ~ (all_30_0 = 0) & implies(all_30_5,
% 85.70/13.92 | all_30_4) = all_30_2 & is_a_theorem(all_30_2) = 0 &
% 85.70/13.92 | is_a_theorem(all_30_4) = all_30_0 & is_a_theorem(all_30_5) = 0 &
% 85.70/13.92 | $i(all_30_2) & ~ modus_ponens) | (modus_ponens & ! [v0: $i] : !
% 85.70/13.92 | [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~
% 85.70/13.92 | $i(v0) | ? [v3: int] : ? [v4: int] : ? [v5: int] : ((v5 = 0 &
% 85.70/13.92 | is_a_theorem(v1) = 0) | ( ~ (v4 = 0) & is_a_theorem(v2) = v4)
% 85.70/13.92 | | ( ~ (v3 = 0) & is_a_theorem(v0) = v3))))
% 85.70/13.92 |
% 85.70/13.92 | DELTA: instantiating (r3) with fresh symbols all_45_0, all_45_1, all_45_2,
% 85.70/13.92 | all_45_3, all_45_4, all_45_5 gives:
% 85.70/13.93 | (14) $i(all_45_4) & $i(all_45_5) & (( ~ (all_45_0 = 0) & or(all_45_4,
% 85.70/13.93 | all_45_5) = all_45_2 & or(all_45_5, all_45_4) = all_45_3 &
% 85.70/13.93 | implies(all_45_3, all_45_2) = all_45_1 & is_a_theorem(all_45_1) =
% 85.70/13.93 | all_45_0 & $i(all_45_1) & $i(all_45_2) & $i(all_45_3) & ~ r3) |
% 85.70/13.93 | (r3 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (or(v1, v0) =
% 85.70/13.93 | v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] :
% 85.70/13.93 | (or(v0, v1) = v3 & implies(v3, v2) = v4 & is_a_theorem(v4) = 0 &
% 85.70/13.93 | $i(v4) & $i(v3))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] :
% 85.70/13.93 | ( ~ (or(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ?
% 85.70/13.93 | [v4: $i] : (or(v1, v0) = v3 & implies(v2, v3) = v4 &
% 85.70/13.93 | is_a_theorem(v4) = 0 & $i(v4) & $i(v3)))))
% 85.70/13.93 |
% 85.70/13.93 | ALPHA: (14) implies:
% 85.70/13.93 | (15) ( ~ (all_45_0 = 0) & or(all_45_4, all_45_5) = all_45_2 & or(all_45_5,
% 85.70/13.93 | all_45_4) = all_45_3 & implies(all_45_3, all_45_2) = all_45_1 &
% 85.70/13.93 | is_a_theorem(all_45_1) = all_45_0 & $i(all_45_1) & $i(all_45_2) &
% 85.70/13.93 | $i(all_45_3) & ~ r3) | (r3 & ! [v0: $i] : ! [v1: $i] : ! [v2:
% 85.70/13.93 | $i] : ( ~ (or(v1, v0) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i]
% 85.70/13.93 | : ? [v4: $i] : (or(v0, v1) = v3 & implies(v3, v2) = v4 &
% 85.70/13.93 | is_a_theorem(v4) = 0 & $i(v4) & $i(v3))) & ! [v0: $i] : ! [v1:
% 85.70/13.93 | $i] : ! [v2: $i] : ( ~ (or(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0)
% 85.70/13.93 | | ? [v3: $i] : ? [v4: $i] : (or(v1, v0) = v3 & implies(v2, v3) =
% 85.70/13.93 | v4 & is_a_theorem(v4) = 0 & $i(v4) & $i(v3))))
% 85.70/13.93 |
% 85.70/13.93 | DELTA: instantiating (r4) with fresh symbols all_53_0, all_53_1, all_53_2,
% 85.70/13.93 | all_53_3, all_53_4, all_53_5, all_53_6, all_53_7, all_53_8 gives:
% 85.70/13.93 | (16) $i(all_53_6) & $i(all_53_7) & $i(all_53_8) & (( ~ (all_53_0 = 0) &
% 85.70/13.93 | or(all_53_7, all_53_3) = all_53_2 & or(all_53_7, all_53_6) =
% 85.70/13.93 | all_53_5 & or(all_53_8, all_53_5) = all_53_4 & or(all_53_8,
% 85.70/13.93 | all_53_6) = all_53_3 & implies(all_53_4, all_53_2) = all_53_1 &
% 85.70/13.93 | is_a_theorem(all_53_1) = all_53_0 & $i(all_53_1) & $i(all_53_2) &
% 85.70/13.93 | $i(all_53_3) & $i(all_53_4) & $i(all_53_5) & ~ r4) | (r4 & !
% 85.70/13.93 | [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] :
% 85.70/13.93 | ( ~ (or(v1, v3) = v4) | ~ (or(v0, v2) = v3) | ~ $i(v2) | ~
% 85.70/13.93 | $i(v1) | ~ $i(v0) | ? [v5: $i] : ? [v6: $i] : ? [v7: $i] :
% 85.70/13.93 | (or(v1, v2) = v5 & or(v0, v5) = v6 & implies(v6, v4) = v7 &
% 85.70/13.93 | is_a_theorem(v7) = 0 & $i(v7) & $i(v6) & $i(v5))) & ! [v0:
% 85.70/13.93 | $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : (
% 85.70/13.93 | ~ (or(v1, v2) = v3) | ~ (or(v0, v3) = v4) | ~ $i(v2) | ~
% 85.70/13.93 | $i(v1) | ~ $i(v0) | ? [v5: $i] : ? [v6: $i] : ? [v7: $i] :
% 85.70/13.93 | (or(v1, v5) = v6 & or(v0, v2) = v5 & implies(v4, v6) = v7 &
% 85.70/13.93 | is_a_theorem(v7) = 0 & $i(v7) & $i(v6) & $i(v5)))))
% 85.70/13.93 |
% 85.70/13.93 | ALPHA: (16) implies:
% 85.70/13.94 | (17) ( ~ (all_53_0 = 0) & or(all_53_7, all_53_3) = all_53_2 & or(all_53_7,
% 85.70/13.94 | all_53_6) = all_53_5 & or(all_53_8, all_53_5) = all_53_4 &
% 85.70/13.94 | or(all_53_8, all_53_6) = all_53_3 & implies(all_53_4, all_53_2) =
% 85.70/13.94 | all_53_1 & is_a_theorem(all_53_1) = all_53_0 & $i(all_53_1) &
% 85.70/13.94 | $i(all_53_2) & $i(all_53_3) & $i(all_53_4) & $i(all_53_5) & ~ r4) |
% 85.70/13.94 | (r4 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4:
% 85.70/13.94 | $i] : ( ~ (or(v1, v3) = v4) | ~ (or(v0, v2) = v3) | ~ $i(v2) |
% 85.70/13.94 | ~ $i(v1) | ~ $i(v0) | ? [v5: $i] : ? [v6: $i] : ? [v7: $i] :
% 85.70/13.94 | (or(v1, v2) = v5 & or(v0, v5) = v6 & implies(v6, v4) = v7 &
% 85.70/13.94 | is_a_theorem(v7) = 0 & $i(v7) & $i(v6) & $i(v5))) & ! [v0: $i]
% 85.70/13.94 | : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( ~
% 85.70/13.94 | (or(v1, v2) = v3) | ~ (or(v0, v3) = v4) | ~ $i(v2) | ~ $i(v1) |
% 85.70/13.94 | ~ $i(v0) | ? [v5: $i] : ? [v6: $i] : ? [v7: $i] : (or(v1, v5)
% 85.70/13.94 | = v6 & or(v0, v2) = v5 & implies(v4, v6) = v7 & is_a_theorem(v7)
% 85.70/13.94 | = 0 & $i(v7) & $i(v6) & $i(v5))))
% 85.70/13.94 |
% 85.70/13.94 | BETA: splitting (7) gives:
% 85.70/13.94 |
% 85.70/13.94 | Case 1:
% 85.70/13.94 | |
% 85.70/13.94 | | (18) ~ (all_4_0 = 0) & and(all_4_3, all_4_3) = all_4_2 &
% 85.70/13.94 | | implies(all_4_3, all_4_2) = all_4_1 & is_a_theorem(all_4_1) =
% 85.70/13.94 | | all_4_0 & $i(all_4_1) & $i(all_4_2) & ~ kn1
% 85.70/13.94 | |
% 85.70/13.94 | | ALPHA: (18) implies:
% 85.70/13.94 | | (19) ~ (all_4_0 = 0)
% 85.70/13.94 | | (20) $i(all_4_2)
% 85.70/13.94 | | (21) is_a_theorem(all_4_1) = all_4_0
% 85.70/13.94 | | (22) implies(all_4_3, all_4_2) = all_4_1
% 85.70/13.94 | | (23) and(all_4_3, all_4_3) = all_4_2
% 85.70/13.94 | |
% 85.70/13.94 | | BETA: splitting (op_or) gives:
% 85.70/13.94 | |
% 85.70/13.94 | | Case 1:
% 85.70/13.94 | | |
% 85.70/13.94 | | | (24) ~ op_or
% 85.70/13.94 | | |
% 85.70/13.94 | | | PRED_UNIFY: (24), (rosser_op_or) imply:
% 85.70/13.94 | | | (25) $false
% 85.70/13.94 | | |
% 85.70/13.94 | | | CLOSE: (25) is inconsistent.
