TSTP Solution File: LCL538+1 by Princess---230619
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%------------------------------------------------------------------------------
% File : Princess---230619
% Problem : LCL538+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 : n012.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 08:11:35 EDT 2023
% Result : Theorem 13.03s 2.58s
% Output : Proof 59.04s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL538+1 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.14 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s
% 0.14/0.35 % Computer : n012.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Aug 25 06:47:26 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.22/0.62 ________ _____
% 0.22/0.62 ___ __ \_________(_)________________________________
% 0.22/0.62 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/
% 0.22/0.62 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ )
% 0.22/0.62 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/
% 0.22/0.62
% 0.22/0.62 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic
% 0.22/0.62 (2023-06-19)
% 0.22/0.62
% 0.22/0.62 (c) Philipp Rümmer, 2009-2023
% 0.22/0.62 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen,
% 0.22/0.62 Amanda Stjerna.
% 0.22/0.62 Free software under BSD-3-Clause.
% 0.22/0.62
% 0.22/0.62 For more information, visit http://www.philipp.ruemmer.org/princess.shtml
% 0.22/0.62
% 0.22/0.62 Loading /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 0.22/0.63 Running up to 7 provers in parallel.
% 0.22/0.65 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1571432423
% 0.22/0.65 Prover 2: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMinimalAndEmpty -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1065072994
% 0.22/0.65 Prover 0: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1042961893
% 0.22/0.65 Prover 3: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1922548996
% 0.22/0.65 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=1868514696
% 0.22/0.65 Prover 5: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=complete -randomSeed=1259561288
% 0.22/0.65 Prover 6: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximalOutermost -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1399714365
% 4.08/1.31 Prover 4: Preprocessing ...
% 4.08/1.31 Prover 1: Preprocessing ...
% 4.08/1.36 Prover 5: Preprocessing ...
% 4.08/1.36 Prover 3: Preprocessing ...
% 4.08/1.36 Prover 0: Preprocessing ...
% 4.08/1.36 Prover 6: Preprocessing ...
% 4.08/1.36 Prover 2: Preprocessing ...
% 10.81/2.27 Prover 5: Proving ...
% 10.81/2.30 Prover 6: Constructing countermodel ...
% 10.81/2.31 Prover 1: Constructing countermodel ...
% 11.39/2.33 Prover 3: Constructing countermodel ...
% 11.39/2.36 Prover 4: Constructing countermodel ...
% 12.41/2.47 Prover 0: Proving ...
% 13.03/2.54 Prover 2: Proving ...
% 13.03/2.58 Prover 3: proved (1940ms)
% 13.03/2.58
% 13.03/2.58 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p
% 13.03/2.58
% 13.03/2.59 Prover 5: stopped
% 13.03/2.59 Prover 6: stopped
% 13.03/2.60 Prover 0: stopped
% 13.03/2.60 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089
% 13.03/2.60 Prover 2: stopped
% 13.03/2.61 Prover 10: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=919308125
% 13.03/2.61 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470
% 13.03/2.61 Prover 11: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1509710984
% 13.03/2.61 Prover 13: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=1138197443
% 14.07/2.70 Prover 8: Preprocessing ...
% 14.07/2.71 Prover 7: Preprocessing ...
% 14.07/2.72 Prover 10: Preprocessing ...
% 14.07/2.73 Prover 11: Preprocessing ...
% 14.07/2.75 Prover 13: Preprocessing ...
% 16.02/3.00 Prover 8: Warning: ignoring some quantifiers
% 16.02/3.03 Prover 13: Warning: ignoring some quantifiers
% 16.02/3.04 Prover 10: Constructing countermodel ...
% 16.68/3.04 Prover 13: Constructing countermodel ...
% 16.68/3.04 Prover 8: Constructing countermodel ...
% 16.68/3.05 Prover 7: Constructing countermodel ...
% 16.68/3.06 Prover 11: Constructing countermodel ...
% 57.41/8.43 Prover 8: Found proof (size 88)
% 57.41/8.43 Prover 8: proved (5809ms)
% 57.41/8.44 Prover 7: stopped
% 57.41/8.44 Prover 11: stopped
% 57.41/8.44 Prover 4: stopped
% 58.10/8.45 Prover 1: stopped
% 58.10/8.46 Prover 10: stopped
% 58.55/8.58 Prover 13: stopped
% 58.55/8.58
% 58.55/8.58 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p
% 58.55/8.58
% 58.55/8.58 % SZS output start Proof for theBenchmark
% 58.55/8.59 Assumptions after simplification:
% 58.55/8.59 ---------------------------------
% 58.55/8.59
% 58.55/8.59 (adjunction)
% 58.80/8.61 (adjunction & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (and(v0, v1) = v2)
% 58.80/8.61 | ~ $i(v1) | ~ $i(v0) | ? [v3: any] : ? [v4: any] : ? [v5: any] :
% 58.80/8.61 (is_a_theorem(v2) = v5 & is_a_theorem(v1) = v4 & is_a_theorem(v0) = v3 & (
