%------------------------------------------------------------------------------ % File : Princess---230619 % Problem : LCL529+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 : n008.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 12:54:17 PM UTC 2026 % Result : Theorem 19.84s 3.36s % Output : Proof 40.81s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.02 % Problem : LCL529+1 : TPTP v9.3.1. Bugfixed v9.2.0. % 0.00/0.04 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s % 0.09/0.36 % Computer : n008.cluster.edu % 0.09/0.36 % Model : x86_64 x86_64 % 0.09/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.09/0.36 % Memory : 8046.5625MB % 0.09/0.36 % OS : Linux 6.8.0-71-generic % 0.09/0.36 % CPULimit : 300 % 0.09/0.36 % WCLimit : 300 % 0.09/0.36 % DateTime : Sat Sep 5 15:33:38 UTC 2026 % 0.09/0.36 % CPUTime : % 0.09/0.38 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. % 0.27/0.56 ________ _____ % 0.27/0.56 ___ __ \_________(_)________________________________ % 0.27/0.56 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/ % 0.27/0.56 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ ) % 0.27/0.56 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/ % 0.52/0.56 % 0.52/0.56 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic % 0.52/0.56 (2023-06-19) % 0.52/0.56 % 0.52/0.56 (c) Philipp Rümmer, 2009-2023 % 0.52/0.56 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen, % 0.52/0.56 Amanda Stjerna. % 0.52/0.56 Free software under BSD-3-Clause. % 0.52/0.56 % 0.52/0.56 For more information, visit http://www.philipp.ruemmer.org/princess.shtml % 0.52/0.56 % 0.52/0.56 Loading /export/starexec/sandbox2/benchmark/theBenchmark.p ... % 0.52/0.57 Running up to 7 provers in parallel. % 0.52/0.58 Prover 3: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1922548996 % 0.52/0.58 Prover 2: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMinimalAndEmpty -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1065072994 % 0.52/0.58 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1571432423 % 0.52/0.58 Prover 0: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1042961893 % 0.52/0.58 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=1868514696 % 0.52/0.58 Prover 5: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=complete -randomSeed=1259561288 % 0.52/0.58 Prover 6: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximalOutermost -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1399714365 % 4.98/1.43 Prover 4: Preprocessing ... % 4.98/1.46 Prover 1: Preprocessing ... % 5.75/1.53 Prover 6: Preprocessing ... % 5.75/1.53 Prover 3: Preprocessing ... % 5.75/1.53 Prover 5: Preprocessing ... % 5.75/1.53 Prover 2: Preprocessing ... % 5.75/1.53 Prover 0: Preprocessing ... % 15.93/2.82 Prover 5: Proving ... % 15.93/2.86 Prover 1: Constructing countermodel ... % 17.97/3.07 Prover 4: Constructing countermodel ... % 17.97/3.08 Prover 3: Constructing countermodel ... % 17.97/3.15 Prover 6: Constructing countermodel ... % 19.05/3.28 Prover 0: Proving ... % 19.84/3.35 Prover 3: proved (2765ms) % 19.84/3.35 % 19.84/3.36 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p % 19.84/3.36 % 19.84/3.36 Prover 0: stopped % 19.84/3.37 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470 % 19.84/3.37 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089 % 19.84/3.37 Prover 5: stopped % 19.84/3.37 Prover 6: stopped % 19.84/3.38 Prover 10: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=919308125 % 19.84/3.38 Prover 11: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1509710984 % 19.84/3.40 Prover 2: Proving ... % 20.62/3.43 Prover 2: stopped % 20.62/3.43 Prover 13: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=1138197443 % 21.47/3.56 Prover 10: Preprocessing ... % 21.47/3.56 Prover 7: Preprocessing ... % 22.21/3.63 Prover 13: Preprocessing ... % 22.21/3.67 Prover 11: Preprocessing ... % 23.01/3.72 Prover 8: Preprocessing ... % 24.59/4.05 Prover 10: Constructing countermodel ... % 25.81/4.12 Prover 7: Constructing countermodel ... % 26.62/4.18 Prover 13: Constructing countermodel ... % 26.94/4.25 Prover 8: Warning: ignoring some quantifiers % 26.94/4.27 Prover 8: Constructing countermodel ... % 29.35/4.52 Prover 11: Constructing countermodel ... % 39.43/5.90 Prover 13: Found proof (size 41) % 39.43/5.90 Prover 13: proved (2461ms) % 39.43/5.90 Prover 4: stopped % 39.43/5.90 Prover 1: stopped % 39.43/5.90 Prover 11: stopped % 40.24/5.90 Prover 7: stopped % 40.24/5.90 Prover 8: stopped % 40.24/5.95 Prover 10: stopped % 40.24/5.95 % 40.24/5.95 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p % 40.24/5.95 % 40.24/5.96 % SZS output start Proof for theBenchmark % 40.24/5.97 Assumptions after simplification: % 40.24/5.97 --------------------------------- % 40.24/5.97 % 40.24/5.97 (and_1) % 40.24/6.02 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ($i(v1) & $i(v0) & % 40.24/6.02 ((and(v0, v1) = v2 & implies(v2, v0) = v3 & $i(v3) & $i(v2) & ~ and_1 & ~ % 40.24/6.02 is_a_theorem(v3)) | (and_1 & ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : % 40.24/6.02 ! [v7: $i] : ( ~ (and(v4, v5) = v6) | ~ (implies(v6, v4) = v7) | ~ % 40.24/6.02 $i(v5) | ~ $i(v4) | is_a_theorem(v7))))) % 40.24/6.02 % 40.24/6.02 (axiom_m2) % 40.24/6.03 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ($i(v1) & $i(v0) & % 40.24/6.03 ((strict_implies(v2, v0) = v3 & and(v0, v1) = v2 & $i(v3) & $i(v2) & ~ % 40.24/6.03 axiom_m2 & ~ is_a_theorem(v3)) | (axiom_m2 & ! [v4: $i] : ! [v5: $i] % 40.24/6.03 : ! [v6: $i] : ! [v7: $i] : ( ~ (strict_implies(v6, v4) = v7) | ~ % 40.24/6.03 (and(v4, v5) = v6) | ~ $i(v5) | ~ $i(v4) | is_a_theorem(v7))))) % 40.24/6.03 % 40.24/6.03 (hilbert_and_1) % 40.24/6.03 and_1 % 40.24/6.03 % 40.24/6.03 (hilbert_or_3) % 40.24/6.03 or_3 % 40.24/6.03 % 40.24/6.03 (km5_necessitation) % 40.24/6.03 necessitation % 40.24/6.03 % 40.24/6.03 (necessitation) % 40.24/6.03 ? [v0: $i] : ? [v1: $i] : ($i(v0) & ((necessarily(v0) = v1 & $i(v1) & % 40.24/6.03 is_a_theorem(v0) & ~ necessitation & ~ is_a_theorem(v1)) | % 40.24/6.03 (necessitation & ! [v2: $i] : ! [v3: $i] : ( ~ (necessarily(v2) = v3) | % 40.24/6.03 ~ $i(v2) | ~ is_a_theorem(v2) | is_a_theorem(v3))))) % 40.24/6.03 % 40.24/6.03 (op_strict_implies) % 40.81/6.04 ~ op_strict_implies | ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ % 40.81/6.04 (strict_implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : % 40.81/6.04 (necessarily(v3) = v2 & implies(v0, v1) = v3 & $i(v3) & $i(v2))) % 