% 85.70/13.94 | | |
% 85.70/13.94 | | Case 2:
% 85.70/13.94 | | |
% 85.70/13.94 | | | (26) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4:
% 85.70/13.94 | | | $i] : ( ~ (and(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) =
% 85.70/13.94 | | | v2) | ~ $i(v1) | ~ $i(v0) | ? [v5: $i] : (or(v0, v1) = v5 &
% 85.70/13.94 | | | not(v4) = v5 & $i(v5))) & ! [v0: $i] : ! [v1: $i] : ! [v2:
% 85.70/13.94 | | | $i] : ( ~ (or(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3:
% 85.70/13.94 | | | $i] : ? [v4: $i] : ? [v5: $i] : (and(v3, v4) = v5 & not(v5)
% 85.70/13.94 | | | = v2 & not(v1) = v4 & not(v0) = v3 & $i(v5) & $i(v4) & $i(v3)
% 85.70/13.94 | | | & $i(v2)))
% 85.70/13.94 | | |
% 85.70/13.94 | | | ALPHA: (26) implies:
% 85.70/13.95 | | | (27) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (or(v0, v1) = v2) |
% 85.70/13.95 | | | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : ? [v5: $i] :
% 85.70/13.95 | | | (and(v3, v4) = v5 & not(v5) = v2 & not(v1) = v4 & not(v0) = v3 &
% 85.70/13.95 | | | $i(v5) & $i(v4) & $i(v3) & $i(v2)))
% 85.70/13.95 | | |
% 85.70/13.95 | | | BETA: splitting (op_implies_or) gives:
% 85.70/13.95 | | |
% 85.70/13.95 | | | Case 1:
% 85.70/13.95 | | | |
% 85.70/13.95 | | | | (28) ~ op_implies_or
% 85.70/13.95 | | | |
% 85.70/13.95 | | | | PRED_UNIFY: (28), (principia_op_implies_or) imply:
% 85.70/13.95 | | | | (29) $false
% 85.70/13.95 | | | |
% 85.70/13.95 | | | | CLOSE: (29) is inconsistent.
% 85.70/13.95 | | | |
% 85.70/13.95 | | | Case 2:
% 85.70/13.95 | | | |
% 85.70/13.95 | | | | (30) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~
% 85.70/13.95 | | | | (or(v2, v1) = v3) | ~ (not(v0) = v2) | ~ $i(v1) | ~ $i(v0)
% 85.70/13.95 | | | | | (implies(v0, v1) = v3 & $i(v3))) & ! [v0: $i] : ! [v1: $i]
% 85.70/13.95 | | | | : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~
% 85.70/13.95 | | | | $i(v0) | ? [v3: $i] : (or(v3, v1) = v2 & not(v0) = v3 &
% 85.70/13.95 | | | | $i(v3) & $i(v2)))
% 85.70/13.95 | | | |
% 85.70/13.95 | | | | ALPHA: (30) implies:
% 85.70/13.95 | | | | (31) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) =
% 85.70/13.95 | | | | v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : (or(v3, v1) = v2
% 85.70/13.95 | | | | & not(v0) = v3 & $i(v3) & $i(v2)))
% 85.70/13.95 | | | | (32) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~
% 85.70/13.95 | | | | (or(v2, v1) = v3) | ~ (not(v0) = v2) | ~ $i(v1) | ~ $i(v0)
% 85.70/13.95 | | | | | (implies(v0, v1) = v3 & $i(v3)))
% 85.70/13.95 | | | |
% 85.70/13.95 | | | | BETA: splitting (op_equiv) gives:
% 85.70/13.95 | | | |
% 85.70/13.95 | | | | Case 1:
% 85.70/13.95 | | | | |
% 85.70/13.95 | | | | | (33) ~ op_equiv
% 85.70/13.95 | | | | |
% 85.70/13.95 | | | | | PRED_UNIFY: (33), (rosser_op_equiv) imply:
% 85.70/13.95 | | | | | (34) $false
% 85.70/13.95 | | | | |
% 85.70/13.95 | | | | | CLOSE: (34) is inconsistent.
% 85.70/13.95 | | | | |
% 85.70/13.95 | | | | Case 2:
% 85.70/13.95 | | | | |
% 85.70/13.95 | | | | | (35) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (equiv(v0, v1) =
% 85.70/13.95 | | | | | v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] :
% 85.70/13.95 | | | | | (and(v3, v4) = v2 & implies(v1, v0) = v4 & implies(v0, v1) =
% 85.70/13.95 | | | | | v3 & $i(v4) & $i(v3) & $i(v2))) & ! [v0: $i] : ! [v1:
% 85.70/13.95 | | | | | $i] : ! [v2: $i] : ( ~ (implies(v1, v0) = v2) | ~ $i(v1) |
% 85.70/13.95 | | | | | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : (and(v4, v2) = v3 &
% 85.70/13.95 | | | | | equiv(v0, v1) = v3 & implies(v0, v1) = v4 & $i(v4) &
% 85.70/13.95 | | | | | $i(v3))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 85.70/13.95 | | | | | (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i]
% 85.70/13.95 | | | | | : ? [v4: $i] : (and(v2, v4) = v3 & equiv(v0, v1) = v3 &
% 85.70/13.95 | | | | | implies(v1, v0) = v4 & $i(v4) & $i(v3)))
% 85.70/13.95 | | | | |
% 85.70/13.95 | | | | | ALPHA: (35) implies:
% 86.01/13.96 | | | | | (36) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1)
% 86.01/13.96 | | | | | = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i]
% 86.01/13.96 | | | | | : (and(v2, v4) = v3 & equiv(v0, v1) = v3 & implies(v1, v0) =
% 86.01/13.96 | | | | | v4 & $i(v4) & $i(v3)))
% 86.01/13.96 | | | | | (37) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v1, v0)
% 86.01/13.96 | | | | | = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i]
% 86.01/13.96 | | | | | : (and(v4, v2) = v3 & equiv(v0, v1) = v3 & implies(v0, v1) =
% 86.01/13.96 | | | | | v4 & $i(v4) & $i(v3)))
% 86.01/13.96 | | | | |
% 86.01/13.96 | | | | | BETA: splitting (op_implies_and) gives:
% 86.01/13.96 | | | | |
% 86.01/13.96 | | | | | Case 1:
% 86.01/13.96 | | | | | |
% 86.01/13.96 | | | | | | (38) ~ op_implies_and
% 86.01/13.96 | | | | | |
% 86.01/13.96 | | | | | | PRED_UNIFY: (38), (rosser_op_implies_and) imply:
% 86.01/13.96 | | | | | | (39) $false
% 86.01/13.96 | | | | | |
% 86.01/13.96 | | | | | | CLOSE: (39) is inconsistent.
% 86.01/13.96 | | | | | |
% 86.01/13.96 | | | | | Case 2:
% 86.01/13.96 | | | | | |
% 86.01/13.96 | | | | | | (40) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~
% 86.01/13.96 | | | | | | (and(v0, v2) = v3) | ~ (not(v1) = v2) | ~ $i(v1) | ~
% 86.01/13.96 | | | | | | $i(v0) | ? [v4: $i] : (not(v3) = v4 & implies(v0, v1) =
% 86.01/13.96 | | | | | | v4 & $i(v4))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i]
% 86.01/13.96 | | | | | | : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ?
% 86.01/13.96 | | | | | | [v3: $i] : ? [v4: $i] : (and(v0, v3) = v4 & not(v4) = v2
% 86.01/13.96 | | | | | | & not(v1) = v3 & $i(v4) & $i(v3) & $i(v2)))
% 86.01/13.96 | | | | | |
% 86.01/13.96 | | | | | | ALPHA: (40) implies:
% 86.01/13.96 | | | | | | (41) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0,
% 86.01/13.96 | | | | | | v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ?