% 58.80/8.61 ~ (v4 = 0) | ~ (v3 = 0) | v5 = 0)))) | ( ~ adjunction & ? [v0: $i] :
% 58.80/8.61 ? [v1: $i] : ? [v2: $i] : ? [v3: int] : ( ~ (v3 = 0) & and(v0, v1) = v2 &
% 58.80/8.61 is_a_theorem(v2) = v3 & is_a_theorem(v1) = 0 & is_a_theorem(v0) = 0 &
% 58.80/8.61 $i(v2) & $i(v1) & $i(v0)))
% 58.80/8.61
% 58.80/8.61 (axiom_M)
% 58.80/8.62 (axiom_M & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (necessarily(v0) =
% 58.80/8.62 v1) | ~ (implies(v1, v0) = v2) | ~ $i(v0) | is_a_theorem(v2) = 0)) | (
% 58.80/8.62 ~ axiom_M & ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: int] : ( ~ (v3
% 58.80/8.62 = 0) & necessarily(v0) = v1 & implies(v1, v0) = v2 & is_a_theorem(v2) =
% 58.80/8.62 v3 & $i(v2) & $i(v1) & $i(v0)))
% 58.80/8.62
% 58.80/8.62 (axiom_m9)
% 58.80/8.62 (axiom_m9 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~
% 58.80/8.62 (possibly(v1) = v2) | ~ (possibly(v0) = v1) | ~ (strict_implies(v2, v1)
% 58.80/8.62 = v3) | ~ $i(v0) | is_a_theorem(v3) = 0)) | ( ~ axiom_m9 & ? [v0: $i]
% 58.80/8.62 : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ? [v4: int] : ( ~ (v4 = 0) &
% 58.80/8.62 possibly(v1) = v2 & possibly(v0) = v1 & strict_implies(v2, v1) = v3 &
% 58.80/8.62 is_a_theorem(v3) = v4 & $i(v3) & $i(v2) & $i(v1) & $i(v0)))
% 58.80/8.62
% 58.80/8.62 (hilbert_modus_ponens)
% 58.80/8.62 modus_ponens
% 58.80/8.62
% 58.80/8.62 (km4b_axiom_M)
% 58.80/8.62 axiom_M
% 58.80/8.62
% 58.80/8.62 (modus_ponens)
% 58.80/8.62 (modus_ponens & ! [v0: $i] : ! [v1: $i] : ! [v2: int] : (v2 = 0 | ~
% 58.80/8.62 (is_a_theorem(v1) = v2) | ~ (is_a_theorem(v0) = 0) | ~ $i(v1) | ~
% 58.80/8.62 $i(v0) | ? [v3: $i] : ? [v4: int] : ( ~ (v4 = 0) & implies(v0, v1) = v3
% 58.80/8.62 & is_a_theorem(v3) = v4 & $i(v3)))) | ( ~ modus_ponens & ? [v0: $i] :
% 58.80/8.62 ? [v1: $i] : ? [v2: $i] : ? [v3: int] : ( ~ (v3 = 0) & implies(v0, v1) =
% 58.80/8.62 v2 & is_a_theorem(v2) = 0 & is_a_theorem(v1) = v3 & is_a_theorem(v0) = 0 &
% 58.80/8.62 $i(v2) & $i(v1) & $i(v0)))
% 58.80/8.62
% 58.80/8.62 (modus_ponens_strict_implies)
% 58.80/8.62 (modus_ponens_strict_implies & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 58.80/8.62 (strict_implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: any] : ?
% 58.80/8.62 [v4: any] : ? [v5: any] : (is_a_theorem(v2) = v4 & is_a_theorem(v1) = v5
% 58.80/8.62 & is_a_theorem(v0) = v3 & ( ~ (v4 = 0) | ~ (v3 = 0) | v5 = 0)))) | ( ~
% 58.80/8.62 modus_ponens_strict_implies & ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ?
% 58.80/8.62 [v3: int] : ( ~ (v3 = 0) & strict_implies(v0, v1) = v2 & is_a_theorem(v2) =
% 58.80/8.62 0 & is_a_theorem(v1) = v3 & is_a_theorem(v0) = 0 & $i(v2) & $i(v1) &
% 58.80/8.62 $i(v0)))
% 58.80/8.62
% 58.80/8.62 (op_strict_implies)
% 58.80/8.63 ~ op_strict_implies | ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 58.80/8.63 (strict_implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] :
% 58.80/8.63 (necessarily(v3) = v2 & implies(v0, v1) = v3 & $i(v3) & $i(v2)))
% 58.80/8.63
% 58.80/8.63 (s1_0_modus_ponens_strict_implies)
% 58.80/8.63 ~ modus_ponens_strict_implies
% 58.80/8.63
% 58.80/8.63 (s1_0_op_strict_implies)
% 58.80/8.63 op_strict_implies
% 58.80/8.63
% 58.80/8.63 (function-axioms)
% 58.80/8.63 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~
% 58.80/8.63 (strict_equiv(v3, v2) = v1) | ~ (strict_equiv(v3, v2) = v0)) & ! [v0: $i]
% 58.80/8.63 : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (strict_implies(v3,
% 58.80/8.63 v2) = v1) | ~ (strict_implies(v3, v2) = v0)) & ! [v0: $i] : ! [v1:
% 58.80/8.63 $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (or(v3, v2) = v1) | ~
% 58.80/8.63 (or(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] :
% 58.80/8.63 (v1 = v0 | ~ (and(v3, v2) = v1) | ~ (and(v3, v2) = v0)) & ! [v0: $i] : !
% 58.80/8.63 [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (equiv(v3, v2) = v1) | ~
% 58.80/8.63 (equiv(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3:
% 58.80/8.63 $i] : (v1 = v0 | ~ (implies(v3, v2) = v1) | ~ (implies(v3, v2) = v0)) & !
% 58.80/8.63 [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (possibly(v2) = v1) | ~
% 58.80/8.63 (possibly(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 |
% 58.80/8.63 ~ (necessarily(v2) = v1) | ~ (necessarily(v2) = v0)) & ! [v0: $i] : !
% 58.80/8.63 [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (not(v2) = v1) | ~ (not(v2) = v0)) &
% 58.80/8.63 ! [v0: MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: $i] : (v1 =
% 58.80/8.63 v0 | ~ (is_a_theorem(v2) = v1) | ~ (is_a_theorem(v2) = v0)) & ? [v0: $i]
% 58.80/8.63 : ? [v1: $i] : ? [v2: $i] : (strict_equiv(v1, v0) = v2 & $i(v2)) & ? [v0:
% 58.80/8.63 $i] : ? [v1: $i] : ? [v2: $i] : (strict_implies(v1, v0) = v2 & $i(v2)) &
% 58.80/8.63 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : (or(v1, v0) = v2 & $i(v2)) & ? [v0:
% 58.80/8.63 $i] : ? [v1: $i] : ? [v2: $i] : (and(v1, v0) = v2 & $i(v2)) & ? [v0: $i]
% 58.80/8.63 : ? [v1: $i] : ? [v2: $i] : (equiv(v1, v0) = v2 & $i(v2)) & ? [v0: $i] : ?
