%------------------------------------------------------------------------------ % File : Princess---230619 % Problem : LCL483+1 : TPTP v9.3.1. Bugfixed v9.2.0. % Transfm : none % Format : tptp % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s % Computer : n006.cluster.edu % Model : x86_64 x86_64 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz % Memory : 8046.5625MB % OS : Linux 6.8.0-71-generic % CPULimit : 300s % WCLimit : 300s % DateTime : Mon Sep 7 12:54:14 PM UTC 2026 % Result : Theorem 17.71s 3.12s % Output : Proof 27.67s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.02 % Problem : LCL483+1 : TPTP v9.3.1. Bugfixed v9.2.0. % 0.00/0.04 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s % 0.07/0.34 % Computer : n006.cluster.edu % 0.07/0.34 % Model : x86_64 x86_64 % 0.07/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.07/0.34 % Memory : 8046.5625MB % 0.07/0.34 % OS : Linux 6.8.0-71-generic % 0.07/0.34 % CPULimit : 300 % 0.07/0.34 % WCLimit : 300 % 0.07/0.34 % DateTime : Sat Sep 5 10:29:46 UTC 2026 % 0.07/0.35 % CPUTime : % 0.11/0.36 OpenJDK 64-Bit Server VM warning: Options -Xverify:none and -noverify were deprecated in JDK 13 and will likely be removed in a future release. % 0.27/0.55 ________ _____ % 0.27/0.55 ___ __ \_________(_)________________________________ % 0.27/0.55 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/ % 0.27/0.55 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ ) % 0.27/0.55 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/ % 0.27/0.55 % 0.27/0.55 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic % 0.27/0.55 (2023-06-19) % 0.27/0.55 % 0.27/0.55 (c) Philipp Rümmer, 2009-2023 % 0.27/0.55 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen, % 0.27/0.55 Amanda Stjerna. % 0.27/0.55 Free software under BSD-3-Clause. % 0.27/0.55 % 0.27/0.55 For more information, visit http://www.philipp.ruemmer.org/princess.shtml % 0.27/0.55 % 0.27/0.55 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ... % 0.53/0.56 Running up to 7 provers in parallel. % 0.53/0.57 Prover 0: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1042961893 % 0.53/0.57 Prover 3: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1922548996 % 0.53/0.57 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1571432423 % 0.53/0.58 Prover 2: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMinimalAndEmpty -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1065072994 % 0.53/0.58 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=1868514696 % 0.53/0.58 Prover 5: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=complete -randomSeed=1259561288 % 0.53/0.58 Prover 6: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximalOutermost -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1399714365 % 4.23/1.30 Prover 1: Preprocessing ... % 4.23/1.31 Prover 4: Preprocessing ... % 4.23/1.36 Prover 2: Preprocessing ... % 4.23/1.36 Prover 5: Preprocessing ... % 4.23/1.36 Prover 3: Preprocessing ... % 4.23/1.36 Prover 0: Preprocessing ... % 4.23/1.36 Prover 6: Preprocessing ... % 10.93/2.20 Prover 6: Constructing countermodel ... % 11.23/2.20 Prover 5: Proving ... % 11.23/2.27 Prover 3: Constructing countermodel ... % 11.23/2.27 Prover 1: Constructing countermodel ... % 12.00/2.30 Prover 4: Constructing countermodel ... % 12.80/2.47 Prover 0: Proving ... % 14.36/2.66 Prover 2: Proving ... % 17.71/3.11 Prover 0: proved (2544ms) % 17.71/3.12 % 17.71/3.12 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % 17.71/3.12 % 17.71/3.12 Prover 6: stopped % 17.71/3.13 Prover 5: stopped % 18.51/3.14 Prover 2: stopped % 18.51/3.14 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470 % 18.51/3.14 Prover 3: stopped % 18.51/3.15 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089 % 18.51/3.15 Prover 10: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=919308125 % 18.51/3.15 Prover 11: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1509710984 % 18.51/3.15 Prover 13: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=1138197443 % 18.51/3.18 Prover 7: Preprocessing ... % 19.84/3.31 Prover 11: Preprocessing ... % 19.84/3.32 Prover 8: Preprocessing ... % 19.84/3.33 Prover 10: Preprocessing ... % 19.84/3.35 Prover 13: Preprocessing ... % 19.84/3.39 Prover 1: gave up % 19.84/3.40 Prover 16: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=completeFrugal -randomSeed=-2043353683 % 21.46/3.53 Prover 7: Constructing countermodel ... % 21.46/3.56 Prover 16: Preprocessing ... % 22.23/3.66 Prover 10: Constructing countermodel ... % 23.03/3.75 Prover 8: Warning: ignoring some quantifiers % 23.03/3.77 Prover 13: Constructing countermodel ... % 23.03/3.77 Prover 8: Constructing countermodel ... % 23.81/3.89 Prover 11: Constructing countermodel ... % 24.60/3.90 Prover 16: Warning: ignoring some quantifiers % 24.60/3.92 Prover 16: Constructing countermodel ... % 25.86/4.11 Prover 4: Found proof (size 83) % 25.86/4.11 Prover 4: proved (3540ms) % 25.86/4.12 Prover 11: stopped % 25.86/4.12 Prover 8: stopped % 25.86/4.12 Prover 13: stopped % 25.86/4.12 Prover 10: stopped % 25.86/4.12 Prover 16: stopped % 25.86/4.12 Prover 7: stopped % 25.86/4.12 % 25.86/4.12 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % 25.86/4.12 % 25.86/4.15 % SZS output start Proof for theBenchmark % 25.86/4.15 Assumptions after simplification: % 25.86/4.15 --------------------------------- % 25.86/4.16 % 25.86/4.16 (hilbert_modus_tollens) % 25.86/4.16 ~ modus_tollens % 25.86/4.16 % 25.86/4.16 (hilbert_op_implies_and) % 25.86/4.16 op_implies_and % 25.86/4.16 % 25.86/4.16 (hilbert_op_or) % 25.86/4.16 op_or % 25.86/4.16 % 26.52/4.16 (modus_ponens) % 26.69/4.21 ? [v0: $i] : ? [v1: $i] : ? [v2: int] : ? [v3: $i] : ? [v4: int] : ? % 26.69/4.21 [v5: int] : ($i(v1) & $i(v0) & ((v4 = 0 & v2 = 0 & ~ (v5 = 0) & implies(v0, % 26.69/4.21 v1) = v3 & is_a_theorem(v3) = 0 & is_a_theorem(v1) = v5 & % 26.69/4.21 is_a_theorem(v0) = 0 & $i(v3) & ~ modus_ponens) | (modus_ponens & ! % 26.69/4.21 [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (implies(v6, v7) = v8) | ~ % 26.69/4.21 $i(v7) | ~ $i(v6) | ? [v9: any] : ? [v10: any] : ? [v11: any] : % 26.69/4.21 (is_a_theorem(v8) = v10 & is_a_theorem(v7) = v11 & is_a_theorem(v6) = % 26.69/4.21 v9 & ( ~ (v10 = 0) | ~ (v9 = 0) | v11 = 0)))))) % 26.69/4.21 % 26.69/4.21 (modus_tollens) % 26.69/4.23 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ? [v4: $i] : ? [v5: % 26.69/4.23 $i] : ? [v6: $i] : ? [v7: int] : ($i(v1) & $i(v0) & (( ~ (v7 = 0) & % 26.69/4.23 not(v1) = v2 & not(v0) = v3 & implies(v4, v5) = v6 & implies(v2, v3) = % 26.69/4.23 v4 & implies(v0, v1) = v5 & is_a_theorem(v6) = v7 & $i(v6) & $i(v5) & % 26.69/4.23 $i(v4) & $i(v3) & $i(v2) & ~ modus_tollens) | (modus_tollens & ! [v8: % 26.69/4.23 $i] : ! [v9: $i] : ! [v10: $i] : ! [v11: $i] : ! [v12: $i] : ( ~ % 26.69/4.23 (not(v9) = v10) | ~ (not(v8) = v11) | ~ (implies(v10, v11) = v12) | % 26.69/4.23 ~ $i(v9) | ~ $i(v8) | ? [v13: $i] : ? [v14: $i] : (implies(v12, % 26.69/4.23 v13) = v14 & implies(v8, v9) = v13 & is_a_theorem(v14) = 0 & % 26.69/4.23 $i(v14) & $i(v13))) & ! [v8: $i] : ! [v9: $i] : ! [v10: $i] : ( ~ % 26.69/4.23 (implies(v8, v9) = v10) | ~ $i(v9) | ~ $i(v8) | ? [v11: $i] : ? % 26.69/4.23 [v12: $i] : ? [v13: $i] : ? [v14: $i] : (not(v9) = v11 & not(v8) = % 26.69/4.23 v12 & implies(v13, v10) = v14 & implies(v11, v12) = v13 & % 26.69/4.23 is_a_theorem(v14) = 0 & $i(v14) & $i(v13) & $i(v12) & $i(v11)))))) % 26.69/4.23 % 26.69/4.23 (op_implies_and) % 26.69/4.24 ~ op_implies_and | ( ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( % 26.69/4.24 ~ (and(v0, v2) = v3) | ~ (not(v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v4: % 26.69/4.24 $i] : (not(v3) = v4 & implies(v0, v1) = v4 & $i(v4))) & ! [v0: $i] : ! % 26.69/4.24 [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) % 26.69/4.24 | ? [v3: $i] : ? [v4: $i] : (and(v0, v3) = v4 & not(v4) = v2 & not(v1) = % 26.69/4.24 v3 & $i(v4) & $i(v3) & $i(v2)))) % 26.69/4.24 % 26.69/4.24 (op_implies_or) % 26.69/4.24 ~ op_implies_or | ( ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( % 26.69/4.24 ~ (or(v2, v1) = v3) | ~ (not(v0) = v2) | ~ $i(v1) | ~ $i(v0) | % 26.69/4.24 (implies(v0, v1) = v3 & $i(v3))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] % 26.69/4.24 : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : (or(v3, % 26.69/4.24 v1) = v2 & not(v0) = v3 & $i(v3) & $i(v2)))) % 26.69/4.24 % 26.69/4.24 (op_or) % 26.69/4.25 ~ op_or | ( ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: % 26.69/4.25 $i] : ( ~ (and(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = v2) | ~ % 26.69/4.25 $i(v1) | ~ $i(v0) | ? [v5: $i] : (or(v0, v1) = v5 & not(v4) = v5 & % 26.69/4.25 $i(v5))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (or(v0, v1) = % 26.69/4.25 v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : ? [v5: $i] : % 26.69/4.25 (and(v3, v4) = v5 & not(v5) = v2 & not(v1) = v4 & not(v0) = v3 & $i(v5) & % 26.69/4.25 $i(v4) & $i(v3) & $i(v2)))) % 26.69/4.25 % 26.69/4.25 (principia_modus_ponens) % 26.69/4.25 modus_ponens % 26.69/4.25 % 26.69/4.25 (principia_op_implies_or) % 26.69/4.25 op_implies_or % 26.69/4.25 % 26.69/4.25 (principia_r3) % 26.69/4.25 r3 % 26.69/4.25 % 26.69/4.25 (principia_r4) % 26.69/4.25 r4 % 26.69/4.25 % 26.69/4.25 (r3) % 26.69/4.26 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ? [v4: $i] : ? [v5: % 26.69/4.26 int] : ($i(v1) & $i(v0) & (( ~ (v5 = 0) & or(v1, v0) = v3 & or(v0, v1) = v2 % 26.69/4.26 & implies(v2, v3) = v4 & is_a_theorem(v4) = v5 & $i(v4) & $i(v3) & % 26.69/4.26 $i(v2) & ~ r3) | (r3 & ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ % 26.69/4.26 (or(v7, v6) = v8) | ~ $i(v7) | ~ $i(v6) | ? [v9: $i] : ? [v10: $i] % 26.69/4.26 : (or(v6, v7) = v9 & implies(v9, v8) = v10 & is_a_theorem(v10) = 0 & % 26.69/4.26 $i(v10) & $i(v9))) & ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ % 26.69/4.26 (or(v6, v7) = v8) | ~ $i(v7) | ~ $i(v6) | ? [v9: $i] : ? [v10: $i] % 26.69/4.26 : (or(v7, v6) = v9 & implies(v8, v9) = v10 & is_a_theorem(v10) = 0 & % 26.69/4.26 $i(v10) & $i(v9)))))) % 26.69/4.26 % 26.69/4.26 (r4) % 26.69/4.27 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ? [v4: $i] : ? [v5: % 26.69/4.27 $i] : ? [v6: $i] : ? [v7: $i] : ? [v8: int] : ($i(v2) & $i(v1) & $i(v0) & % 26.69/4.27 (( ~ (v8 = 0) & or(v1, v5) = v6 & or(v1, v2) = v3 & or(v0, v3) = v4 & or(v0, % 26.69/4.27 v2) = v5 & implies(v4, v6) = v7 & is_a_theorem(v7) = v8 & $i(v7) & % 26.69/4.27 $i(v6) & $i(v5) & $i(v4) & $i(v3) & ~ r4) | (r4 & ! [v9: $i] : ! % 26.69/4.27 [v10: $i] : ! [v11: $i] : ! [v12: $i] : ! [v13: $i] : ( ~ (or(v10, % 26.69/4.27 v12) = v13) | ~ (or(v9, v11) = v12) | ~ $i(v11) | ~ $i(v10) | % 26.69/4.27 ~ $i(v9) | ? [v14: $i] : ? [v15: $i] : ? [v16: $i] : (or(v10, v11) % 26.69/4.27 = v14 & or(v9, v14) = v15 & implies(v15, v13) = v16 & % 26.69/4.27 is_a_theorem(v16) = 0 & $i(v16) & $i(v15) & $i(v14))) & ! [v9: $i] % 26.69/4.27 : ! [v10: $i] : ! [v11: $i] : ! [v12: $i] : ! [v13: $i] : ( ~ % 26.69/4.27 (or(v10, v11) = v12) | ~ (or(v9, v12) = v13) | ~ $i(v11) | ~ % 26.69/4.27 $i(v10) | ~ $i(v9) | ? [v14: $i] : ? [v15: $i] : ? [v16: $i] : % 26.69/4.27 (or(v10, v14) = v15 & or(v9, v11) = v14 & implies(v13, v15) = v16 & % 26.69/4.27 is_a_theorem(v16) = 0 & $i(v16) & $i(v15) & $i(v14)))))) % 26.69/4.27 % 26.69/4.27 (function-axioms) % 26.69/4.27 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (or(v3, % 26.69/4.27 v2) = v1) | ~ (or(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: % 26.69/4.27 $i] : ! [v3: $i] : (v1 = v0 | ~ (and(v3, v2) = v1) | ~ (and(v3, v2) = % 26.69/4.27 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | % 26.69/4.27 ~ (equiv(v3, v2) = v1) | ~ (equiv(v3, v2) = v0)) & ! [v0: $i] : ! [v1: % 26.69/4.27 $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (implies(v3, v2) = v1) | ~ % 26.69/4.27 (implies(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 % 26.69/4.27 | ~ (not(v2) = v1) | ~ (not(v2) = v0)) & ! [v0: MultipleValueBool] : ! % 26.69/4.27 [v1: MultipleValueBool] : ! [v2: $i] : (v1 = v0 | ~ (is_a_theorem(v2) = v1) % 26.69/4.27 | ~ (is_a_theorem(v2) = v0)) % 26.69/4.27 % 26.69/4.27 Further assumptions not needed in the proof: % 26.69/4.27 -------------------------------------------- % 26.69/4.27 and_1, and_2, and_3, cn1, cn2, cn3, equivalence_1, equivalence_2, equivalence_3, % 26.69/4.27 hilbert_op_equiv, implies_1, implies_2, implies_3, kn1, kn2, kn3, op_and, % 26.69/4.27 op_equiv, or_1, or_2, or_3, principia_op_and, principia_op_equiv, principia_r1, % 26.69/4.27 principia_r2, principia_r5, r1, r2, r5, substitution_of_equivalents, % 26.69/4.27 use_substitution_of_equivalents % 26.69/4.27 % 26.69/4.27 Those formulas are unsatisfiable: % 26.69/4.27 --------------------------------- % 26.69/4.27 % 26.69/4.27 Begin of proof % 26.69/4.28 | % 26.69/4.28 | ALPHA: (function-axioms) implies: % 26.69/4.28 | (1) ! [v0: MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: $i] : % 26.69/4.28 | (v1 = v0 | ~ (is_a_theorem(v2) = v1) | ~ (is_a_theorem(v2) = v0)) % 26.69/4.28 | (2) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (not(v2) = v1) % 26.69/4.28 | | ~ (not(v2) = v0)) % 26.69/4.28 | (3) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ % 26.69/4.28 | (implies(v3, v2) = v1) | ~ (implies(v3, v2) = v0)) % 26.69/4.28 | (4) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ % 26.69/4.28 | (or(v3, v2) = v1) | ~ (or(v3, v2) = v0)) % 26.69/4.28 | % 26.69/4.28 | DELTA: instantiating (modus_ponens) with fresh symbols all_32_0, all_32_1, % 26.69/4.28 | all_32_2, all_32_3, all_32_4, all_32_5 gives: % 26.69/4.29 | (5) $i(all_32_4) & $i(all_32_5) & ((all_32_1 = 0 & all_32_3 = 0 & ~ % 26.69/4.29 | (all_32_0 = 0) & implies(all_32_5, all_32_4) = all_32_2 & % 26.69/4.29 | is_a_theorem(all_32_2) = 0 & is_a_theorem(all_32_4) = all_32_0 & % 26.69/4.29 | is_a_theorem(all_32_5) = 0 & $i(all_32_2) & ~ modus_ponens) | % 26.69/4.29 | (modus_ponens & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ % 26.69/4.29 | (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: any] : % 26.69/4.29 | ? [v4: any] : ? [v5: any] : (is_a_theorem(v2) = v4 & % 26.69/4.29 | is_a_theorem(v1) = v5 & is_a_theorem(v0) = v3 & ( ~ (v4 = 0) | % 26.69/4.29 | ~ (v3 = 0) | v5 = 0))))) % 26.69/4.29 | % 26.69/4.29 | ALPHA: (5) implies: % 26.69/4.29 | (6) (all_32_1 = 0 & all_32_3 = 0 & ~ (all_32_0 = 0) & implies(all_32_5, % 26.69/4.29 | all_32_4) = all_32_2 & is_a_theorem(all_32_2) = 0 & % 26.69/4.29 | is_a_theorem(all_32_4) = all_32_0 & is_a_theorem(all_32_5) = 0 & % 26.69/4.29 | $i(all_32_2) & ~ modus_ponens) | (modus_ponens & ! [v0: $i] : ! % 26.69/4.29 | [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~ % 26.69/4.29 | $i(v0) | ? [v3: any] : ? [v4: any] : ? [v5: any] : % 26.69/4.29 | (is_a_theorem(v2) = v4 & is_a_theorem(v1) = v5 & is_a_theorem(v0) = % 26.69/4.29 | v3 & ( ~ (v4 = 0) | ~ (v3 = 0) | v5 = 0)))) % 26.69/4.29 | % 26.69/4.29 | DELTA: instantiating (r3) with fresh symbols all_36_0, all_36_1, all_36_2, % 26.69/4.29 | all_36_3, all_36_4, all_36_5 gives: % 27.18/4.30 | (7) $i(all_36_4) & $i(all_36_5) & (( ~ (all_36_0 = 0) & or(all_36_4, % 27.18/4.30 | all_36_5) = all_36_2 & or(all_36_5, all_36_4) = all_36_3 & % 27.18/4.30 | implies(all_36_3, all_36_2) = all_36_1 & is_a_theorem(all_36_1) = % 27.18/4.30 | all_36_0 & $i(all_36_1) & $i(all_36_2) & $i(all_36_3) & ~ r3) | % 27.18/4.30 | (r3 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (or(v1, v0) = v2) % 27.18/4.30 | | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : (or(v0, v1) % 27.18/4.30 | = v3 & implies(v3, v2) = v4 & is_a_theorem(v4) = 0 & $i(v4) & % 27.18/4.30 | $i(v3))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ % 27.18/4.30 | (or(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: % 27.18/4.30 | $i] : (or(v1, v0) = v3 & implies(v2, v3) = v4 & % 27.18/4.30 | is_a_theorem(v4) = 0 & $i(v4) & $i(v3))))) % 27.18/4.30 | % 27.18/4.30 | ALPHA: (7) implies: % 27.18/4.31 | (8) ( ~ (all_36_0 = 0) & or(all_36_4, all_36_5) = all_36_2 & or(all_36_5, % 27.18/4.31 | all_36_4) = all_36_3 & implies(all_36_3, all_36_2) = all_36_1 & % 27.18/4.31 | is_a_theorem(all_36_1) = all_36_0 & $i(all_36_1) & $i(all_36_2) & % 27.18/4.31 | $i(all_36_3) & ~ r3) | (r3 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] % 27.18/4.31 | : ( ~ (or(v1, v0) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? % 27.18/4.31 | [v4: $i] : (or(v0, v1) = v3 & implies(v3, v2) = v4 & % 27.18/4.31 | is_a_theorem(v4) = 0 & $i(v4) & $i(v3))) & ! [v0: $i] : ! [v1: % 27.18/4.31 | $i] : ! [v2: $i] : ( ~ (or(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | % 27.18/4.31 | ? [v3: $i] : ? [v4: $i] : (or(v1, v0) = v3 & implies(v2, v3) = v4 % 27.18/4.31 | & is_a_theorem(v4) = 0 & $i(v4) & $i(v3)))) % 27.18/4.31 | % 27.18/4.31 | DELTA: instantiating (modus_tollens) with fresh symbols all_50_0, all_50_1, % 27.18/4.31 | all_50_2, all_50_3, all_50_4, all_50_5, all_50_6, all_50_7 gives: % 27.18/4.32 | (9) $i(all_50_6) & $i(all_50_7) & (( ~ (all_50_0 = 0) & not(all_50_6) = % 27.18/4.32 | all_50_5 & not(all_50_7) = all_50_4 & implies(all_50_3, all_50_2) = % 27.18/4.32 | all_50_1 & implies(all_50_5, all_50_4) = all_50_3 & % 27.18/4.32 | implies(all_50_7, all_50_6) = all_50_2 & is_a_theorem(all_50_1) = % 27.18/4.32 | all_50_0 & $i(all_50_1) & $i(all_50_2) & $i(all_50_3) & % 27.18/4.32 | $i(all_50_4) & $i(all_50_5) & ~ modus_tollens) | (modus_tollens & % 27.18/4.32 | ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] % 27.18/4.32 | : ( ~ (not(v1) = v2) | ~ (not(v0) = v3) | ~ (implies(v2, v3) = % 27.18/4.32 | v4) | ~ $i(v1) | ~ $i(v0) | ? [v5: $i] : ? [v6: $i] : % 27.18/4.32 | (implies(v4, v5) = v6 & implies(v0, v1) = v5 & is_a_theorem(v6) = % 27.18/4.32 | 0 & $i(v6) & $i(v5))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] % 27.18/4.32 | : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] % 27.18/4.32 | : ? [v4: $i] : ? [v5: $i] : ? [v6: $i] : (not(v1) = v3 & % 27.18/4.32 | not(v0) = v4 & implies(v5, v2) = v6 & implies(v3, v4) = v5 & % 27.18/4.32 | is_a_theorem(v6) = 0 & $i(v6) & $i(v5) & $i(v4) & $i(v3))))) % 27.18/4.32 | % 27.18/4.32 | ALPHA: (9) implies: % 27.18/4.32 | (10) $i(all_50_7) % 27.18/4.32 | (11) $i(all_50_6) % 27.18/4.33 | (12) ( ~ (all_50_0 = 0) & not(all_50_6) = all_50_5 & not(all_50_7) = % 27.18/4.33 | all_50_4 & implies(all_50_3, all_50_2) = all_50_1 & % 27.18/4.33 | implies(all_50_5, all_50_4) = all_50_3 & implies(all_50_7, all_50_6) % 27.18/4.33 | = all_50_2 & is_a_theorem(all_50_1) = all_50_0 & $i(all_50_1) & % 27.18/4.33 | $i(all_50_2) & $i(all_50_3) & $i(all_50_4) & $i(all_50_5) & ~ % 27.18/4.33 | modus_tollens) | (modus_tollens & ! [v0: $i] : ! [v1: $i] : ! % 27.18/4.33 | [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( ~ (not(v1) = v2) | ~ % 27.18/4.33 | (not(v0) = v3) | ~ (implies(v2, v3) = v4) | ~ $i(v1) | ~ $i(v0) % 27.18/4.33 | | ? [v5: $i] : ? [v6: $i] : (implies(v4, v5) = v6 & implies(v0, % 27.18/4.33 | v1) = v5 & is_a_theorem(v6) = 0 & $i(v6) & $i(v5))) & ! [v0: % 27.18/4.33 | $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~ % 27.18/4.33 | $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : ? [v5: $i] : ? % 27.18/4.33 | [v6: $i] : (not(v1) = v3 & not(v0) = v4 & implies(v5, v2) = v6 & % 27.18/4.33 | implies(v3, v4) = v5 & is_a_theorem(v6) = 0 & $i(v6) & $i(v5) & % 27.18/4.33 | $i(v4) & $i(v3)))) % 27.18/4.33 | % 27.18/4.33 | DELTA: instantiating (r4) with fresh symbols all_52_0, all_52_1, all_52_2, % 27.18/4.33 | all_52_3, all_52_4, all_52_5, all_52_6, all_52_7, all_52_8 gives: % 27.18/4.34 | (13) $i(all_52_6) & $i(all_52_7) & $i(all_52_8) & (( ~ (all_52_0 = 0) & % 27.18/4.34 | or(all_52_7, all_52_3) = all_52_2 & or(all_52_7, all_52_6) = % 27.18/4.34 | all_52_5 & or(all_52_8, all_52_5) = all_52_4 & or(all_52_8, % 27.18/4.34 | all_52_6) = all_52_3 & implies(all_52_4, all_52_2) = all_52_1 & % 27.18/4.34 | is_a_theorem(all_52_1) = all_52_0 & $i(all_52_1) & $i(all_52_2) & % 27.18/4.34 | $i(all_52_3) & $i(all_52_4) & $i(all_52_5) & ~ r4) | (r4 & ! % 27.18/4.34 | [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : % 27.18/4.34 | ( ~ (or(v1, v3) = v4) | ~ (or(v0, v2) = v3) | ~ $i(v2) | ~ % 27.18/4.34 | $i(v1) | ~ $i(v0) | ? [v5: $i] : ? [v6: $i] : ? [v7: $i] : % 27.18/4.34 | (or(v1, v2) = v5 & or(v0, v5) = v6 & implies(v6, v4) = v7 & % 27.18/4.34 | is_a_theorem(v7) = 0 & $i(v7) & $i(v6) & $i(v5))) & ! [v0: % 27.18/4.34 | $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( % 27.18/4.34 | ~ (or(v1, v2) = v3) | ~ (or(v0, v3) = v4) | ~ $i(v2) | ~ % 27.18/4.34 | $i(v1) | ~ $i(v0) | ? [v5: $i] : ? [v6: $i] : ? [v7: $i] : % 27.18/4.34 | (or(v1, v5) = v6 & or(v0, v2) = v5 & implies(v4, v6) = v7 & % 27.18/4.34 | is_a_theorem(v7) = 0 & $i(v7) & $i(v6) & $i(v5))))) % 27.18/4.34 | % 27.18/4.34 | ALPHA: (13) implies: % 27.18/4.35 | (14) ( ~ (all_52_0 = 0) & or(all_52_7, all_52_3) = all_52_2 & or(all_52_7, % 27.18/4.35 | all_52_6) = all_52_5 & or(all_52_8, all_52_5) = all_52_4 & % 27.18/4.35 | or(all_52_8, all_52_6) = all_52_3 & implies(all_52_4, all_52_2) = % 27.18/4.35 | all_52_1 & is_a_theorem(all_52_1) = all_52_0 & $i(all_52_1) & % 27.18/4.35 | $i(all_52_2) & $i(all_52_3) & $i(all_52_4) & $i(all_52_5) & ~ r4) | % 27.18/4.35 | (r4 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: % 27.18/4.35 | $i] : ( ~ (or(v1, v3) = v4) | ~ (or(v0, v2) = v3) | ~ $i(v2) | % 27.18/4.35 | ~ $i(v1) | ~ $i(v0) | ? [v5: $i] : ? [v6: $i] : ? [v7: $i] : % 27.18/4.35 | (or(v1, v2) = v5 & or(v0, v5) = v6 & implies(v6, v4) = v7 & % 27.18/4.35 | is_a_theorem(v7) = 0 & $i(v7) & $i(v6) & $i(v5))) & ! [v0: $i] % 27.18/4.35 | : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( ~ % 27.18/4.35 | (or(v1, v2) = v3) | ~ (or(v0, v3) = v4) | ~ $i(v2) | ~ $i(v1) | % 27.18/4.35 | ~ $i(v0) | ? [v5: $i] : ? [v6: $i] : ? [v7: $i] : (or(v1, v5) % 27.18/4.35 | = v6 & or(v0, v2) = v5 & implies(v4, v6) = v7 & is_a_theorem(v7) % 27.18/4.35 | = 0 & $i(v7) & $i(v6) & $i(v5)))) % 27.18/4.35 | % 27.18/4.35 | BETA: splitting (12) gives: % 27.18/4.35 | % 27.18/4.35 | Case 1: % 27.18/4.35 | | % 27.18/4.35 | | (15) ~ (all_50_0 = 0) & not(all_50_6) = all_50_5 & not(all_50_7) = % 27.18/4.35 | | all_50_4 & implies(all_50_3, all_50_2) = all_50_1 & % 27.18/4.35 | | implies(all_50_5, all_50_4) = all_50_3 & implies(all_50_7, all_50_6) % 27.18/4.35 | | = all_50_2 & is_a_theorem(all_50_1) = all_50_0 & $i(all_50_1) & % 27.18/4.35 | | $i(all_50_2) & $i(all_50_3) & $i(all_50_4) & $i(all_50_5) & ~ % 27.18/4.35 | | modus_tollens % 27.18/4.35 | | % 27.18/4.35 | | ALPHA: (15) implies: % 27.18/4.35 | | (16) ~ (all_50_0 = 0) % 27.18/4.35 | | (17) $i(all_50_5) % 27.18/4.35 | | (18) $i(all_50_4) % 27.18/4.35 | | (19) $i(all_50_3) % 27.18/4.35 | | (20) $i(all_50_2) % 27.18/4.35 | | (21) is_a_theorem(all_50_1) = all_50_0 % 27.18/4.36 | | (22) implies(all_50_7, all_50_6) = all_50_2 % 27.18/4.36 | | (23) implies(all_50_5, all_50_4) = all_50_3 % 27.18/4.36 | | (24) implies(all_50_3, all_50_2) = all_50_1 % 27.18/4.36 | | (25) not(all_50_7) = all_50_4 % 27.18/4.36 | | (26) not(all_50_6) = all_50_5 % 27.18/4.36 | | % 27.18/4.36 | | BETA: splitting (op_or) gives: % 27.18/4.36 | | % 27.18/4.36 | | Case 1: % 27.18/4.36 | | | % 27.18/4.36 | | | (27) ~ op_or % 27.18/4.36 | | | % 27.18/4.36 | | | PRED_UNIFY: (27), (hilbert_op_or) imply: % 27.18/4.36 | | | (28) $false % 27.18/4.36 | | | % 27.18/4.36 | | | CLOSE: (28) is inconsistent. % 27.18/4.36 | | | % 27.18/4.36 | | Case 2: % 27.18/4.36 | | | % 27.18/4.37 | | | (29) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: % 27.18/4.37 | | | $i] : ( ~ (and(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = % 27.18/4.37 | | | v2) | ~ $i(v1) | ~ $i(v0) | ? [v5: $i] : (or(v0, v1) = v5 & % 27.18/4.37 | | | not(v4) = v5 & $i(v5))) & ! [v0: $i] : ! [v1: $i] : ! [v2: % 27.18/4.37 | | | $i] : ( ~ (or(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: % 27.18/4.37 | | | $i] : ? [v4: $i] : ? [v5: $i] : (and(v3, v4) = v5 & not(v5) % 27.18/4.37 | | | = v2 & not(v1) = v4 & not(v0) = v3 & $i(v5) & $i(v4) & $i(v3) % 27.18/4.37 | | | & $i(v2))) % 27.18/4.37 | | | % 27.18/4.37 | | | ALPHA: (29) implies: % 27.18/4.37 | | | (30) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: % 27.18/4.37 | | | $i] : ( ~ (and(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = % 27.18/4.37 | | | v2) | ~ $i(v1) | ~ $i(v0) | ? [v5: $i] : (or(v0, v1) = v5 & % 27.18/4.37 | | | not(v4) = v5 & $i(v5))) % 27.18/4.37 | | | % 27.18/4.37 | | | BETA: splitting (op_implies_and) gives: % 27.18/4.37 | | | % 27.18/4.37 | | | Case 1: % 27.18/4.37 | | | | % 27.18/4.37 | | | | (31) ~ op_implies_and % 27.18/4.37 | | | | % 27.18/4.37 | | | | PRED_UNIFY: (31), (hilbert_op_implies_and) imply: % 27.18/4.37 | | | | (32) $false % 27.18/4.37 | | | | % 27.18/4.37 | | | | CLOSE: (32) is inconsistent. % 27.18/4.37 | | | | % 27.18/4.37 | | | Case 2: % 27.18/4.37 | | | | % 27.18/4.38 | | | | (33) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~ % 27.18/4.38 | | | | (and(v0, v2) = v3) | ~ (not(v1) = v2) | ~ $i(v1) | ~ $i(v0) % 27.18/4.38 | | | | | ? [v4: $i] : (not(v3) = v4 & implies(v0, v1) = v4 & % 27.18/4.38 | | | | $i(v4))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ % 27.18/4.38 | | | | (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : % 27.18/4.38 | | | | ? [v4: $i] : (and(v0, v3) = v4 & not(v4) = v2 & not(v1) = v3 % 27.18/4.38 | | | | & $i(v4) & $i(v3) & $i(v2))) % 27.18/4.38 | | | | % 27.18/4.38 | | | | ALPHA: (33) implies: % 27.18/4.38 | | | | (34) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = % 27.18/4.38 | | | | v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : % 27.18/4.38 | | | | (and(v0, v3) = v4 & not(v4) = v2 & not(v1) = v3 & $i(v4) & % 27.18/4.38 | | | | $i(v3) & $i(v2))) % 27.18/4.38 | | | | % 27.18/4.38 | | | | BETA: splitting (op_implies_or) gives: % 27.18/4.38 | | | | % 27.18/4.38 | | | | Case 1: % 27.18/4.38 | | | | | % 27.18/4.38 | | | | | (35) ~ op_implies_or % 27.18/4.38 | | | | | % 27.18/4.38 | | | | | PRED_UNIFY: (35), (principia_op_implies_or) imply: % 27.18/4.38 | | | | | (36) $false % 27.18/4.38 | | | | | % 27.18/4.38 | | | | | CLOSE: (36) is inconsistent. % 27.18/4.38 | | | | | % 27.18/4.38 | | | | Case 2: % 27.18/4.38 | | | | | % 27.18/4.38 | | | | | (37) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~ % 27.18/4.38 | | | | | (or(v2, v1) = v3) | ~ (not(v0) = v2) | ~ $i(v1) | ~ % 27.18/4.38 | | | | | $i(v0) | (implies(v0, v1) = v3 & $i(v3))) & ! [v0: $i] : ! % 27.18/4.38 | | | | | [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~ % 27.18/4.38 | | | | | $i(v1) | ~ $i(v0) | ? [v3: $i] : (or(v3, v1) = v2 & % 27.18/4.38 | | | | | not(v0) = v3 & $i(v3) & $i(v2))) % 27.18/4.38 | | | | | % 27.18/4.38 | | | | | ALPHA: (37) implies: % 27.18/4.39 | | | | | (38) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, v1) % 27.18/4.39 | | | | | = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : (or(v3, v1) % 27.18/4.39 | | | | | = v2 & not(v0) = v3 & $i(v3) & $i(v2))) % 27.18/4.39 | | | | | % 27.18/4.39 | | | | | BETA: splitting (6) gives: % 27.18/4.39 | | | | | % 27.18/4.39 | | | | | Case 1: % 27.18/4.39 | | | | | | % 27.18/4.39 | | | | | | (39) all_32_1 = 0 & all_32_3 = 0 & ~ (all_32_0 = 0) & % 27.18/4.39 | | | | | | implies(all_32_5, all_32_4) = all_32_2 & % 27.18/4.39 | | | | | | is_a_theorem(all_32_2) = 0 & is_a_theorem(all_32_4) = % 27.18/4.39 | | | | | | all_32_0 & is_a_theorem(all_32_5) = 0 & $i(all_32_2) & ~ % 27.18/4.39 | | | | | | modus_ponens % 27.18/4.39 | | | | | | % 27.18/4.39 | | | | | | ALPHA: (39) implies: % 27.18/4.39 | | | | | | (40) ~ modus_ponens % 27.18/4.39 | | | | | | % 27.18/4.39 | | | | | | PRED_UNIFY: (40), (principia_modus_ponens) imply: % 27.18/4.39 | | | | | | (41) $false % 27.18/4.39 | | | | | | % 27.18/4.39 | | | | | | CLOSE: (41) is inconsistent. % 27.18/4.39 | | | | | | % 27.18/4.39 | | | | | Case 2: % 27.18/4.39 | | | | | | % 27.18/4.39 | | | | | | (42) modus_ponens & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ % 27.18/4.39 | | | | | | (implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: % 27.18/4.39 | | | | | | any] : ? [v4: any] : ? [v5: any] : (is_a_theorem(v2) = % 27.18/4.39 | | | | | | v4 & is_a_theorem(v1) = v5 & is_a_theorem(v0) = v3 & ( ~ % 27.18/4.39 | | | | | | (v4 = 0) | ~ (v3 = 0) | v5 = 0))) % 27.18/4.39 | | | | | | % 27.18/4.39 | | | | | | ALPHA: (42) implies: % 27.18/4.39 | | | | | | (43) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (implies(v0, % 27.18/4.39 | | | | | | v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: any] : ? % 27.18/4.39 | | | | | | [v4: any] : ? [v5: any] : (is_a_theorem(v2) = v4 & % 27.18/4.39 | | | | | | is_a_theorem(v1) = v5 & is_a_theorem(v0) = v3 & ( ~ (v4 % 27.18/4.39 | | | | | | = 0) | ~ (v3 = 0) | v5 = 0))) % 27.18/4.39 | | | | | | % 27.18/4.39 | | | | | | BETA: splitting (8) gives: % 27.18/4.39 | | | | | | % 27.18/4.39 | | | | | | Case 1: % 27.18/4.39 | | | | | | | % 27.18/4.40 | | | | | | | (44) ~ (all_36_0 = 0) & or(all_36_4, all_36_5) = all_36_2 & % 27.18/4.40 | | | | | | | or(all_36_5, all_36_4) = all_36_3 & implies(all_36_3, % 27.18/4.40 | | | | | | | all_36_2) = all_36_1 & is_a_theorem(all_36_1) = all_36_0 % 27.18/4.40 | | | | | | | & $i(all_36_1) & $i(all_36_2) & $i(all_36_3) & ~ r3 % 27.18/4.40 | | | | | | | % 27.18/4.40 | | | | | | | ALPHA: (44) implies: % 27.67/4.40 | | | | | | | (45) ~ r3 % 27.67/4.40 | | | | | | | % 27.67/4.40 | | | | | | | PRED_UNIFY: (45), (principia_r3) imply: % 27.67/4.40 | | | | | | | (46) $false % 27.67/4.40 | | | | | | | % 27.67/4.40 | | | | | | | CLOSE: (46) is inconsistent. % 27.67/4.40 | | | | | | | % 27.67/4.40 | | | | | | Case 2: % 27.67/4.40 | | | | | | | % 27.67/4.40 | | | | | | | (47) r3 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (or(v1, % 27.67/4.40 | | | | | | | v0) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? % 27.67/4.40 | | | | | | | [v4: $i] : (or(v0, v1) = v3 & implies(v3, v2) = v4 & % 27.67/4.40 | | | | | | | is_a_theorem(v4) = 0 & $i(v4) & $i(v3))) & ! [v0: $i] % 27.67/4.40 | | | | | | | : ! [v1: $i] : ! [v2: $i] : ( ~ (or(v0, v1) = v2) | ~ % 27.67/4.40 | | | | | | | $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : (or(v1, % 27.67/4.40 | | | | | | | v0) = v3 & implies(v2, v3) = v4 & is_a_theorem(v4) = % 27.67/4.40 | | | | | | | 0 & $i(v4) & $i(v3))) % 27.67/4.40 | | | | | | | % 27.67/4.40 | | | | | | | ALPHA: (47) implies: % 27.67/4.40 | | | | | | | (48) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (or(v0, v1) % 27.67/4.40 | | | | | | | = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: % 27.67/4.40 | | | | | | | $i] : (or(v1, v0) = v3 & implies(v2, v3) = v4 & % 27.67/4.40 | | | | | | | is_a_theorem(v4) = 0 & $i(v4) & $i(v3))) % 27.67/4.41 | | | | | | | (49) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (or(v1, v0) % 27.67/4.41 | | | | | | | = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: % 27.67/4.41 | | | | | | | $i] : (or(v0, v1) = v3 & implies(v3, v2) = v4 & % 27.67/4.41 | | | | | | | is_a_theorem(v4) = 0 & $i(v4) & $i(v3))) % 27.67/4.41 | | | | | | | % 27.67/4.41 | | | | | | | BETA: splitting (14) gives: % 27.67/4.41 | | | | | | | % 27.67/4.41 | | | | | | | Case 1: % 27.67/4.41 | | | | | | | | % 27.67/4.41 | | | | | | | | (50) ~ (all_52_0 = 0) & or(all_52_7, all_52_3) = all_52_2 & % 27.67/4.41 | | | | | | | | or(all_52_7, all_52_6) = all_52_5 & or(all_52_8, % 27.67/4.41 | | | | | | | | all_52_5) = all_52_4 & or(all_52_8, all_52_6) = % 27.67/4.41 | | | | | | | | all_52_3 & implies(all_52_4, all_52_2) = all_52_1 & % 27.67/4.41 | | | | | | | | is_a_theorem(all_52_1) = all_52_0 & $i(all_52_1) & % 27.67/4.41 | | | | | | | | $i(all_52_2) & $i(all_52_3) & $i(all_52_4) & % 27.67/4.41 | | | | | | | | $i(all_52_5) & ~ r4 % 27.67/4.41 | | | | | | | | % 27.67/4.41 | | | | | | | | ALPHA: (50) implies: % 27.67/4.41 | | | | | | | | (51) ~ r4 % 27.67/4.41 | | | | | | | | % 27.67/4.41 | | | | | | | | PRED_UNIFY: (51), (principia_r4) imply: % 27.67/4.41 | | | | | | | | (52) $false % 27.67/4.41 | | | | | | | | % 27.67/4.41 | | | | | | | | CLOSE: (52) is inconsistent. % 27.67/4.41 | | | | | | | | % 27.67/4.41 | | | | | | | Case 2: % 27.67/4.41 | | | | | | | | % 27.67/4.41 | | | | | | | | % 27.67/4.41 | | | | | | | | GROUND_INST: instantiating (34) with all_50_7, all_50_6, % 27.67/4.41 | | | | | | | | all_50_2, simplifying with (10), (11), (22) gives: % 27.67/4.41 | | | | | | | | (53) ? [v0: $i] : ? [v1: $i] : (and(all_50_7, v0) = v1 & % 27.67/4.41 | | | | | | | | not(v1) = all_50_2 & not(all_50_6) = v0 & $i(v1) & % 27.67/4.41 | | | | | | | | $i(v0) & $i(all_50_2)) % 27.67/4.41 | | | | | | | | % 27.67/4.41 | | | | | | | | GROUND_INST: instantiating (38) with all_50_7, all_50_6, % 27.67/4.41 | | | | | | | | all_50_2, simplifying with (10), (11), (22) gives: % 27.67/4.42 | | | | | | | | (54) ? [v0: $i] : (or(v0, all_50_6) = all_50_2 & % 27.67/4.42 | | | | | | | | not(all_50_7) = v0 & $i(v0) & $i(all_50_2)) % 27.67/4.42 | | | | | | | | % 27.67/4.42 | | | | | | | | GROUND_INST: instantiating (34) with all_50_5, all_50_4, % 27.67/4.42 | | | | | | | | all_50_3, simplifying with (17), (18), (23) gives: % 27.67/4.42 | | | | | | | | (55) ? [v0: $i] : ? [v1: $i] : (and(all_50_5, v0) = v1 & % 27.67/4.42 | | | | | | | | not(v1) = all_50_3 & not(all_50_4) = v0 & $i(v1) & % 27.67/4.42 | | | | | | | | $i(v0) & $i(all_50_3)) % 27.67/4.42 | | | | | | | | % 27.67/4.42 | | | | | | | | GROUND_INST: instantiating (43) with all_50_3, all_50_2, % 27.67/4.42 | | | | | | | | all_50_1, simplifying with (19), (20), (24) gives: % 27.67/4.42 | | | | | | | | (56) ? [v0: any] : ? [v1: any] : ? [v2: any] : % 27.67/4.42 | | | | | | | | (is_a_theorem(all_50_1) = v1 & is_a_theorem(all_50_2) = % 27.67/4.42 | | | | | | | | v2 & is_a_theorem(all_50_3) = v0 & ( ~ (v1 = 0) | ~ % 27.67/4.42 | | | | | | | | (v0 = 0) | v2 = 0)) % 27.67/4.42 | | | | | | | | % 27.67/4.42 | | | | | | | | DELTA: instantiating (54) with fresh symbol all_117_0 gives: % 27.67/4.42 | | | | | | | | (57) or(all_117_0, all_50_6) = all_50_2 & not(all_50_7) = % 27.67/4.42 | | | | | | | | all_117_0 & $i(all_117_0) & $i(all_50_2) % 27.67/4.42 | | | | | | | | % 27.67/4.42 | | | | | | | | ALPHA: (57) implies: % 27.67/4.42 | | | | | | | | (58) $i(all_117_0) % 27.67/4.42 | | | | | | | | (59) not(all_50_7) = all_117_0 % 27.67/4.42 | | | | | | | | (60) or(all_117_0, all_50_6) = all_50_2 % 27.67/4.42 | | | | | | | | % 27.67/4.42 | | | | | | | | DELTA: instantiating (53) with fresh symbols all_131_0, % 27.67/4.42 | | | | | | | | all_131_1 gives: % 27.67/4.42 | | | | | | | | (61) and(all_50_7, all_131_1) = all_131_0 & not(all_131_0) = % 27.67/4.42 | | | | | | | | all_50_2 & not(all_50_6) = all_131_1 & $i(all_131_0) & % 27.67/4.42 | | | | | | | | $i(all_131_1) & $i(all_50_2) % 27.67/4.42 | | | | | | | | % 27.67/4.42 | | | | | | | | ALPHA: (61) implies: % 27.67/4.42 | | | | | | | | (62) not(all_50_6) = all_131_1 % 27.67/4.42 | | | | | | | | % 27.67/4.42 | | | | | | | | DELTA: instantiating (56) with fresh symbols all_135_0, % 27.67/4.42 | | | | | | | | all_135_1, all_135_2 gives: % 27.67/4.42 | | | | | | | | (63) is_a_theorem(all_50_1) = all_135_1 & % 27.67/4.42 | | | | | | | | is_a_theorem(all_50_2) = all_135_0 & % 27.67/4.42 | | | | | | | | is_a_theorem(all_50_3) = all_135_2 & ( ~ (all_135_1 = 0) % 27.67/4.42 | | | | | | | | | ~ (all_135_2 = 0) | all_135_0 = 0) % 27.67/4.42 | | | | | | | | % 27.67/4.42 | | | | | | | | ALPHA: (63) implies: % 27.67/4.43 | | | | | | | | (64) is_a_theorem(all_50_1) = all_135_1 % 27.67/4.43 | | | | | | | | % 27.67/4.43 | | | | | | | | DELTA: instantiating (55) with fresh symbols all_141_0, % 27.67/4.43 | | | | | | | | all_141_1 gives: % 27.67/4.43 | | | | | | | | (65) and(all_50_5, all_141_1) = all_141_0 & not(all_141_0) = % 27.67/4.43 | | | | | | | | all_50_3 & not(all_50_4) = all_141_1 & $i(all_141_0) & % 27.67/4.43 | | | | | | | | $i(all_141_1) & $i(all_50_3) % 27.67/4.43 | | | | | | | | % 27.67/4.43 | | | | | | | | ALPHA: (65) implies: % 27.67/4.43 | | | | | | | | (66) not(all_50_4) = all_141_1 % 27.67/4.43 | | | | | | | | (67) not(all_141_0) = all_50_3 % 27.67/4.43 | | | | | | | | (68) and(all_50_5, all_141_1) = all_141_0 % 27.67/4.43 | | | | | | | | % 27.67/4.43 | | | | | | | | GROUND_INST: instantiating (1) with all_50_0, all_135_1, % 27.67/4.43 | | | | | | | | all_50_1, simplifying with (21), (64) gives: % 27.67/4.43 | | | | | | | | (69) all_135_1 = all_50_0 % 27.67/4.43 | | | | | | | | % 27.67/4.43 | | | | | | | | GROUND_INST: instantiating (2) with all_50_4, all_117_0, % 27.67/4.43 | | | | | | | | all_50_7, simplifying with (25), (59) gives: % 27.67/4.43 | | | | | | | | (70) all_117_0 = all_50_4 % 27.67/4.43 | | | | | | | | % 27.67/4.43 | | | | | | | | GROUND_INST: instantiating (2) with all_50_5, all_131_1, % 27.67/4.43 | | | | | | | | all_50_6, simplifying with (26), (62) gives: % 27.67/4.43 | | | | | | | | (71) all_131_1 = all_50_5 % 27.67/4.43 | | | | | | | | % 27.67/4.43 | | | | | | | | REDUCE: (60), (70) imply: % 27.67/4.43 | | | | | | | | (72) or(all_50_4, all_50_6) = all_50_2 % 27.67/4.43 | | | | | | | | % 27.67/4.43 | | | | | | | | GROUND_INST: instantiating (30) with all_50_6, all_50_4, % 27.67/4.43 | | | | | | | | all_50_5, all_141_1, all_141_0, simplifying with % 27.67/4.43 | | | | | | | | (11), (18), (26), (66), (68) gives: % 27.67/4.43 | | | | | | | | (73) ? [v0: $i] : (or(all_50_6, all_50_4) = v0 & % 27.67/4.43 | | | | | | | | not(all_141_0) = v0 & $i(v0)) % 27.67/4.43 | | | | | | | | % 27.67/4.43 | | | | | | | | GROUND_INST: instantiating (49) with all_50_6, all_50_4, % 27.67/4.43 | | | | | | | | all_50_2, simplifying with (11), (18), (72) gives: % 27.67/4.44 | | | | | | | | (74) ? [v0: $i] : ? [v1: $i] : (or(all_50_6, all_50_4) = v0 % 27.67/4.44 | | | | | | | | & implies(v0, all_50_2) = v1 & is_a_theorem(v1) = 0 & % 27.67/4.44 | | | | | | | | $i(v1) & $i(v0)) % 27.67/4.44 | | | | | | | | % 27.67/4.44 | | | | | | | | GROUND_INST: instantiating (48) with all_50_4, all_50_6, % 27.67/4.44 | | | | | | | | all_50_2, simplifying with (11), (18), (72) gives: % 27.67/4.44 | | | | | | | | (75) ? [v0: $i] : ? [v1: $i] : (or(all_50_6, all_50_4) = v0 % 27.67/4.44 | | | | | | | | & implies(all_50_2, v0) = v1 & is_a_theorem(v1) = 0 & % 27.67/4.44 | | | | | | | | $i(v1) & $i(v0)) % 27.67/4.44 | | | | | | | | % 