% 86.01/13.96 | | | | | | [v4: $i] : (and(v0, v3) = v4 & not(v4) = v2 & not(v1) = v3
% 86.01/13.96 | | | | | | & $i(v4) & $i(v3) & $i(v2)))
% 86.01/13.96 | | | | | |
% 86.01/13.96 | | | | | | BETA: splitting (9) gives:
% 86.01/13.96 | | | | | |
% 86.01/13.96 | | | | | | Case 1:
% 86.01/13.96 | | | | | | |
% 86.01/13.96 | | | | | | | (42) ~ (all_6_0 = 0) & or(all_6_3, all_6_3) = all_6_2 &
% 86.01/13.96 | | | | | | | implies(all_6_2, all_6_3) = all_6_1 &
% 86.01/13.96 | | | | | | | is_a_theorem(all_6_1) = all_6_0 & $i(all_6_1) &
% 86.01/13.96 | | | | | | | $i(all_6_2) & ~ r1
% 86.01/13.96 | | | | | | |
% 86.01/13.96 | | | | | | | ALPHA: (42) implies:
% 86.01/13.96 | | | | | | | (43) ~ r1
% 86.01/13.96 | | | | | | |
% 86.01/13.96 | | | | | | | PRED_UNIFY: (43), (principia_r1) imply:
% 86.01/13.96 | | | | | | | (44) $false
% 86.01/13.96 | | | | | | |
% 86.01/13.96 | | | | | | | CLOSE: (44) is inconsistent.
% 86.01/13.96 | | | | | | |
% 86.01/13.96 | | | | | | Case 2:
% 86.01/13.96 | | | | | | |
% 86.01/13.96 | | | | | | | (45) r1 & ! [v0: $i] : ! [v1: $i] : ( ~ (or(v0, v0) = v1) |
% 86.01/13.96 | | | | | | | ~ $i(v0) | ? [v2: $i] : (implies(v1, v0) = v2 &
% 86.01/13.96 | | | | | | | is_a_theorem(v2) = 0 & $i(v2)))
% 86.01/13.96 | | | | | | |
% 86.01/13.96 | | | | | | | ALPHA: (45) implies:
% 86.01/13.96 | | | | | | | (46) ! [v0: $i] : ! [v1: $i] : ( ~ (or(v0, v0) = v1) | ~
% 86.01/13.96 | | | | | | | $i(v0) | ? [v2: $i] : (implies(v1, v0) = v2 &
% 86.01/13.96 | | | | | | | is_a_theorem(v2) = 0 & $i(v2)))
% 86.01/13.96 | | | | | | |
% 86.01/13.96 | | | | | | | BETA: splitting (11) gives:
% 86.01/13.96 | | | | | | |
% 86.01/13.96 | | | | | | | Case 1:
% 86.01/13.96 | | | | | | | |
% 86.01/13.96 | | | | | | | | (47) ~ (all_20_0 = 0) & or(all_20_4, all_20_3) = all_20_2 &
% 86.01/13.96 | | | | | | | | implies(all_20_3, all_20_2) = all_20_1 &
% 86.01/13.96 | | | | | | | | is_a_theorem(all_20_1) = all_20_0 & $i(all_20_1) &
% 86.01/13.96 | | | | | | | | $i(all_20_2) & ~ r2
% 86.01/13.96 | | | | | | | |
% 86.01/13.96 | | | | | | | | ALPHA: (47) implies:
% 86.01/13.97 | | | | | | | | (48) ~ r2
% 86.01/13.97 | | | | | | | |
% 86.01/13.97 | | | | | | | | PRED_UNIFY: (48), (principia_r2) imply:
% 86.01/13.97 | | | | | | | | (49) $false
% 86.01/13.97 | | | | | | | |
% 86.01/13.97 | | | | | | | | CLOSE: (49) is inconsistent.
% 86.01/13.97 | | | | | | | |
% 86.01/13.97 | | | | | | | Case 2:
% 86.01/13.97 | | | | | | | |
% 86.01/13.97 | | | | | | | | (50) r2 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 86.01/13.97 | | | | | | | | (or(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3:
% 86.01/13.97 | | | | | | | | $i] : (implies(v1, v2) = v3 & is_a_theorem(v3) = 0 &
% 86.01/13.97 | | | | | | | | $i(v3)))
% 86.01/13.97 | | | | | | | |
% 86.01/13.97 | | | | | | | | ALPHA: (50) implies:
% 86.01/13.97 | | | | | | | | (51) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (or(v0,
% 86.01/13.97 | | | | | | | | v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] :
% 86.01/13.97 | | | | | | | | (implies(v1, v2) = v3 & is_a_theorem(v3) = 0 &
% 86.01/13.97 | | | | | | | | $i(v3)))
% 86.01/13.97 | | | | | | | |
% 86.01/13.97 | | | | | | | | BETA: splitting (13) gives:
% 86.01/13.97 | | | | | | | |
% 86.01/13.97 | | | | | | | | Case 1:
% 86.01/13.97 | | | | | | | | |
% 86.01/13.97 | | | | | | | | | (52) all_30_1 = 0 & all_30_3 = 0 & ~ (all_30_0 = 0) &
% 86.01/13.97 | | | | | | | | | implies(all_30_5, all_30_4) = all_30_2 &
% 86.01/13.97 | | | | | | | | | is_a_theorem(all_30_2) = 0 & is_a_theorem(all_30_4) =
% 86.01/13.97 | | | | | | | | | all_30_0 & is_a_theorem(all_30_5) = 0 & $i(all_30_2) &
% 86.01/13.97 | | | | | | | | | ~ modus_ponens
% 86.01/13.97 | | | | | | | | |
% 86.01/13.97 | | | | | | | | | ALPHA: (52) implies:
% 86.01/13.97 | | | | | | | | | (53) ~ modus_ponens
% 86.01/13.97 | | | | | | | | |
% 86.01/13.97 | | | | | | | | | PRED_UNIFY: (53), (principia_modus_ponens) imply:
% 86.01/13.97 | | | | | | | | | (54) $false
% 86.01/13.97 | | | | | | | | |
% 86.01/13.97 | | | | | | | | | CLOSE: (54) is inconsistent.
% 86.01/13.97 | | | | | | | | |
% 86.01/13.97 | | | | | | | | Case 2:
% 86.01/13.97 | | | | | | | | |
% 86.01/13.97 | | | | | | | | | (55) modus_ponens & ! [v0: $i] : ! [v1: $i] : ! [v2: $i]
% 86.01/13.97 | | | | | | | | | : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) |
% 86.01/13.97 | | | | | | | | | ? [v3: int] : ? [v4: int] : ? [v5: int] : ((v5 =
% 86.01/13.97 | | | | | | | | | 0 & is_a_theorem(v1) = 0) | ( ~ (v4 = 0) &
% 86.01/13.97 | | | | | | | | | is_a_theorem(v2) = v4) | ( ~ (v3 = 0) &
% 86.01/13.97 | | | | | | | | | is_a_theorem(v0) = v3)))
% 86.01/13.97 | | | | | | | | |
% 86.01/13.97 | | | | | | | | | ALPHA: (55) implies:
% 86.01/13.97 | | | | | | | | | (56) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 86.01/13.97 | | | | | | | | | (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ?
% 86.01/13.97 | | | | | | | | | [v3: int] : ? [v4: int] : ? [v5: int] : ((v5 = 0 &
% 86.01/13.97 | | | | | | | | | is_a_theorem(v1) = 0) | ( ~ (v4 = 0) &
% 86.01/13.97 | | | | | | | | | is_a_theorem(v2) = v4) | ( ~ (v3 = 0) &
% 86.01/13.97 | | | | | | | | | is_a_theorem(v0) = v3)))
% 86.01/13.97 | | | | | | | | |
% 86.01/13.97 | | | | | | | | | BETA: splitting (op_and) gives:
% 86.01/13.97 | | | | | | | | |
% 86.01/13.97 | | | | | | | | | Case 1:
% 86.01/13.97 | | | | | | | | | |
% 86.01/13.97 | | | | | | | | | | (57) ~ op_and
% 86.01/13.97 | | | | | | | | | |
% 86.01/13.97 | | | | | | | | | | PRED_UNIFY: (57), (principia_op_and) imply:
% 86.01/13.97 | | | | | | | | | | (58) $false
% 86.01/13.97 | | | | | | | | | |
% 86.01/13.97 | | | | | | | | | | CLOSE: (58) is inconsistent.