% 58.80/8.63 [v1: $i] : ? [v2: $i] : (implies(v1, v0) = v2 & $i(v2)) & ? [v0: $i] : ?
% 58.80/8.63 [v1: MultipleValueBool] : (is_a_theorem(v0) = v1) & ? [v0: $i] : ? [v1: $i]
% 58.80/8.63 : (possibly(v0) = v1 & $i(v1)) & ? [v0: $i] : ? [v1: $i] : (necessarily(v0)
% 58.80/8.63 = v1 & $i(v1)) & ? [v0: $i] : ? [v1: $i] : (not(v0) = v1 & $i(v1))
% 58.80/8.63
% 58.80/8.63 Further assumptions not needed in the proof:
% 58.80/8.63 --------------------------------------------
% 58.80/8.63 and_1, and_2, and_3, axiom_4, axiom_5, axiom_B, axiom_K, axiom_m1, axiom_m10,
% 58.80/8.63 axiom_m2, axiom_m3, axiom_m4, axiom_m5, axiom_m6, axiom_m7, axiom_m8, axiom_s1,
% 58.80/8.63 axiom_s2, axiom_s3, axiom_s4, cn1, cn2, cn3, equivalence_1, equivalence_2,
% 58.80/8.63 equivalence_3, hilbert_and_1, hilbert_and_2, hilbert_and_3,
% 58.80/8.63 hilbert_equivalence_1, hilbert_equivalence_2, hilbert_equivalence_3,
% 58.80/8.63 hilbert_implies_1, hilbert_implies_2, hilbert_implies_3, hilbert_modus_tollens,
% 58.80/8.63 hilbert_op_equiv, hilbert_op_implies_and, hilbert_op_or, hilbert_or_1,
% 58.80/8.63 hilbert_or_2, hilbert_or_3, implies_1, implies_2, implies_3, km4b_axiom_4,
% 58.80/8.63 km4b_axiom_B, km4b_axiom_K, km4b_necessitation, km4b_op_possibly, kn1, kn2, kn3,
% 58.80/8.63 modus_tollens, necessitation, op_and, op_equiv, op_implies_and, op_implies_or,
% 58.80/8.63 op_necessarily, op_or, op_possibly, op_strict_equiv, or_1, or_2, or_3, r1, r2,
% 58.80/8.63 r3, r4, r5, s1_0_op_equiv, s1_0_op_implies, s1_0_op_or, s1_0_op_possibly,
% 58.80/8.63 s1_0_op_strict_equiv, substitution_of_equivalents, substitution_strict_equiv
% 58.80/8.63
% 58.80/8.63 Those formulas are unsatisfiable:
% 58.80/8.63 ---------------------------------
% 58.80/8.63
% 58.80/8.63 Begin of proof
% 58.80/8.63 |
% 58.80/8.63 | ALPHA: (function-axioms) implies:
% 58.80/8.63 | (1) ! [v0: MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: $i] :
% 58.80/8.63 | (v1 = v0 | ~ (is_a_theorem(v2) = v1) | ~ (is_a_theorem(v2) = v0))
% 58.80/8.63 | (2) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~
% 58.80/8.63 | (implies(v3, v2) = v1) | ~ (implies(v3, v2) = v0))
% 58.80/8.63 |
% 58.80/8.63 | BETA: splitting (op_strict_implies) gives:
% 58.80/8.63 |
% 58.80/8.63 | Case 1:
% 58.80/8.63 | |
% 58.80/8.63 | | (3) ~ op_strict_implies
% 58.80/8.64 | |
% 58.80/8.64 | | PRED_UNIFY: (3), (s1_0_op_strict_implies) imply:
% 58.80/8.64 | | (4) $false
% 58.80/8.64 | |
% 58.80/8.64 | | CLOSE: (4) is inconsistent.
% 58.80/8.64 | |
% 58.80/8.64 | Case 2:
% 58.80/8.64 | |
% 58.80/8.64 | | (5) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (strict_implies(v0, v1)
% 58.80/8.64 | | = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : (necessarily(v3) =
% 58.80/8.64 | | v2 & implies(v0, v1) = v3 & $i(v3) & $i(v2)))
% 58.80/8.64 | |
% 58.80/8.64 | | BETA: splitting (modus_ponens) gives:
% 58.80/8.64 | |
% 58.80/8.64 | | Case 1:
% 58.80/8.64 | | |
% 58.80/8.64 | | | (6) modus_ponens & ! [v0: $i] : ! [v1: $i] : ! [v2: int] : (v2 = 0 |
% 58.80/8.64 | | | ~ (is_a_theorem(v1) = v2) | ~ (is_a_theorem(v0) = 0) | ~
% 58.80/8.64 | | | $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: int] : ( ~ (v4 = 0) &
% 58.80/8.64 | | | implies(v0, v1) = v3 & is_a_theorem(v3) = v4 & $i(v3)))
% 58.80/8.64 | | |
% 58.80/8.64 | | | ALPHA: (6) implies:
% 58.80/8.64 | | | (7) ! [v0: $i] : ! [v1: $i] : ! [v2: int] : (v2 = 0 | ~
% 58.80/8.64 | | | (is_a_theorem(v1) = v2) | ~ (is_a_theorem(v0) = 0) | ~ $i(v1) |
% 58.80/8.64 | | | ~ $i(v0) | ? [v3: $i] : ? [v4: int] : ( ~ (v4 = 0) &
% 58.80/8.64 | | | implies(v0, v1) = v3 & is_a_theorem(v3) = v4 & $i(v3)))
% 58.80/8.64 | | |
% 58.80/8.64 | | | BETA: splitting (axiom_M) gives:
% 58.80/8.64 | | |
% 58.80/8.64 | | | Case 1:
% 58.80/8.64 | | | |
% 58.80/8.64 | | | | (8) axiom_M & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 58.80/8.64 | | | | (necessarily(v0) = v1) | ~ (implies(v1, v0) = v2) | ~ $i(v0)
% 58.80/8.64 | | | | | is_a_theorem(v2) = 0)
% 58.80/8.64 | | | |
% 58.80/8.64 | | | | ALPHA: (8) implies:
% 58.80/8.64 | | | | (9) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (necessarily(v0) =
% 58.80/8.64 | | | | v1) | ~ (implies(v1, v0) = v2) | ~ $i(v0) |
% 58.80/8.64 | | | | is_a_theorem(v2) = 0)
% 58.80/8.64 | | | |
% 58.80/8.64 | | | | BETA: splitting (modus_ponens_strict_implies) gives:
% 58.80/8.64 | | | |
% 58.80/8.64 | | | | Case 1:
% 58.80/8.64 | | | | |
% 58.80/8.64 | | | | | (10) modus_ponens_strict_implies & ! [v0: $i] : ! [v1: $i] : !