27.67/4.44 | | | | | | | | DELTA: instantiating (73) with fresh symbol all_214_0 gives: % 27.67/4.44 | | | | | | | | (76) or(all_50_6, all_50_4) = all_214_0 & not(all_141_0) = % 27.67/4.44 | | | | | | | | all_214_0 & $i(all_214_0) % 27.67/4.44 | | | | | | | | % 27.67/4.44 | | | | | | | | ALPHA: (76) implies: % 27.67/4.44 | | | | | | | | (77) not(all_141_0) = all_214_0 % 27.67/4.44 | | | | | | | | (78) or(all_50_6, all_50_4) = all_214_0 % 27.67/4.44 | | | | | | | | % 27.67/4.44 | | | | | | | | DELTA: instantiating (74) with fresh symbols all_230_0, % 27.67/4.44 | | | | | | | | all_230_1 gives: % 27.67/4.44 | | | | | | | | (79) or(all_50_6, all_50_4) = all_230_1 & implies(all_230_1, % 27.67/4.44 | | | | | | | | all_50_2) = all_230_0 & is_a_theorem(all_230_0) = 0 & % 27.67/4.44 | | | | | | | | $i(all_230_0) & $i(all_230_1) % 27.67/4.44 | | | | | | | | % 27.67/4.44 | | | | | | | | ALPHA: (79) implies: % 27.67/4.44 | | | | | | | | (80) is_a_theorem(all_230_0) = 0 % 27.67/4.44 | | | | | | | | (81) implies(all_230_1, all_50_2) = all_230_0 % 27.67/4.44 | | | | | | | | (82) or(all_50_6, all_50_4) = all_230_1 % 27.67/4.44 | | | | | | | | % 27.67/4.44 | | | | | | | | DELTA: instantiating (75) with fresh symbols all_232_0, % 27.67/4.44 | | | | | | | | all_232_1 gives: % 27.67/4.44 | | | | | | | | (83) or(all_50_6, all_50_4) = all_232_1 & implies(all_50_2, % 27.67/4.44 | | | | | | | | all_232_1) = all_232_0 & is_a_theorem(all_232_0) = 0 & % 27.67/4.44 | | | | | | | | $i(all_232_0) & $i(all_232_1) % 27.67/4.44 | | | | | | | | % 27.67/4.44 | | | | | | | | ALPHA: (83) implies: % 27.67/4.44 | | | | | | | | (84) or(all_50_6, all_50_4) = all_232_1 % 27.67/4.44 | | | | | | | | % 27.67/4.44 | | | | | | | | GROUND_INST: instantiating (2) with all_50_3, all_214_0, % 27.67/4.44 | | | | | | | | all_141_0, simplifying with (67), (77) gives: % 27.67/4.44 | | | | | | | | (85) all_214_0 = all_50_3 % 27.67/4.44 | | | | | | | | % 27.67/4.44 | | | | | | | | GROUND_INST: instantiating (4) with all_230_1, all_232_1, % 27.67/4.44 | | | | | | | | all_50_4, all_50_6, simplifying with (82), (84) % 27.67/4.44 | | | | | | | | gives: % 27.67/4.45 | | | | | | | | (86) all_232_1 = all_230_1 % 27.67/4.45 | | | | | | | | % 27.67/4.45 | | | | | | | | GROUND_INST: instantiating (4) with all_214_0, all_232_1, % 27.67/4.45 | | | | | | | | all_50_4, all_50_6, simplifying with (78), (84) % 27.67/4.45 | | | | | | | | gives: % 27.67/4.45 | | | | | | | | (87) all_232_1 = all_214_0 % 27.67/4.45 | | | | | | | | % 27.67/4.45 | | | | | | | | COMBINE_EQS: (86), (87) imply: % 27.67/4.45 | | | | | | | | (88) all_230_1 = all_214_0 % 27.67/4.45 | | | | | | | | % 27.67/4.45 | | | | | | | | COMBINE_EQS: (85), (88) imply: % 27.67/4.45 | | | | | | | | (89) all_230_1 = all_50_3 % 27.67/4.45 | | | | | | | | % 27.67/4.45 | | | | | | | | REDUCE: (81), (89) imply: % 27.67/4.45 | | | | | | | | (90) implies(all_50_3, all_50_2) = all_230_0 % 27.67/4.45 | | | | | | | | % 27.67/4.45 | | | | | | | | GROUND_INST: instantiating (3) with all_50_1, all_230_0, % 27.67/4.45 | | | | | | | | all_50_2, all_50_3, simplifying with (24), (90) % 27.67/4.45 | | | | | | | | gives: % 27.67/4.45 | | | | | | | | (91) all_230_0 = all_50_1 % 27.67/4.45 | | | | | | | | % 27.67/4.45 | | | | | | | | REDUCE: (80), (91) imply: % 27.67/4.45 | | | | | | | | (92) is_a_theorem(all_50_1) = 0 % 27.67/4.45 | | | | | | | | % 27.67/4.45 | | | | | | | | GROUND_INST: instantiating (1) with all_50_0, 0, all_50_1, % 27.67/4.45 | | | | | | | | simplifying with (21), (92) gives: % 27.67/4.45 | | | | | | | | (93) all_50_0 = 0 % 27.67/4.45 | | | | | | | | % 27.67/4.45 | | | | | | | | REDUCE: (16), (93) imply: % 27.67/4.45 | | | | | | | | (94) $false % 27.67/4.45 | | | | | | | | % 27.67/4.45 | | | | | | | | CLOSE: (94) is inconsistent. % 27.67/4.45 | | | | | | | | % 27.67/4.45 | | | | | | | End of split % 27.67/4.45 | | | | | | | % 27.67/4.45 | | | | | | End of split % 27.67/4.45 | | | | | | % 27.67/4.45 | | | | | End of split % 27.67/4.45 | | | | | % 27.67/4.45 | | | | End of split % 27.67/4.45 | | | | % 27.67/4.45 | | | End of split % 27.67/4.45 | | | % 27.67/4.45 | | End of split % 27.67/4.45 | | % 27.67/4.45 | Case 2: % 27.67/4.45 | | % 27.67/4.46 | | (95) modus_tollens & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: % 27.67/4.46 | | $i] : ! [v4: $i] : ( ~ (not(v1) = v2) | ~ (not(v0) = v3) | ~ % 27.67/4.46 | | (implies(v2, v3) = v4) | ~ $i(v1) | ~ $i(v0) | ? [v5: $i] : ? % 27.67/4.46 | | [v6: $i] : (implies(v4, v5) = v6 & implies(v0, v1) = v5 & % 27.67/4.46 | | is_a_theorem(v6) = 0 & $i(v6) & $i(v5))) & ! [v0: $i] : ! [v1: % 27.67/4.46 | | $i] : ! [v2: $i] : ( ~ (implies(v0, v1) = v2) | ~ $i(v1) | ~ % 27.67/4.46 | | $i(v0) | ? [v3: $i] : ? [v4: $i] : ? [v5: $i] : ? [v6: $i] : % 27.67/4.46 | | (not(v1) = v3 & not(v0) = v4 & implies(v5, v2) = v6 & implies(v3, % 27.67/4.46 | | v4) = v5 & is_a_theorem(v6) = 0 & $i(v6) & $i(v5) & $i(v4) & % 27.67/4.46 | | $i(v3))) % 27.67/4.46 | | % 27.67/4.46 | | ALPHA: (95) implies: % 27.67/4.46 | | (96) modus_tollens % 27.67/4.46 | | % 27.67/4.46 | | PRED_UNIFY: (96), (hilbert_modus_tollens) imply: % 27.67/4.46 | | (97) $false % 27.67/4.46 | | % 27.67/4.46 | | CLOSE: (97) is inconsistent. % 27.67/4.46 | | % 27.67/4.46 | End of split % 27.67/4.46 | % 27.67/4.46 End of proof % 27.67/4.46 % SZS output end Proof for theBenchmark % 27.67/4.46 % 27.67/4.46 3905ms %------------------------------------------------------------------------------