% 86.01/13.97 | | | | | | | | | |
% 86.01/13.97 | | | | | | | | | Case 2:
% 86.01/13.97 | | | | | | | | | |
% 86.01/13.97 | | | | | | | | | | (59) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3:
% 86.01/13.97 | | | | | | | | | | $i] : ! [v4: $i] : ( ~ (or(v2, v3) = v4) | ~
% 86.01/13.97 | | | | | | | | | | (not(v1) = v3) | ~ (not(v0) = v2) | ~ $i(v1) |
% 86.01/13.97 | | | | | | | | | | ~ $i(v0) | ? [v5: $i] : (and(v0, v1) = v5 &
% 86.01/13.97 | | | | | | | | | | not(v4) = v5 & $i(v5))) & ! [v0: $i] : ! [v1:
% 86.01/13.97 | | | | | | | | | | $i] : ! [v2: $i] : ( ~ (and(v0, v1) = v2) | ~
% 86.01/13.97 | | | | | | | | | | $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] :
% 86.01/13.97 | | | | | | | | | | ? [v5: $i] : (or(v3, v4) = v5 & not(v5) = v2 &
% 86.01/13.97 | | | | | | | | | | not(v1) = v4 & not(v0) = v3 & $i(v5) & $i(v4) &
% 86.01/13.97 | | | | | | | | | | $i(v3) & $i(v2)))
% 86.01/13.97 | | | | | | | | | |
% 86.01/13.97 | | | | | | | | | | ALPHA: (59) implies:
% 86.01/13.97 | | | | | | | | | | (60) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 86.01/13.97 | | | | | | | | | | (and(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ?
% 86.01/13.97 | | | | | | | | | | [v3: $i] : ? [v4: $i] : ? [v5: $i] : (or(v3, v4)
% 86.01/13.97 | | | | | | | | | | = v5 & not(v5) = v2 & not(v1) = v4 & not(v0) =
% 86.01/13.97 | | | | | | | | | | v3 & $i(v5) & $i(v4) & $i(v3) & $i(v2)))
% 86.01/13.97 | | | | | | | | | |
% 86.01/13.97 | | | | | | | | | | BETA: splitting (15) gives:
% 86.01/13.97 | | | | | | | | | |
% 86.01/13.97 | | | | | | | | | | Case 1:
% 86.01/13.97 | | | | | | | | | | |
% 86.01/13.97 | | | | | | | | | | | (61) ~ (all_45_0 = 0) & or(all_45_4, all_45_5) =
% 86.01/13.97 | | | | | | | | | | | all_45_2 & or(all_45_5, all_45_4) = all_45_3 &
% 86.01/13.97 | | | | | | | | | | | implies(all_45_3, all_45_2) = all_45_1 &
% 86.01/13.97 | | | | | | | | | | | is_a_theorem(all_45_1) = all_45_0 & $i(all_45_1) &
% 86.01/13.97 | | | | | | | | | | | $i(all_45_2) & $i(all_45_3) & ~ r3
% 86.01/13.97 | | | | | | | | | | |
% 86.01/13.97 | | | | | | | | | | | ALPHA: (61) implies:
% 86.01/13.97 | | | | | | | | | | | (62) ~ r3
% 86.01/13.97 | | | | | | | | | | |
% 86.01/13.97 | | | | | | | | | | | PRED_UNIFY: (62), (principia_r3) imply:
% 86.01/13.97 | | | | | | | | | | | (63) $false
% 86.01/13.97 | | | | | | | | | | |
% 86.01/13.97 | | | | | | | | | | | CLOSE: (63) is inconsistent.
% 86.01/13.97 | | | | | | | | | | |
% 86.01/13.97 | | | | | | | | | | Case 2:
% 86.01/13.97 | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | (64) r3 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 86.01/13.98 | | | | | | | | | | | (or(v1, v0) = v2) | ~ $i(v1) | ~ $i(v0) | ?
% 86.01/13.98 | | | | | | | | | | | [v3: $i] : ? [v4: $i] : (or(v0, v1) = v3 &
% 86.01/13.98 | | | | | | | | | | | implies(v3, v2) = v4 & is_a_theorem(v4) = 0 &
% 86.01/13.98 | | | | | | | | | | | $i(v4) & $i(v3))) & ! [v0: $i] : ! [v1: $i]
% 86.01/13.98 | | | | | | | | | | | : ! [v2: $i] : ( ~ (or(v0, v1) = v2) | ~ $i(v1)
% 86.01/13.98 | | | | | | | | | | | | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] :
% 86.01/13.98 | | | | | | | | | | | (or(v1, v0) = v3 & implies(v2, v3) = v4 &
% 86.01/13.98 | | | | | | | | | | | is_a_theorem(v4) = 0 & $i(v4) & $i(v3)))
% 86.01/13.98 | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | ALPHA: (64) implies:
% 86.01/13.98 | | | | | | | | | | | (65) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 86.01/13.98 | | | | | | | | | | | (or(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ?
% 86.01/13.98 | | | | | | | | | | | [v3: $i] : ? [v4: $i] : (or(v1, v0) = v3 &
% 86.01/13.98 | | | | | | | | | | | implies(v2, v3) = v4 & is_a_theorem(v4) = 0 &
% 86.01/13.98 | | | | | | | | | | | $i(v4) & $i(v3)))
% 86.01/13.98 | | | | | | | | | | | (66) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 86.01/13.98 | | | | | | | | | | | (or(v1, v0) = v2) | ~ $i(v1) | ~ $i(v0) | ?
% 86.01/13.98 | | | | | | | | | | | [v3: $i] : ? [v4: $i] : (or(v0, v1) = v3 &
% 86.01/13.98 | | | | | | | | | | | implies(v3, v2) = v4 & is_a_theorem(v4) = 0 &
% 86.01/13.98 | | | | | | | | | | | $i(v4) & $i(v3)))
% 86.01/13.98 | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | BETA: splitting (17) gives:
% 86.01/13.98 | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | Case 1:
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | | (67) ~ (all_53_0 = 0) & or(all_53_7, all_53_3) =
% 86.01/13.98 | | | | | | | | | | | | all_53_2 & or(all_53_7, all_53_6) = all_53_5 &
% 86.01/13.98 | | | | | | | | | | | | or(all_53_8, all_53_5) = all_53_4 & or(all_53_8,
% 86.01/13.98 | | | | | | | | | | | | all_53_6) = all_53_3 & implies(all_53_4,
% 86.01/13.98 | | | | | | | | | | | | all_53_2) = all_53_1 & is_a_theorem(all_53_1) =
% 86.01/13.98 | | | | | | | | | | | | all_53_0 & $i(all_53_1) & $i(all_53_2) &
% 86.01/13.98 | | | | | | | | | | | | $i(all_53_3) & $i(all_53_4) & $i(all_53_5) & ~ r4
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | | ALPHA: (67) implies:
% 86.01/13.98 | | | | | | | | | | | | (68) ~ r4
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | | PRED_UNIFY: (68), (principia_r4) imply:
% 86.01/13.98 | | | | | | | | | | | | (69) $false
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | | CLOSE: (69) is inconsistent.