% 58.80/8.64 | | | | | [v2: $i] : ( ~ (strict_implies(v0, v1) = v2) | ~ $i(v1) | ~
% 58.80/8.64 | | | | | $i(v0) | ? [v3: any] : ? [v4: any] : ? [v5: any] :
% 58.80/8.64 | | | | | (is_a_theorem(v2) = v4 & is_a_theorem(v1) = v5 &
% 58.80/8.64 | | | | | is_a_theorem(v0) = v3 & ( ~ (v4 = 0) | ~ (v3 = 0) | v5 =
% 58.80/8.64 | | | | | 0)))
% 58.80/8.64 | | | | |
% 58.80/8.64 | | | | | ALPHA: (10) implies:
% 58.80/8.64 | | | | | (11) modus_ponens_strict_implies
% 58.80/8.64 | | | | |
% 58.80/8.64 | | | | | PRED_UNIFY: (11), (s1_0_modus_ponens_strict_implies) imply:
% 58.80/8.64 | | | | | (12) $false
% 58.80/8.64 | | | | |
% 58.80/8.64 | | | | | CLOSE: (12) is inconsistent.
% 58.80/8.64 | | | | |
% 58.80/8.64 | | | | Case 2:
% 58.80/8.64 | | | | |
% 58.80/8.64 | | | | | (13) ~ modus_ponens_strict_implies & ? [v0: $i] : ? [v1: $i] :
% 58.80/8.64 | | | | | ? [v2: $i] : ? [v3: int] : ( ~ (v3 = 0) & strict_implies(v0,
% 58.80/8.64 | | | | | v1) = v2 & is_a_theorem(v2) = 0 & is_a_theorem(v1) = v3 &
% 58.80/8.64 | | | | | is_a_theorem(v0) = 0 & $i(v2) & $i(v1) & $i(v0))
% 58.80/8.64 | | | | |
% 58.80/8.64 | | | | | ALPHA: (13) implies:
% 58.80/8.64 | | | | | (14) ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: int] : ( ~
% 58.80/8.64 | | | | | (v3 = 0) & strict_implies(v0, v1) = v2 & is_a_theorem(v2) =
% 58.80/8.64 | | | | | 0 & is_a_theorem(v1) = v3 & is_a_theorem(v0) = 0 & $i(v2) &
% 58.80/8.64 | | | | | $i(v1) & $i(v0))
% 58.80/8.64 | | | | |
% 58.80/8.64 | | | | | DELTA: instantiating (14) with fresh symbols all_129_0, all_129_1,
% 58.80/8.64 | | | | | all_129_2, all_129_3 gives:
% 58.80/8.64 | | | | | (15) ~ (all_129_0 = 0) & strict_implies(all_129_3, all_129_2) =
% 58.80/8.64 | | | | | all_129_1 & is_a_theorem(all_129_1) = 0 &
% 58.80/8.64 | | | | | is_a_theorem(all_129_2) = all_129_0 & is_a_theorem(all_129_3)
% 58.80/8.64 | | | | | = 0 & $i(all_129_1) & $i(all_129_2) & $i(all_129_3)
% 58.80/8.64 | | | | |
% 58.80/8.64 | | | | | ALPHA: (15) implies:
% 58.80/8.64 | | | | | (16) ~ (all_129_0 = 0)
% 58.80/8.64 | | | | | (17) $i(all_129_3)
% 58.80/8.64 | | | | | (18) $i(all_129_2)
% 58.80/8.64 | | | | | (19) is_a_theorem(all_129_3) = 0
% 58.80/8.64 | | | | | (20) is_a_theorem(all_129_2) = all_129_0
% 58.80/8.64 | | | | | (21) is_a_theorem(all_129_1) = 0
% 58.80/8.65 | | | | | (22) strict_implies(all_129_3, all_129_2) = all_129_1
% 58.80/8.65 | | | | |
% 58.80/8.65 | | | | | GROUND_INST: instantiating (7) with all_129_3, all_129_2, all_129_0,
% 58.80/8.65 | | | | | simplifying with (17), (18), (19), (20) gives:
% 58.80/8.65 | | | | | (23) all_129_0 = 0 | ? [v0: $i] : ? [v1: int] : ( ~ (v1 = 0) &
% 58.80/8.65 | | | | | implies(all_129_3, all_129_2) = v0 & is_a_theorem(v0) = v1 &
% 58.80/8.65 | | | | | $i(v0))
% 58.80/8.65 | | | | |
% 58.80/8.65 | | | | | GROUND_INST: instantiating (5) with all_129_3, all_129_2, all_129_1,
% 58.80/8.65 | | | | | simplifying with (17), (18), (22) gives:
% 58.80/8.65 | | | | | (24) ? [v0: $i] : (necessarily(v0) = all_129_1 &
% 58.80/8.65 | | | | | implies(all_129_3, all_129_2) = v0 & $i(v0) & $i(all_129_1))
% 58.80/8.65 | | | | |
% 58.80/8.65 | | | | | DELTA: instantiating (24) with fresh symbol all_136_0 gives:
% 58.80/8.65 | | | | | (25) necessarily(all_136_0) = all_129_1 & implies(all_129_3,
% 58.80/8.65 | | | | | all_129_2) = all_136_0 & $i(all_136_0) & $i(all_129_1)
% 58.80/8.65 | | | | |
% 58.80/8.65 | | | | | ALPHA: (25) implies:
% 58.80/8.65 | | | | | (26) $i(all_129_1)
% 58.80/8.65 | | | | | (27) implies(all_129_3, all_129_2) = all_136_0
% 58.80/8.65 | | | | | (28) necessarily(all_136_0) = all_129_1
% 58.80/8.65 | | | | |
% 58.80/8.65 | | | | | BETA: splitting (23) gives:
% 58.80/8.65 | | | | |
% 58.80/8.65 | | | | | Case 1:
% 58.80/8.65 | | | | | |
% 58.80/8.65 | | | | | | (29) all_129_0 = 0
% 58.80/8.65 | | | | | |
% 58.80/8.65 | | | | | | REDUCE: (16), (29) imply:
% 58.80/8.65 | | | | | | (30) $false
% 58.80/8.65 | | | | | |
% 58.80/8.65 | | | | | | CLOSE: (30) is inconsistent.
% 58.80/8.65 | | | | | |
% 58.80/8.65 | | | | | Case 2:
% 58.80/8.65 | | | | | |
% 58.80/8.65 | | | | | | (31) ? [v0: $i] : ? [v1: int] : ( ~ (v1 = 0) &
% 58.80/8.65 | | | | | | implies(all_129_3, all_129_2) = v0 & is_a_theorem(v0) = v1
% 58.80/8.65 | | | | | | & $i(v0))
% 58.80/8.65 | | | | | |
% 58.80/8.65 | | | | | | DELTA: instantiating (31) with fresh symbols all_142_0, all_142_1
% 58.80/8.65 | | | | | | gives:
% 58.80/8.65 | | | | | | (32) ~ (all_142_0 = 0) & implies(all_129_3, all_129_2) =
% 58.80/8.65 | | | | | | all_142_1 & is_a_theorem(all_142_1) = all_142_0 &
% 58.80/8.65 | | | | | | $i(all_142_1)
% 58.80/8.65 | | | | | |
% 58.80/8.65 | | | | | | ALPHA: (32) implies:
% 58.80/8.65 | | | | | | (33) ~ (all_142_0 = 0)
% 58.80/8.65 | | | | | | (34) $i(all_142_1)
% 58.80/8.65 | | | | | | (35) is_a_theorem(all_142_1) = all_142_0
% 58.80/8.65 | | | | | | (36) implies(all_129_3, all_129_2) = all_142_1
% 58.80/8.65 | | | | | |
% 58.80/8.65 | | | | | | GROUND_INST: instantiating (2) with all_136_0, all_142_1, all_129_2,
% 58.80/8.65 | | | | | | all_129_3, simplifying with (27), (36) gives:
% 58.80/8.65 | | | | | | (37) all_142_1 = all_136_0
% 58.80/8.65 | | | | | |
% 58.80/8.65 | | | | | | REDUCE: (35), (37) imply:
% 58.80/8.65 | | | | | | (38) is_a_theorem(all_136_0) = all_142_0
% 58.80/8.65 | | | | | |
% 58.80/8.65 | | | | | | REDUCE: (34), (37) imply:
% 58.80/8.65 | | | | | | (39) $i(all_136_0)
% 58.80/8.65 | | | | | |
% 58.80/8.65 | | | | | | BETA: splitting (axiom_m9) gives:
% 58.80/8.65 | | | | | |
% 58.80/8.65 | | | | | | Case 1:
% 58.80/8.65 | | | | | | |
% 58.80/8.65 | | | | | | |
% 58.80/8.65 | | | | | | | GROUND_INST: instantiating (7) with all_129_1, all_136_0,
% 58.80/8.65 | | | | | | | all_142_0, simplifying with (21), (26), (38), (39)
% 58.80/8.65 | | | | | | | gives:
% 58.80/8.65 | | | | | | | (40) all_142_0 = 0 | ? [v0: $i] : ? [v1: int] : ( ~ (v1 = 0)
% 58.80/8.65 | | | | | | | & implies(all_129_1, all_136_0) = v0 & is_a_theorem(v0)
% 58.80/8.65 | | | | | | | = v1 & $i(v0))
% 58.80/8.65 | | | | | | |
% 58.80/8.65 | | | | | | | REF_CLOSE: (1), (9), (28), (33), (39), (40) are inconsistent by
% 58.80/8.65 | | | | | | | sub-proof #1.
% 58.80/8.65 | | | | | | |
% 58.80/8.65 | | | | | | Case 2:
% 58.80/8.65 | | | | | | |
% 58.80/8.65 | | | | | | | (41) ~ axiom_m9 & ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ?
% 58.80/8.65 | | | | | | | [v3: $i] : ? [v4: int] : ( ~ (v4 = 0) & possibly(v1) = v2
% 58.80/8.65 | | | | | | | & possibly(v0) = v1 & strict_implies(v2, v1) = v3 &
% 58.80/8.65 | | | | | | | is_a_theorem(v3) = v4 & $i(v3) & $i(v2) & $i(v1) &
% 58.80/8.65 | | | | | | | $i(v0))
% 58.80/8.65 | | | | | | |
% 58.80/8.65 | | | | | | | ALPHA: (41) implies:
% 58.80/8.65 | | | | | | | (42) ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ?