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | Case 2:
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | | GROUND_INST: instantiating (37) with all_4_2, all_4_3, all_4_1,
% 86.01/13.98 | | | | | | | | | | | | simplifying with (6), (20), (22) gives:
% 86.01/13.98 | | | | | | | | | | | | (70) ? [v0: $i] : ? [v1: $i] : (and(v1, all_4_1) = v0
% 86.01/13.98 | | | | | | | | | | | | & equiv(all_4_2, all_4_3) = v0 &
% 86.01/13.98 | | | | | | | | | | | | implies(all_4_2, all_4_3) = v1 & $i(v1) &
% 86.01/13.98 | | | | | | | | | | | | $i(v0))
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | | GROUND_INST: instantiating (36) with all_4_3, all_4_2, all_4_1,
% 86.01/13.98 | | | | | | | | | | | | simplifying with (6), (20), (22) gives:
% 86.01/13.98 | | | | | | | | | | | | (71) ? [v0: $i] : ? [v1: $i] : (and(all_4_1, v1) = v0
% 86.01/13.98 | | | | | | | | | | | | & equiv(all_4_3, all_4_2) = v0 &
% 86.01/13.98 | | | | | | | | | | | | implies(all_4_2, all_4_3) = v1 & $i(v1) &
% 86.01/13.98 | | | | | | | | | | | | $i(v0))
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | | GROUND_INST: instantiating (41) with all_4_3, all_4_2, all_4_1,
% 86.01/13.98 | | | | | | | | | | | | simplifying with (6), (20), (22) gives:
% 86.01/13.98 | | | | | | | | | | | | (72) ? [v0: $i] : ? [v1: $i] : (and(all_4_3, v0) = v1
% 86.01/13.98 | | | | | | | | | | | | & not(v1) = all_4_1 & not(all_4_2) = v0 & $i(v1)
% 86.01/13.98 | | | | | | | | | | | | & $i(v0) & $i(all_4_1))
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | | GROUND_INST: instantiating (31) with all_4_3, all_4_2, all_4_1,
% 86.01/13.98 | | | | | | | | | | | | simplifying with (6), (20), (22) gives:
% 86.01/13.98 | | | | | | | | | | | | (73) ? [v0: $i] : (or(v0, all_4_2) = all_4_1 &
% 86.01/13.98 | | | | | | | | | | | | not(all_4_3) = v0 & $i(v0) & $i(all_4_1))
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | | GROUND_INST: instantiating (60) with all_4_3, all_4_3, all_4_2,
% 86.01/13.98 | | | | | | | | | | | | simplifying with (6), (23) gives:
% 86.01/13.98 | | | | | | | | | | | | (74) ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : (or(v0,
% 86.01/13.98 | | | | | | | | | | | | v1) = v2 & not(v2) = all_4_2 & not(all_4_3) =
% 86.01/13.98 | | | | | | | | | | | | v1 & not(all_4_3) = v0 & $i(v2) & $i(v1) &
% 86.01/13.98 | | | | | | | | | | | | $i(v0) & $i(all_4_2))
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | | DELTA: instantiating (73) with fresh symbol all_111_0
% 86.01/13.98 | | | | | | | | | | | | gives:
% 86.01/13.98 | | | | | | | | | | | | (75) or(all_111_0, all_4_2) = all_4_1 & not(all_4_3) =
% 86.01/13.98 | | | | | | | | | | | | all_111_0 & $i(all_111_0) & $i(all_4_1)
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | | ALPHA: (75) implies:
% 86.01/13.98 | | | | | | | | | | | | (76) not(all_4_3) = all_111_0
% 86.01/13.98 | | | | | | | | | | | | (77) or(all_111_0, all_4_2) = all_4_1
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.98 | | | | | | | | | | | | DELTA: instantiating (71) with fresh symbols all_113_0,
% 86.01/13.98 | | | | | | | | | | | | all_113_1 gives:
% 86.01/13.98 | | | | | | | | | | | | (78) and(all_4_1, all_113_0) = all_113_1 &
% 86.01/13.98 | | | | | | | | | | | | equiv(all_4_3, all_4_2) = all_113_1 &
% 86.01/13.98 | | | | | | | | | | | | implies(all_4_2, all_4_3) = all_113_0 &
% 86.01/13.98 | | | | | | | | | | | | $i(all_113_0) & $i(all_113_1)
% 86.01/13.98 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | ALPHA: (78) implies:
% 86.01/13.99 | | | | | | | | | | | | (79) implies(all_4_2, all_4_3) = all_113_0
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | DELTA: instantiating (70) with fresh symbols all_115_0,
% 86.01/13.99 | | | | | | | | | | | | all_115_1 gives:
% 86.01/13.99 | | | | | | | | | | | | (80) and(all_115_0, all_4_1) = all_115_1 &
% 86.01/13.99 | | | | | | | | | | | | equiv(all_4_2, all_4_3) = all_115_1 &
% 86.01/13.99 | | | | | | | | | | | | implies(all_4_2, all_4_3) = all_115_0 &
% 86.01/13.99 | | | | | | | | | | | | $i(all_115_0) & $i(all_115_1)
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | ALPHA: (80) implies:
% 86.01/13.99 | | | | | | | | | | | | (81) implies(all_4_2, all_4_3) = all_115_0
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | DELTA: instantiating (72) with fresh symbols all_118_0,
% 86.01/13.99 | | | | | | | | | | | | all_118_1 gives:
% 86.01/13.99 | | | | | | | | | | | | (82) and(all_4_3, all_118_1) = all_118_0 &
% 86.01/13.99 | | | | | | | | | | | | not(all_118_0) = all_4_1 & not(all_4_2) =
% 86.01/13.99 | | | | | | | | | | | | all_118_1 & $i(all_118_0) & $i(all_118_1) &
% 86.01/13.99 | | | | | | | | | | | | $i(all_4_1)
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | ALPHA: (82) implies:
% 86.01/13.99 | | | | | | | | | | | | (83) $i(all_118_1)
% 86.01/13.99 | | | | | | | | | | | | (84) and(all_4_3, all_118_1) = all_118_0
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | DELTA: instantiating (74) with fresh symbols all_120_0,
% 86.01/13.99 | | | | | | | | | | | | all_120_1, all_120_2 gives:
% 86.01/13.99 | | | | | | | | | | | | (85) or(all_120_2, all_120_1) = all_120_0 &
% 86.01/13.99 | | | | | | | | | | | | not(all_120_0) = all_4_2 & not(all_4_3) =
% 86.01/13.99 | | | | | | | | | | | | all_120_1 & not(all_4_3) = all_120_2 &
% 86.01/13.99 | | | | | | | | | | | | $i(all_120_0) & $i(all_120_1) & $i(all_120_2) &
% 86.01/13.99 | | | | | | | | | | | | $i(all_4_2)
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | ALPHA: (85) implies:
% 86.01/13.99 | | | | | | | | | | | | (86) $i(all_120_2)
% 86.01/13.99 | | | | | | | | | | | | (87) not(all_4_3) = all_120_2
% 86.01/13.99 | | | | | | | | | | | | (88) not(all_4_3) = all_120_1
% 86.01/13.99 | | | | | | | | | | | | (89) not(all_120_0) = all_4_2
% 86.01/13.99 | | | | | | | | | | | | (90) or(all_120_2, all_120_1) = all_120_0
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | GROUND_INST: instantiating (3) with all_113_0, all_115_0,
% 86.01/13.99 | | | | | | | | | | | | all_4_3, all_4_2, simplifying with (79), (81)
% 86.01/13.99 | | | | | | | | | | | | gives:
% 86.01/13.99 | | | | | | | | | | | | (91) all_115_0 = all_113_0
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | GROUND_INST: instantiating (2) with all_120_2, all_120_1,
% 86.01/13.99 | | | | | | | | | | | | all_4_3, simplifying with (87), (88) gives:
% 86.01/13.99 | | | | | | | | | | | | (92) all_120_1 = all_120_2
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | GROUND_INST: instantiating (2) with all_111_0, all_120_1,
% 86.01/13.99 | | | | | | | | | | | | all_4_3, simplifying with (76), (88) gives:
% 86.01/13.99 | | | | | | | | | | | | (93) all_120_1 = all_111_0
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | COMBINE_EQS: (92), (93) imply:
% 86.01/13.99 | | | | | | | | | | | | (94) all_120_2 = all_111_0
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | SIMP: (94) implies:
% 86.01/13.99 | | | | | | | | | | | | (95) all_120_2 = all_111_0
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | REDUCE: (90), (93), (95) imply:
% 86.01/13.99 | | | | | | | | | | | | (96) or(all_111_0, all_111_0) = all_120_0
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | REDUCE: (86), (95) imply:
% 86.01/13.99 | | | | | | | | | | | | (97) $i(all_111_0)
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | GROUND_INST: instantiating (41) with all_4_2, all_4_3,
% 86.01/13.99 | | | | | | | | | | | | all_113_0, simplifying with (6), (20), (79) gives:
% 86.01/13.99 | | | | | | | | | | | | (98) ? [v0: $i] : ? [v1: $i] : (and(all_4_2, v0) = v1
% 86.01/13.99 | | | | | | | | | | | | & not(v1) = all_113_0 & not(all_4_3) = v0 &
% 86.01/13.99 | | | | | | | | | | | | $i(v1) & $i(v0) & $i(all_113_0))
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | GROUND_INST: instantiating (60) with all_4_3, all_118_1,
% 86.01/13.99 | | | | | | | | | | | | all_118_0, simplifying with (6), (83), (84) gives:
% 86.01/13.99 | | | | | | | | | | | | (99) ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : (or(v0,
% 86.01/13.99 | | | | | | | | | | | | v1) = v2 & not(v2) = all_118_0 &
% 86.01/13.99 | | | | | | | | | | | | not(all_118_1) = v1 & not(all_4_3) = v0 & $i(v2)
% 86.01/13.99 | | | | | | | | | | | | & $i(v1) & $i(v0) & $i(all_118_0))
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | GROUND_INST: instantiating (66) with all_4_2, all_111_0,
% 86.01/13.99 | | | | | | | | | | | | all_4_1, simplifying with (20), (77), (97) gives:
% 86.01/13.99 | | | | | | | | | | | | (100) ? [v0: $i] : ? [v1: $i] : (or(all_4_2,
% 86.01/13.99 | | | | | | | | | | | | all_111_0) = v0 & implies(v0, all_4_1) = v1 &
% 86.01/13.99 | | | | | | | | | | | | is_a_theorem(v1) = 0 & $i(v1) & $i(v0))
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | GROUND_INST: instantiating (27) with all_111_0, all_4_2,
% 86.01/13.99 | | | | | | | | | | | | all_4_1, simplifying with (20), (77), (97) gives:
% 86.01/13.99 | | | | | | | | | | | | (101) ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : (and(v0,
% 86.01/13.99 | | | | | | | | | | | | v1) = v2 & not(v2) = all_4_1 & not(all_111_0)
% 86.01/13.99 | | | | | | | | | | | | = v0 & not(all_4_2) = v1 & $i(v2) & $i(v1) &
% 86.01/13.99 | | | | | | | | | | | | $i(v0) & $i(all_4_1))
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | GROUND_INST: instantiating (65) with all_111_0, all_4_2,
% 86.01/13.99 | | | | | | | | | | | | all_4_1, simplifying with (20), (77), (97) gives:
% 86.01/13.99 | | | | | | | | | | | | (102) ? [v0: $i] : ? [v1: $i] : (or(all_4_2,
% 86.01/13.99 | | | | | | | | | | | | all_111_0) = v0 & implies(all_4_1, v0) = v1 &
% 86.01/13.99 | | | | | | | | | | | | is_a_theorem(v1) = 0 & $i(v1) & $i(v0))
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | GROUND_INST: instantiating (46) with all_111_0, all_120_0,
% 86.01/13.99 | | | | | | | | | | | | simplifying with (96), (97) gives:
% 86.01/13.99 | | | | | | | | | | | | (103) ? [v0: $i] : (implies(all_120_0, all_111_0) = v0
% 86.01/13.99 | | | | | | | | | | | | & is_a_theorem(v0) = 0 & $i(v0))
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | GROUND_INST: instantiating (66) with all_111_0, all_111_0,
% 86.01/13.99 | | | | | | | | | | | | all_120_0, simplifying with (96), (97) gives:
% 86.01/13.99 | | | | | | | | | | | | (104) ? [v0: $i] : ? [v1: $i] : (or(all_111_0,
% 86.01/13.99 | | | | | | | | | | | | all_111_0) = v0 & implies(v0, all_120_0) = v1
% 86.01/13.99 | | | | | | | | | | | | & is_a_theorem(v1) = 0 & $i(v1) & $i(v0))
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | GROUND_INST: instantiating (65) with all_111_0, all_111_0,
% 86.01/13.99 | | | | | | | | | | | | all_120_0, simplifying with (96), (97) gives:
% 86.01/13.99 | | | | | | | | | | | | (105) ? [v0: $i] : ? [v1: $i] : (or(all_111_0,
% 86.01/13.99 | | | | | | | | | | | | all_111_0) = v0 & implies(all_120_0, v0) = v1
% 86.01/13.99 | | | | | | | | | | | | & is_a_theorem(v1) = 0 & $i(v1) & $i(v0))
% 86.01/13.99 | | | | | | | | | | | |
% 86.01/13.99 | | | | | | | | | | | | GROUND_INST: instantiating (51) with all_111_0, all_111_0,
% 86.01/13.99 | | | | | | | | | | | | all_120_0, simplifying with (96), (97) gives:
% 86.01/14.00 | | | | | | | | | | | | (106) ? [v0: $i] : (implies(all_111_0, all_120_0) = v0
% 86.01/14.00 | | | | | | | | | | | | & is_a_theorem(v0) = 0 & $i(v0))
% 86.01/14.00 | | | | | | | | | | | |
% 86.01/14.00 | | | | | | | | | | | | DELTA: instantiating (106) with fresh symbol all_161_0
% 86.01/14.00 | | | | | | | | | | | | gives:
% 86.01/14.00 | | | | | | | | | | | | (107) implies(all_111_0, all_120_0) = all_161_0 &
% 86.01/14.00 | | | | | | | | | | | | is_a_theorem(all_161_0) = 0 & $i(all_161_0)
% 86.01/14.00 | | | | | | | | | | | |
% 86.01/14.00 | | | | | | | | | | | | ALPHA: (107) implies:
% 86.01/14.00 | | | | | | | | | | | | (108) implies(all_111_0, all_120_0) = all_161_0
% 86.01/14.00 | | | | | | | | | | | |
% 86.01/14.00 | | | | | | | | | | | | DELTA: instantiating (103) with fresh symbol all_167_0
% 86.01/14.00 | | | | | | | | | | | | gives:
% 86.01/14.00 | | | | | | | | | | | | (109) implies(all_120_0, all_111_0) = all_167_0 &
% 86.01/14.00 | | | | | | | | | | | | is_a_theorem(all_167_0) = 0 & $i(all_167_0)
% 86.01/14.00 | | | | | | | | | | | |
% 86.01/14.00 | | | | | | | | | | | | ALPHA: (109) implies:
% 86.01/14.00 | | | | | | | | | | | | (110) is_a_theorem(all_167_0) = 0
% 86.01/14.00 | | | | | | | | | | | | (111) implies(all_120_0, all_111_0) = all_167_0
% 86.01/14.00 | | | | | | | | | | | |
% 86.01/14.00 | | | | | | | | | | | | DELTA: instantiating (105) with fresh symbols all_177_0,
% 86.01/14.00 | | | | | | | | | | | | all_177_1 gives:
% 86.01/14.00 | | | | | | | | | | | | (112) or(all_111_0, all_111_0) = all_177_1 &
% 86.01/14.00 | | | | | | | | | | | | implies(all_120_0, all_177_1) = all_177_0 &
% 86.01/14.00 | | | | | | | | | | | | is_a_theorem(all_177_0) = 0 & $i(all_177_0) &
% 86.01/14.00 | | | | | | | | | | | | $i(all_177_1)
% 86.01/14.00 | | | | | | | | | | | |
% 86.01/14.00 | | | | | | | | | | | | ALPHA: (112) implies:
% 86.01/14.00 | | | | | | | | | | | | (113) $i(all_177_1)
% 86.01/14.00 | | | | | | | | | | | | (114) or(all_111_0, all_111_0) = all_177_1
% 86.01/14.00 | | | | | | | | | | | |
% 86.01/14.00 | | | | | | | | | | | | DELTA: instantiating (102) with fresh symbols all_187_0,
% 86.01/14.00 | | | | | | | | | | | | all_187_1 gives:
% 86.01/14.00 | | | | | | | | | | | | (115) or(all_4_2, all_111_0) = all_187_1 &
% 86.01/14.00 | | | | | | | | | | | | implies(all_4_1, all_187_1) = all_187_0 &
% 86.01/14.00 | | | | | | | | | | | | is_a_theorem(all_187_0) = 0 & $i(all_187_0) &
% 86.01/14.00 | | | | | | | | | | | | $i(all_187_1)
% 86.01/14.00 | | | | | | | | | | | |
% 86.01/14.00 | | | | | | | | | | | | ALPHA: (115) implies:
% 86.01/14.00 | | | | | | | | | | | | (116) or(all_4_2, all_111_0) = all_187_1
% 86.01/14.00 | | | | | | | | | | | |
% 86.01/14.00 | | | | | | | | | | | | DELTA: instantiating (104) with fresh symbols all_189_0,
% 86.01/14.00 | | | | | | | | | | | | all_189_1 gives:
% 86.01/14.00 | | | | | | | | | | | | (117) or(all_111_0, all_111_0) = all_189_1 &
% 86.01/14.00 | | | | | | | | | | | | implies(all_189_1, all_120_0) = all_189_0 &
% 86.01/14.00 | | | | | | | | | | | | is_a_theorem(all_189_0) = 0 & $i(all_189_0) &
% 86.01/14.00 | | | | | | | | | | | | $i(all_189_1)
% 86.01/14.00 | | | | | | | | | | | |
% 86.01/14.00 | | | | | | | | | | | | ALPHA: (117) implies:
% 86.01/14.00 | | | | | | | | | | | | (118) or(all_111_0, all_111_0) = all_189_1
% 86.01/14.00 | | | | | | | | | | | |
% 86.01/14.00 | | | | | | | | | | | | DELTA: instantiating (100) with fresh symbols all_191_0,
% 86.01/14.00 | | | | | | | | | | | | all_191_1 gives:
% 86.01/14.00 | | | | | | | | | | | | (119) or(all_4_2, all_111_0) = all_191_1 &
% 86.01/14.00 | | | | | | | | | | | | implies(all_191_1, all_4_1) = all_191_0 &
% 86.01/14.00 | | | | | | | | | | | | is_a_theorem(all_191_0) = 0 & $i(all_191_0) &
% 86.01/14.00 | | | | | | | | | | | | $i(all_191_1)
% 86.01/14.00 | | | | | | | | | | | |
% 86.01/14.00 | | | | | | | | | | | | ALPHA: (119) implies:
% 86.01/14.00 | | | | | | | | | | | | (120) $i(all_191_1)
% 86.01/14.00 | | | | | | | | | | | | (121) is_a_theorem(all_191_0) = 0
% 86.01/14.00 | | | | | | | | | | | | (122) implies(all_191_1, all_4_1) = all_191_0