% 58.80/8.65 | | | | | | | [v4: int] : ( ~ (v4 = 0) & possibly(v1) = v2 &
% 58.80/8.65 | | | | | | | possibly(v0) = v1 & strict_implies(v2, v1) = v3 &
% 58.80/8.65 | | | | | | | is_a_theorem(v3) = v4 & $i(v3) & $i(v2) & $i(v1) &
% 58.80/8.65 | | | | | | | $i(v0))
% 58.80/8.65 | | | | | | |
% 58.80/8.65 | | | | | | | DELTA: instantiating (42) with fresh symbols all_235_0, all_235_1,
% 58.80/8.65 | | | | | | | all_235_2, all_235_3, all_235_4 gives:
% 59.04/8.65 | | | | | | | (43) ~ (all_235_0 = 0) & possibly(all_235_3) = all_235_2 &
% 59.04/8.65 | | | | | | | possibly(all_235_4) = all_235_3 &
% 59.04/8.65 | | | | | | | strict_implies(all_235_2, all_235_3) = all_235_1 &
% 59.04/8.65 | | | | | | | is_a_theorem(all_235_1) = all_235_0 & $i(all_235_1) &
% 59.04/8.65 | | | | | | | $i(all_235_2) & $i(all_235_3) & $i(all_235_4)
% 59.04/8.65 | | | | | | |
% 59.04/8.65 | | | | | | | ALPHA: (43) implies:
% 59.04/8.65 | | | | | | | (44) ~ (all_235_0 = 0)
% 59.04/8.65 | | | | | | | (45) $i(all_235_1)
% 59.04/8.66 | | | | | | | (46) is_a_theorem(all_235_1) = all_235_0
% 59.04/8.66 | | | | | | |
% 59.04/8.66 | | | | | | | BETA: splitting (adjunction) gives:
% 59.04/8.66 | | | | | | |
% 59.04/8.66 | | | | | | | Case 1:
% 59.04/8.66 | | | | | | | |
% 59.04/8.66 | | | | | | | |
% 59.04/8.66 | | | | | | | | GROUND_INST: instantiating (7) with all_129_1, all_136_0,
% 59.04/8.66 | | | | | | | | all_142_0, simplifying with (21), (26), (38), (39)
% 59.04/8.66 | | | | | | | | gives:
% 59.04/8.66 | | | | | | | | (47) all_142_0 = 0 | ? [v0: $i] : ? [v1: int] : ( ~ (v1 =
% 59.04/8.66 | | | | | | | | 0) & implies(all_129_1, all_136_0) = v0 &
% 59.04/8.66 | | | | | | | | is_a_theorem(v0) = v1 & $i(v0))
% 59.04/8.66 | | | | | | | |
% 59.04/8.66 | | | | | | | | REF_CLOSE: (1), (9), (28), (33), (39), (47) are inconsistent by
% 59.04/8.66 | | | | | | | | sub-proof #1.
% 59.04/8.66 | | | | | | | |
% 59.04/8.66 | | | | | | | Case 2:
% 59.04/8.66 | | | | | | | |
% 59.04/8.66 | | | | | | | | (48) ~ adjunction & ? [v0: $i] : ? [v1: $i] : ? [v2: $i]
% 59.04/8.66 | | | | | | | | : ? [v3: int] : ( ~ (v3 = 0) & and(v0, v1) = v2 &
% 59.04/8.66 | | | | | | | | is_a_theorem(v2) = v3 & is_a_theorem(v1) = 0 &
% 59.04/8.66 | | | | | | | | is_a_theorem(v0) = 0 & $i(v2) & $i(v1) & $i(v0))
% 59.04/8.66 | | | | | | | |
% 59.04/8.66 | | | | | | | | ALPHA: (48) implies:
% 59.04/8.66 | | | | | | | | (49) ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: int] :
% 59.04/8.66 | | | | | | | | ( ~ (v3 = 0) & and(v0, v1) = v2 & is_a_theorem(v2) = v3
% 59.04/8.66 | | | | | | | | & is_a_theorem(v1) = 0 & is_a_theorem(v0) = 0 & $i(v2)
% 59.04/8.66 | | | | | | | | & $i(v1) & $i(v0))
% 59.04/8.66 | | | | | | | |
% 59.04/8.66 | | | | | | | | DELTA: instantiating (49) with fresh symbols all_347_0,
% 59.04/8.66 | | | | | | | | all_347_1, all_347_2, all_347_3 gives:
% 59.04/8.66 | | | | | | | | (50) ~ (all_347_0 = 0) & and(all_347_3, all_347_2) =
% 59.04/8.66 | | | | | | | | all_347_1 & is_a_theorem(all_347_1) = all_347_0 &
% 59.04/8.66 | | | | | | | | is_a_theorem(all_347_2) = 0 & is_a_theorem(all_347_3) =
% 59.04/8.66 | | | | | | | | 0 & $i(all_347_1) & $i(all_347_2) & $i(all_347_3)
% 59.04/8.66 | | | | | | | |
% 59.04/8.66 | | | | | | | | ALPHA: (50) implies:
% 59.04/8.66 | | | | | | | | (51) $i(all_347_3)
% 59.04/8.66 | | | | | | | | (52) is_a_theorem(all_347_3) = 0
% 59.04/8.66 | | | | | | | |
% 59.04/8.66 | | | | | | | | GROUND_INST: instantiating (7) with all_129_1, all_136_0,
% 59.04/8.66 | | | | | | | | all_142_0, simplifying with (21), (26), (38), (39)
% 59.04/8.66 | | | | | | | | gives:
% 59.04/8.66 | | | | | | | | (53) all_142_0 = 0 | ? [v0: $i] : ? [v1: int] : ( ~ (v1 =
% 59.04/8.66 | | | | | | | | 0) & implies(all_129_1, all_136_0) = v0 &
% 59.04/8.66 | | | | | | | | is_a_theorem(v0) = v1 & $i(v0))
% 59.04/8.66 | | | | | | | |
% 59.04/8.66 | | | | | | | | GROUND_INST: instantiating (7) with all_347_3, all_235_1,
% 59.04/8.66 | | | | | | | | all_235_0, simplifying with (45), (46), (51), (52)
% 59.04/8.66 | | | | | | | | gives:
% 59.04/8.66 | | | | | | | | (54) all_235_0 = 0 | ? [v0: $i] : ? [v1: int] : ( ~ (v1 =
% 59.04/8.66 | | | | | | | | 0) & implies(all_347_3, all_235_1) = v0 &
% 59.04/8.66 | | | | | | | | is_a_theorem(v0) = v1 & $i(v0))
% 59.04/8.66 | | | | | | | |
% 59.04/8.66 | | | | | | | | BETA: splitting (53) gives:
% 59.04/8.66 | | | | | | | |
% 59.04/8.66 | | | | | | | | Case 1:
% 59.04/8.66 | | | | | | | | |
% 59.04/8.66 | | | | | | | | | (55) all_142_0 = 0
% 59.04/8.66 | | | | | | | | |
% 59.04/8.66 | | | | | | | | | REDUCE: (33), (55) imply:
% 59.04/8.66 | | | | | | | | | (56) $false
% 59.04/8.66 | | | | | | | | |
% 59.04/8.66 | | | | | | | | | CLOSE: (56) is inconsistent.