% 86.01/14.00 | | | | | | | | | | | | (123) or(all_4_2, all_111_0) = all_191_1
% 86.01/14.00 | | | | | | | | | | | |
% 86.01/14.00 | | | | | | | | | | | | DELTA: instantiating (98) with fresh symbols all_202_0,
% 86.01/14.00 | | | | | | | | | | | | all_202_1 gives:
% 86.01/14.00 | | | | | | | | | | | | (124) and(all_4_2, all_202_1) = all_202_0 &
% 86.01/14.00 | | | | | | | | | | | | not(all_202_0) = all_113_0 & not(all_4_3) =
% 86.01/14.00 | | | | | | | | | | | | all_202_1 & $i(all_202_0) & $i(all_202_1) &
% 86.01/14.00 | | | | | | | | | | | | $i(all_113_0)
% 86.01/14.00 | | | | | | | | | | | |
% 86.01/14.00 | | | | | | | | | | | | ALPHA: (124) implies:
% 86.22/14.00 | | | | | | | | | | | | (125) $i(all_202_1)
% 86.22/14.00 | | | | | | | | | | | | (126) not(all_4_3) = all_202_1
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | DELTA: instantiating (101) with fresh symbols all_211_0,
% 86.22/14.00 | | | | | | | | | | | | all_211_1, all_211_2 gives:
% 86.22/14.00 | | | | | | | | | | | | (127) and(all_211_2, all_211_1) = all_211_0 &
% 86.22/14.00 | | | | | | | | | | | | not(all_211_0) = all_4_1 & not(all_111_0) =
% 86.22/14.00 | | | | | | | | | | | | all_211_2 & not(all_4_2) = all_211_1 &
% 86.22/14.00 | | | | | | | | | | | | $i(all_211_0) & $i(all_211_1) & $i(all_211_2) &
% 86.22/14.00 | | | | | | | | | | | | $i(all_4_1)
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | ALPHA: (127) implies:
% 86.22/14.00 | | | | | | | | | | | | (128) $i(all_4_1)
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | DELTA: instantiating (99) with fresh symbols all_221_0,
% 86.22/14.00 | | | | | | | | | | | | all_221_1, all_221_2 gives:
% 86.22/14.00 | | | | | | | | | | | | (129) or(all_221_2, all_221_1) = all_221_0 &
% 86.22/14.00 | | | | | | | | | | | | not(all_221_0) = all_118_0 & not(all_118_1) =
% 86.22/14.00 | | | | | | | | | | | | all_221_1 & not(all_4_3) = all_221_2 &
% 86.22/14.00 | | | | | | | | | | | | $i(all_221_0) & $i(all_221_1) & $i(all_221_2) &
% 86.22/14.00 | | | | | | | | | | | | $i(all_118_0)
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | ALPHA: (129) implies:
% 86.22/14.00 | | | | | | | | | | | | (130) not(all_4_3) = all_221_2
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | GROUND_INST: instantiating (2) with all_111_0, all_221_2,
% 86.22/14.00 | | | | | | | | | | | | all_4_3, simplifying with (76), (130) gives:
% 86.22/14.00 | | | | | | | | | | | | (131) all_221_2 = all_111_0
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | GROUND_INST: instantiating (2) with all_202_1, all_221_2,
% 86.22/14.00 | | | | | | | | | | | | all_4_3, simplifying with (126), (130) gives:
% 86.22/14.00 | | | | | | | | | | | | (132) all_221_2 = all_202_1
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | GROUND_INST: instantiating (4) with all_187_1, all_191_1,
% 86.22/14.00 | | | | | | | | | | | | all_111_0, all_4_2, simplifying with (116), (123)
% 86.22/14.00 | | | | | | | | | | | | gives:
% 86.22/14.00 | | | | | | | | | | | | (133) all_191_1 = all_187_1
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | GROUND_INST: instantiating (4) with all_120_0, all_189_1,
% 86.22/14.00 | | | | | | | | | | | | all_111_0, all_111_0, simplifying with (96), (118)
% 86.22/14.00 | | | | | | | | | | | | gives:
% 86.22/14.00 | | | | | | | | | | | | (134) all_189_1 = all_120_0
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | GROUND_INST: instantiating (4) with all_177_1, all_189_1,
% 86.22/14.00 | | | | | | | | | | | | all_111_0, all_111_0, simplifying with (114),
% 86.22/14.00 | | | | | | | | | | | | (118) gives:
% 86.22/14.00 | | | | | | | | | | | | (135) all_189_1 = all_177_1
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | COMBINE_EQS: (131), (132) imply:
% 86.22/14.00 | | | | | | | | | | | | (136) all_202_1 = all_111_0
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | COMBINE_EQS: (134), (135) imply:
% 86.22/14.00 | | | | | | | | | | | | (137) all_177_1 = all_120_0
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | SIMP: (137) implies:
% 86.22/14.00 | | | | | | | | | | | | (138) all_177_1 = all_120_0
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | REDUCE: (122), (133) imply:
% 86.22/14.00 | | | | | | | | | | | | (139) implies(all_187_1, all_4_1) = all_191_0
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | REDUCE: (120), (133) imply:
% 86.22/14.00 | | | | | | | | | | | | (140) $i(all_187_1)
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | REDUCE: (113), (138) imply:
% 86.22/14.00 | | | | | | | | | | | | (141) $i(all_120_0)
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | GROUND_INST: instantiating (37) with all_120_0, all_111_0,
% 86.22/14.00 | | | | | | | | | | | | all_161_0, simplifying with (97), (108), (141)
% 86.22/14.00 | | | | | | | | | | | | gives:
% 86.22/14.00 | | | | | | | | | | | | (142) ? [v0: $i] : ? [v1: $i] : (and(v1, all_161_0) =
% 86.22/14.00 | | | | | | | | | | | | v0 & equiv(all_120_0, all_111_0) = v0 &
% 86.22/14.00 | | | | | | | | | | | | implies(all_120_0, all_111_0) = v1 & $i(v1) &
% 86.22/14.00 | | | | | | | | | | | | $i(v0))
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | GROUND_INST: instantiating (36) with all_111_0, all_120_0,
% 86.22/14.00 | | | | | | | | | | | | all_161_0, simplifying with (97), (108), (141)
% 86.22/14.00 | | | | | | | | | | | | gives:
% 86.22/14.00 | | | | | | | | | | | | (143) ? [v0: $i] : ? [v1: $i] : (and(all_161_0, v1) =
% 86.22/14.00 | | | | | | | | | | | | v0 & equiv(all_111_0, all_120_0) = v0 &
% 86.22/14.00 | | | | | | | | | | | | implies(all_120_0, all_111_0) = v1 & $i(v1) &
% 86.22/14.00 | | | | | | | | | | | | $i(v0))
% 86.22/14.00 | | | | | | | | | | | |
% 86.22/14.00 | | | | | | | | | | | | GROUND_INST: instantiating (56) with all_187_1, all_4_1,
% 86.22/14.00 | | | | | | | | | | | | all_191_0, simplifying with (128), (139), (140)
% 86.22/14.00 | | | | | | | | | | | | gives:
% 86.22/14.01 | | | | | | | | | | | | (144) ? [v0: int] : ? [v1: int] : ? [v2: int] : ((v2
% 86.22/14.01 | | | | | | | | | | | | = 0 & is_a_theorem(all_4_1) = 0) | ( ~ (v1 =
% 86.22/14.01 | | | | | | | | | | | | 0) & is_a_theorem(all_191_0) = v1) | ( ~ (v0
% 86.22/14.01 | | | | | | | | | | | | = 0) & is_a_theorem(all_187_1) = v0))
% 86.22/14.01 | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | GROUND_INST: instantiating (32) with all_120_0, all_111_0,
% 86.22/14.01 | | | | | | | | | | | | all_4_2, all_187_1, simplifying with (89), (97),
% 86.22/14.01 | | | | | | | | | | | | (116), (141) gives:
% 86.22/14.01 | | | | | | | | | | | | (145) implies(all_120_0, all_111_0) = all_187_1 &
% 86.22/14.01 | | | | | | | | | | | | $i(all_187_1)
% 86.22/14.01 | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | ALPHA: (145) implies:
% 86.22/14.01 | | | | | | | | | | | | (146) implies(all_120_0, all_111_0) = all_187_1
% 86.22/14.01 | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | DELTA: instantiating (143) with fresh symbols all_1005_0,
% 86.22/14.01 | | | | | | | | | | | | all_1005_1 gives:
% 86.22/14.01 | | | | | | | | | | | | (147) and(all_161_0, all_1005_0) = all_1005_1 &
% 86.22/14.01 | | | | | | | | | | | | equiv(all_111_0, all_120_0) = all_1005_1 &
% 86.22/14.01 | | | | | | | | | | | | implies(all_120_0, all_111_0) = all_1005_0 &
% 86.22/14.01 | | | | | | | | | | | | $i(all_1005_0) & $i(all_1005_1)
% 86.22/14.01 | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | ALPHA: (147) implies:
% 86.22/14.01 | | | | | | | | | | | | (148) implies(all_120_0, all_111_0) = all_1005_0
% 86.22/14.01 | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | DELTA: instantiating (142) with fresh symbols all_1053_0,