% 59.04/8.66 | | | | | | | | |
% 59.04/8.66 | | | | | | | | Case 2:
% 59.04/8.66 | | | | | | | | |
% 59.04/8.66 | | | | | | | | | (57) ? [v0: $i] : ? [v1: int] : ( ~ (v1 = 0) &
% 59.04/8.66 | | | | | | | | | implies(all_129_1, all_136_0) = v0 &
% 59.04/8.66 | | | | | | | | | is_a_theorem(v0) = v1 & $i(v0))
% 59.04/8.66 | | | | | | | | |
% 59.04/8.66 | | | | | | | | | DELTA: instantiating (57) with fresh symbols all_545_0,
% 59.04/8.66 | | | | | | | | | all_545_1 gives:
% 59.04/8.66 | | | | | | | | | (58) ~ (all_545_0 = 0) & implies(all_129_1, all_136_0) =
% 59.04/8.66 | | | | | | | | | all_545_1 & is_a_theorem(all_545_1) = all_545_0 &
% 59.04/8.66 | | | | | | | | | $i(all_545_1)
% 59.04/8.66 | | | | | | | | |
% 59.04/8.66 | | | | | | | | | ALPHA: (58) implies:
% 59.04/8.66 | | | | | | | | | (59) ~ (all_545_0 = 0)
% 59.04/8.66 | | | | | | | | | (60) is_a_theorem(all_545_1) = all_545_0
% 59.04/8.66 | | | | | | | | | (61) implies(all_129_1, all_136_0) = all_545_1
% 59.04/8.66 | | | | | | | | |
% 59.04/8.66 | | | | | | | | | BETA: splitting (54) gives:
% 59.04/8.66 | | | | | | | | |
% 59.04/8.66 | | | | | | | | | Case 1:
% 59.04/8.66 | | | | | | | | | |
% 59.04/8.66 | | | | | | | | | | (62) all_235_0 = 0
% 59.04/8.66 | | | | | | | | | |
% 59.04/8.66 | | | | | | | | | | REDUCE: (44), (62) imply:
% 59.04/8.66 | | | | | | | | | | (63) $false
% 59.04/8.66 | | | | | | | | | |
% 59.04/8.66 | | | | | | | | | | CLOSE: (63) is inconsistent.
% 59.04/8.66 | | | | | | | | | |
% 59.04/8.66 | | | | | | | | | Case 2:
% 59.04/8.66 | | | | | | | | | |
% 59.04/8.66 | | | | | | | | | |
% 59.04/8.66 | | | | | | | | | | GROUND_INST: instantiating (9) with all_136_0, all_129_1,
% 59.04/8.66 | | | | | | | | | | all_545_1, simplifying with (28), (39), (61)
% 59.04/8.66 | | | | | | | | | | gives:
% 59.04/8.66 | | | | | | | | | | (64) is_a_theorem(all_545_1) = 0
% 59.04/8.66 | | | | | | | | | |
% 59.04/8.66 | | | | | | | | | | GROUND_INST: instantiating (1) with all_545_0, 0, all_545_1,
% 59.04/8.66 | | | | | | | | | | simplifying with (60), (64) gives:
% 59.04/8.66 | | | | | | | | | | (65) all_545_0 = 0
% 59.04/8.66 | | | | | | | | | |
% 59.04/8.66 | | | | | | | | | | REDUCE: (59), (65) imply:
% 59.04/8.66 | | | | | | | | | | (66) $false
% 59.04/8.66 | | | | | | | | | |
% 59.04/8.66 | | | | | | | | | | CLOSE: (66) is inconsistent.
% 59.04/8.66 | | | | | | | | | |
% 59.04/8.66 | | | | | | | | | End of split
% 59.04/8.66 | | | | | | | | |
% 59.04/8.67 | | | | | | | | End of split
% 59.04/8.67 | | | | | | | |
% 59.04/8.67 | | | | | | | End of split
% 59.04/8.67 | | | | | | |
% 59.04/8.67 | | | | | | End of split
% 59.04/8.67 | | | | | |
% 59.04/8.67 | | | | | End of split
% 59.04/8.67 | | | | |
% 59.04/8.67 | | | | End of split
% 59.04/8.67 | | | |
% 59.04/8.67 | | | Case 2:
% 59.04/8.67 | | | |
% 59.04/8.67 | | | | (67) ~ axiom_M & ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3:
% 59.04/8.67 | | | | int] : ( ~ (v3 = 0) & necessarily(v0) = v1 & implies(v1, v0) =
% 59.04/8.67 | | | | v2 & is_a_theorem(v2) = v3 & $i(v2) & $i(v1) & $i(v0))
% 59.04/8.67 | | | |
% 59.04/8.67 | | | | ALPHA: (67) implies:
% 59.04/8.67 | | | | (68) ~ axiom_M
% 59.04/8.67 | | | |
% 59.04/8.67 | | | | PRED_UNIFY: (68), (km4b_axiom_M) imply:
% 59.04/8.67 | | | | (69) $false
% 59.04/8.67 | | | |
% 59.04/8.67 | | | | CLOSE: (69) is inconsistent.