% 86.22/14.01 | | | | | | | | | | | | all_1053_1 gives:
% 86.22/14.01 | | | | | | | | | | | | (149) and(all_1053_0, all_161_0) = all_1053_1 &
% 86.22/14.01 | | | | | | | | | | | | equiv(all_120_0, all_111_0) = all_1053_1 &
% 86.22/14.01 | | | | | | | | | | | | implies(all_120_0, all_111_0) = all_1053_0 &
% 86.22/14.01 | | | | | | | | | | | | $i(all_1053_0) & $i(all_1053_1)
% 86.22/14.01 | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | ALPHA: (149) implies:
% 86.22/14.01 | | | | | | | | | | | | (150) implies(all_120_0, all_111_0) = all_1053_0
% 86.22/14.01 | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | DELTA: instantiating (144) with fresh symbols all_1081_0,
% 86.22/14.01 | | | | | | | | | | | | all_1081_1, all_1081_2 gives:
% 86.22/14.01 | | | | | | | | | | | | (151) (all_1081_0 = 0 & is_a_theorem(all_4_1) = 0) | ( ~
% 86.22/14.01 | | | | | | | | | | | | (all_1081_1 = 0) & is_a_theorem(all_191_0) =
% 86.22/14.01 | | | | | | | | | | | | all_1081_1) | ( ~ (all_1081_2 = 0) &
% 86.22/14.01 | | | | | | | | | | | | is_a_theorem(all_187_1) = all_1081_2)
% 86.22/14.01 | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | BETA: splitting (151) gives:
% 86.22/14.01 | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | Case 1:
% 86.22/14.01 | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | (152) all_1081_0 = 0 & is_a_theorem(all_4_1) = 0
% 86.22/14.01 | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | ALPHA: (152) implies:
% 86.22/14.01 | | | | | | | | | | | | | (153) is_a_theorem(all_4_1) = 0
% 86.22/14.01 | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | GROUND_INST: instantiating (1) with all_4_0, 0, all_4_1,
% 86.22/14.01 | | | | | | | | | | | | | simplifying with (21), (153) gives:
% 86.22/14.01 | | | | | | | | | | | | | (154) all_4_0 = 0
% 86.22/14.01 | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | REDUCE: (19), (154) imply:
% 86.22/14.01 | | | | | | | | | | | | | (155) $false
% 86.22/14.01 | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | CLOSE: (155) is inconsistent.
% 86.22/14.01 | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | Case 2:
% 86.22/14.01 | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | (156) ( ~ (all_1081_1 = 0) & is_a_theorem(all_191_0) =
% 86.22/14.01 | | | | | | | | | | | | | all_1081_1) | ( ~ (all_1081_2 = 0) &
% 86.22/14.01 | | | | | | | | | | | | | is_a_theorem(all_187_1) = all_1081_2)
% 86.22/14.01 | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | BETA: splitting (156) gives:
% 86.22/14.01 | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | Case 1:
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | (157) ~ (all_1081_1 = 0) & is_a_theorem(all_191_0) =
% 86.22/14.01 | | | | | | | | | | | | | | all_1081_1
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | ALPHA: (157) implies:
% 86.22/14.01 | | | | | | | | | | | | | | (158) ~ (all_1081_1 = 0)
% 86.22/14.01 | | | | | | | | | | | | | | (159) is_a_theorem(all_191_0) = all_1081_1
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | GROUND_INST: instantiating (1) with 0, all_1081_1, all_191_0,
% 86.22/14.01 | | | | | | | | | | | | | | simplifying with (121), (159) gives:
% 86.22/14.01 | | | | | | | | | | | | | | (160) all_1081_1 = 0
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | REDUCE: (158), (160) imply:
% 86.22/14.01 | | | | | | | | | | | | | | (161) $false
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | CLOSE: (161) is inconsistent.
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | Case 2:
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | (162) ~ (all_1081_2 = 0) & is_a_theorem(all_187_1) =
% 86.22/14.01 | | | | | | | | | | | | | | all_1081_2
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | ALPHA: (162) implies:
% 86.22/14.01 | | | | | | | | | | | | | | (163) ~ (all_1081_2 = 0)
% 86.22/14.01 | | | | | | | | | | | | | | (164) is_a_theorem(all_187_1) = all_1081_2
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | GROUND_INST: instantiating (3) with all_167_0, all_1005_0,
% 86.22/14.01 | | | | | | | | | | | | | | all_111_0, all_120_0, simplifying with (111),
% 86.22/14.01 | | | | | | | | | | | | | | (148) gives:
% 86.22/14.01 | | | | | | | | | | | | | | (165) all_1005_0 = all_167_0
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | GROUND_INST: instantiating (3) with all_1005_0, all_1053_0,
% 86.22/14.01 | | | | | | | | | | | | | | all_111_0, all_120_0, simplifying with (148),
% 86.22/14.01 | | | | | | | | | | | | | | (150) gives:
% 86.22/14.01 | | | | | | | | | | | | | | (166) all_1053_0 = all_1005_0
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | GROUND_INST: instantiating (3) with all_187_1, all_1053_0,
% 86.22/14.01 | | | | | | | | | | | | | | all_111_0, all_120_0, simplifying with (146),
% 86.22/14.01 | | | | | | | | | | | | | | (150) gives:
% 86.22/14.01 | | | | | | | | | | | | | | (167) all_1053_0 = all_187_1
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | COMBINE_EQS: (166), (167) imply:
% 86.22/14.01 | | | | | | | | | | | | | | (168) all_1005_0 = all_187_1
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | SIMP: (168) implies:
% 86.22/14.01 | | | | | | | | | | | | | | (169) all_1005_0 = all_187_1
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | COMBINE_EQS: (165), (169) imply:
% 86.22/14.01 | | | | | | | | | | | | | | (170) all_187_1 = all_167_0
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | REDUCE: (164), (170) imply:
% 86.22/14.01 | | | | | | | | | | | | | | (171) is_a_theorem(all_167_0) = all_1081_2
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | GROUND_INST: instantiating (1) with 0, all_1081_2, all_167_0,
% 86.22/14.01 | | | | | | | | | | | | | | simplifying with (110), (171) gives:
% 86.22/14.01 | | | | | | | | | | | | | | (172) all_1081_2 = 0
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | REDUCE: (163), (172) imply:
% 86.22/14.01 | | | | | | | | | | | | | | (173) $false
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | | CLOSE: (173) is inconsistent.
% 86.22/14.01 | | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | | End of split
% 86.22/14.01 | | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | | End of split
% 86.22/14.01 | | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | | End of split
% 86.22/14.01 | | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | | End of split
% 86.22/14.01 | | | | | | | | | |
% 86.22/14.01 | | | | | | | | | End of split
% 86.22/14.01 | | | | | | | | |
% 86.22/14.01 | | | | | | | | End of split
% 86.22/14.01 | | | | | | | |
% 86.22/14.01 | | | | | | | End of split
% 86.22/14.01 | | | | | | |
% 86.22/14.01 | | | | | | End of split
% 86.22/14.01 | | | | | |
% 86.22/14.01 | | | | | End of split
% 86.22/14.01 | | | | |
% 86.22/14.01 | | | | End of split
% 86.22/14.01 | | | |
% 86.22/14.01 | | | End of split
% 86.22/14.01 | | |
% 86.22/14.01 | | End of split
% 86.22/14.01 | |
% 86.22/14.01 | Case 2:
% 86.22/14.01 | |
% 86.22/14.01 | | (174) kn1 & ! [v0: $i] : ! [v1: $i] : ( ~ (and(v0, v0) = v1) | ~
% 86.22/14.01 | | $i(v0) | ? [v2: $i] : (implies(v0, v1) = v2 & is_a_theorem(v2) =
% 86.22/14.01 | | 0 & $i(v2)))
% 86.22/14.01 | |
% 86.22/14.01 | | ALPHA: (174) implies:
% 86.22/14.01 | | (175) kn1
% 86.22/14.01 | |
% 86.22/14.01 | | PRED_UNIFY: (175), (rosser_kn1) imply:
% 86.22/14.01 | | (176) $false
% 86.22/14.01 | |
% 86.22/14.01 | | CLOSE: (176) is inconsistent.
% 86.22/14.01 | |
% 86.22/14.01 | End of split
% 86.22/14.01 |
% 86.22/14.01 End of proof
% 86.22/14.02 % SZS output end Proof for theBenchmark
% 86.22/14.02
% 86.22/14.02 13430ms
%------------------------------------------------------------------------------