% 59.04/8.67 | | | |
% 59.04/8.67 | | | End of split
% 59.04/8.67 | | |
% 59.04/8.67 | | Case 2:
% 59.04/8.67 | | |
% 59.04/8.67 | | | (70) ~ modus_ponens & ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ?
% 59.04/8.67 | | | [v3: int] : ( ~ (v3 = 0) & implies(v0, v1) = v2 & is_a_theorem(v2)
% 59.04/8.67 | | | = 0 & is_a_theorem(v1) = v3 & is_a_theorem(v0) = 0 & $i(v2) &
% 59.04/8.67 | | | $i(v1) & $i(v0))
% 59.04/8.67 | | |
% 59.04/8.67 | | | ALPHA: (70) implies:
% 59.04/8.67 | | | (71) ~ modus_ponens
% 59.04/8.67 | | |
% 59.04/8.67 | | | PRED_UNIFY: (71), (hilbert_modus_ponens) imply:
% 59.04/8.67 | | | (72) $false
% 59.04/8.67 | | |
% 59.04/8.67 | | | CLOSE: (72) is inconsistent.
% 59.04/8.67 | | |
% 59.04/8.67 | | End of split
% 59.04/8.67 | |
% 59.04/8.67 | End of split
% 59.04/8.67 |
% 59.04/8.67 End of proof
% 59.04/8.67
% 59.04/8.67 Sub-proof #1 shows that the following formulas are inconsistent:
% 59.04/8.67 ----------------------------------------------------------------
% 59.04/8.67 (1) ~ (all_142_0 = 0)
% 59.04/8.67 (2) ! [v0: MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: $i] :
% 59.04/8.67 (v1 = v0 | ~ (is_a_theorem(v2) = v1) | ~ (is_a_theorem(v2) = v0))
% 59.04/8.67 (3) all_142_0 = 0 | ? [v0: $i] : ? [v1: int] : ( ~ (v1 = 0) &
% 59.04/8.67 implies(all_129_1, all_136_0) = v0 & is_a_theorem(v0) = v1 & $i(v0))
% 59.04/8.67 (4) necessarily(all_136_0) = all_129_1
% 59.04/8.67 (5) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (necessarily(v0) = v1) | ~
% 59.04/8.67 (implies(v1, v0) = v2) | ~ $i(v0) | is_a_theorem(v2) = 0)
% 59.04/8.67 (6) $i(all_136_0)
% 59.04/8.67
% 59.04/8.67 Begin of proof
% 59.04/8.67 |
% 59.04/8.67 | BETA: splitting (3) gives:
% 59.04/8.67 |
% 59.04/8.67 | Case 1:
% 59.04/8.67 | |
% 59.04/8.67 | | (7) all_142_0 = 0
% 59.04/8.67 | |
% 59.04/8.67 | | REDUCE: (1), (7) imply:
% 59.04/8.67 | | (8) $false
% 59.04/8.67 | |
% 59.04/8.67 | | CLOSE: (8) is inconsistent.
% 59.04/8.67 | |
% 59.04/8.67 | Case 2:
% 59.04/8.67 | |
% 59.04/8.67 | | (9) ? [v0: $i] : ? [v1: int] : ( ~ (v1 = 0) & implies(all_129_1,
% 59.04/8.67 | | all_136_0) = v0 & is_a_theorem(v0) = v1 & $i(v0))
% 59.04/8.67 | |
% 59.04/8.67 | | DELTA: instantiating (9) with fresh symbols all_590_0, all_590_1 gives:
% 59.04/8.67 | | (10) ~ (all_590_0 = 0) & implies(all_129_1, all_136_0) = all_590_1 &
% 59.04/8.67 | | is_a_theorem(all_590_1) = all_590_0 & $i(all_590_1)
% 59.04/8.67 | |
% 59.04/8.67 | | ALPHA: (10) implies:
% 59.04/8.67 | | (11) ~ (all_590_0 = 0)
% 59.04/8.67 | | (12) is_a_theorem(all_590_1) = all_590_0
% 59.04/8.67 | | (13) implies(all_129_1, all_136_0) = all_590_1
% 59.04/8.67 | |
% 59.04/8.67 | | GROUND_INST: instantiating (5) with all_136_0, all_129_1, all_590_1,
% 59.04/8.67 | | simplifying with (4), (6), (13) gives:
% 59.04/8.67 | | (14) is_a_theorem(all_590_1) = 0
% 59.04/8.67 | |
% 59.04/8.67 | | GROUND_INST: instantiating (2) with all_590_0, 0, all_590_1, simplifying
% 59.04/8.67 | | with (12), (14) gives:
% 59.04/8.67 | | (15) all_590_0 = 0
% 59.04/8.67 | |
% 59.04/8.67 | | REDUCE: (11), (15) imply:
% 59.04/8.67 | | (16) $false
% 59.04/8.67 | |
% 59.04/8.67 | | CLOSE: (16) is inconsistent.
% 59.04/8.67 | |
% 59.04/8.67 | End of split
% 59.04/8.67 |
% 59.04/8.67 End of proof
% 59.04/8.67 % SZS output end Proof for theBenchmark
% 59.04/8.67
% 59.04/8.67 8057ms
%------------------------------------------------------------------------------