40.81/6.04 % 40.81/6.04 (or_3) % 40.81/6.04 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ? [v4: $i] : ? [v5: % 40.81/6.04 $i] : ? [v6: $i] : ? [v7: $i] : ? [v8: $i] : ($i(v2) & $i(v1) & $i(v0) & % 40.81/6.04 ((or(v0, v1) = v5 & implies(v5, v2) = v6 & implies(v4, v6) = v7 & % 40.81/6.04 implies(v3, v7) = v8 & implies(v1, v2) = v4 & implies(v0, v2) = v3 & % 40.81/6.04 $i(v8) & $i(v7) & $i(v6) & $i(v5) & $i(v4) & $i(v3) & ~ or_3 & ~ % 40.81/6.04 is_a_theorem(v8)) | (or_3 & ! [v9: $i] : ! [v10: $i] : ! [v11: $i] : % 40.81/6.04 ! [v12: $i] : ! [v13: $i] : ! [v14: $i] : ! [v15: $i] : ! [v16: $i] % 40.81/6.04 : ! [v17: $i] : ( ~ (or(v9, v10) = v14) | ~ (implies(v14, v11) = v15) % 40.81/6.04 | ~ (implies(v13, v15) = v16) | ~ (implies(v12, v16) = v17) | ~ % 40.81/6.04 (implies(v10, v11) = v13) | ~ (implies(v9, v11) = v12) | ~ $i(v11) | % 40.81/6.05 ~ $i(v10) | ~ $i(v9) | is_a_theorem(v17))))) % 40.81/6.05 % 40.81/6.05 (s1_0_axiom_m2) % 40.81/6.05 ~ axiom_m2 % 40.81/6.05 % 40.81/6.05 (s1_0_op_strict_implies) % 40.81/6.05 op_strict_implies % 40.81/6.05 % 40.81/6.05 Further assumptions not needed in the proof: % 40.81/6.05 -------------------------------------------- % 40.81/6.05 adjunction, and_2, and_3, axiom_4, axiom_5, axiom_B, axiom_K, axiom_M, axiom_m1, % 40.81/6.05 axiom_m10, axiom_m3, axiom_m4, axiom_m5, axiom_m6, axiom_m7, axiom_m8, axiom_m9, % 40.81/6.05 axiom_s1, axiom_s2, axiom_s3, axiom_s4, cn1, cn2, cn3, equivalence_1, % 40.81/6.05 equivalence_2, equivalence_3, hilbert_and_2, hilbert_and_3, % 40.81/6.05 hilbert_equivalence_1, hilbert_equivalence_2, hilbert_equivalence_3, % 40.81/6.05 hilbert_implies_1, hilbert_implies_2, hilbert_implies_3, hilbert_modus_ponens, % 40.81/6.05 hilbert_modus_tollens, hilbert_op_equiv, hilbert_op_implies_and, hilbert_op_or, % 40.81/6.05 hilbert_or_1, hilbert_or_2, implies_1, implies_2, implies_3, km5_axiom_5, % 40.81/6.05 km5_axiom_K, km5_axiom_M, km5_op_possibly, kn1, kn2, kn3, modus_ponens, % 40.81/6.05 modus_ponens_strict_implies, modus_tollens, op_and, op_equiv, op_implies_and, % 40.81/6.05 op_implies_or, op_necessarily, op_or, op_possibly, op_strict_equiv, or_1, or_2, % 40.81/6.05 r1, r2, r3, r4, r5, s1_0_op_equiv, s1_0_op_implies, s1_0_op_or, % 40.81/6.05 s1_0_op_possibly, s1_0_op_strict_equiv, substitution_of_equivalents, % 40.81/6.05 substitution_strict_equiv, use_substitution_of_equivalents % 40.81/6.05 % 40.81/6.05 Those formulas are unsatisfiable: % 40.81/6.05 --------------------------------- % 40.81/6.05 % 40.81/6.05 Begin of proof % 40.81/6.05 | % 40.81/6.05 | DELTA: instantiating (necessitation) with fresh symbols all_4_0, all_4_1 % 40.81/6.05 | gives: % 40.81/6.05 | (1) $i(all_4_1) & ((necessarily(all_4_1) = all_4_0 & $i(all_4_0) & % 40.81/6.05 | is_a_theorem(all_4_1) & ~ necessitation & ~ % 40.81/6.05 | is_a_theorem(all_4_0)) | (necessitation & ! [v0: $i] : ! [v1: $i] % 40.81/6.05 | : ( ~ (necessarily(v0) = v1) | ~ $i(v0) | ~ is_a_theorem(v0) | % 40.81/6.05 | is_a_theorem(v1)))) % 40.81/6.05 | % 40.81/6.05 | ALPHA: (1) implies: % 40.81/6.06 | (2) (necessarily(all_4_1) = all_4_0 & $i(all_4_0) & is_a_theorem(all_4_1) & % 40.81/6.06 | ~ necessitation & ~ is_a_theorem(all_4_0)) | (necessitation & ! % 40.81/6.06 | [v0: $i] : ! [v1: $i] : ( ~ (necessarily(v0) = v1) | ~ $i(v0) | ~ % 40.81/6.06 | is_a_theorem(v0) | is_a_theorem(v1))) % 40.81/6.06 | % 40.81/6.06 | DELTA: instantiating (axiom_m2) with fresh symbols all_22_0, all_22_1, % 40.81/6.06 | all_22_2, all_22_3 gives: % 40.81/6.06 | (3) $i(all_22_2) & $i(all_22_3) & ((strict_implies(all_22_1, all_22_3) = % 40.81/6.06 | all_22_0 & and(all_22_3, all_22_2) = all_22_1 & $i(all_22_0) & % 40.81/6.06 | $i(all_22_1) & ~ axiom_m2 & ~ is_a_theorem(all_22_0)) | (axiom_m2 % 40.81/6.06 | & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~ % 40.81/6.06 | (strict_implies(v2, v0) = v3) | ~ (and(v0, v1) = v2) | ~ $i(v1) % 40.81/6.06 | | ~ $i(v0) | is_a_theorem(v3)))) % 40.81/6.06 | % 40.81/6.06 | ALPHA: (3) implies: % 40.81/6.06 | (4) $i(all_22_3) % 40.81/6.06 | (5) $i(all_22_2) % 40.81/6.06 | (6) (strict_implies(all_22_1, all_22_3) = all_22_0 & and(all_22_3, % 40.81/6.06 | all_22_2) = all_22_1 & $i(all_22_0) & $i(all_22_1) & ~ axiom_m2 & % 40.81/6.06 | ~ is_a_theorem(all_22_0)) | (axiom_m2 & ! [v0: $i] : ! [v1: $i] : % 40.81/6.06 | ! [v2: $i] : ! [v3: $i] : ( ~ (strict_implies(v2, v0) = v3) | ~ % 40.81/6.06 | (and(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | is_a_theorem(v3))) % 40.81/6.06 | % 40.81/6.06 | DELTA: instantiating (and_1) with fresh symbols all_26_0, all_26_1, all_26_2, % 40.81/6.06 | all_26_3 gives: % 40.81/6.06 | (7) $i(all_26_2) & $i(all_26_3) & ((and(all_26_3, all_26_2) = all_26_1 & % 40.81/6.06 | implies(all_26_1, all_26_3) = all_26_0 & $i(all_26_0) & % 40.81/6.06 | $i(all_26_1) & ~ and_1 & ~ is_a_theorem(all_26_0)) | (and_1 & ! % 40.81/6.06 | [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~ (and(v0, % 40.81/6.06 | v1) = v2) | ~ (implies(v2, v0) = v3) | ~ $i(v1) | ~ $i(v0) % 40.81/6.06 | | is_a_theorem(v3)))) % 40.81/6.06 | % 40.81/6.06 | ALPHA: (7) implies: % 40.81/6.07 | (8) (and(all_26_3, all_26_2) = all_26_1 & implies(all_26_1, all_26_3) = % 40.81/6.07 | all_26_0 & $i(all_26_0) & $i(all_26_1) & ~ and_1 & ~ % 40.81/6.07 | is_a_theorem(all_26_0)) | (and_1 & ! [v0: $i] : ! [v1: $i] : ! % 40.81/6.07 | [v2: $i] : ! [v3: $i] : ( ~ (and(v0, v1) = v2) | ~ (implies(v2, v0) % 40.81/6.07 | = v3) | ~ $i(v1) | ~ $i(v0) | is_a_theorem(v3))) % 40.81/6.07 | % 40.81/6.07 | DELTA: instantiating (or_3) with fresh symbols all_94_0, all_94_1, all_94_2, % 40.81/6.07 | all_94_3, all_94_4, all_94_5, all_94_6, all_94_7, all_94_8 gives: % 40.81/6.07 | (9) $i(all_94_6) & $i(all_94_7) & $i(all_94_8) & ((or(all_94_8, all_94_7) = % 40.81/6.07 | all_94_3 & implies(all_94_3, all_94_6) = all_94_2 & % 40.81/6.07 | implies(all_94_4, all_94_2) = all_94_1 & implies(all_94_5, % 40.81/6.07 | all_94_1) = all_94_0 & implies(all_94_7, all_94_6) = all_94_4 & % 40.81/6.07 | implies(all_94_8, all_94_6) = all_94_5 & $i(all_94_0) & % 40.81/6.07 | $i(all_94_1) & $i(all_94_2) & $i(all_94_3) & $i(all_94_4) & % 40.81/6.07 | $i(all_94_5) & ~ or_3 & ~ is_a_theorem(all_94_0)) | (or_3 & ! % 40.81/6.07 | [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : % 40.81/6.07 | ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (or(v0, % 40.81/6.07 | v1) = v5) | ~ (implies(v5, v2) = v6) | ~ (implies(v4, v6) = % 40.81/6.07 | v7) | ~ (implies(v3, v7) = v8) | ~ (implies(v1, v2) = v4) | % 40.81/6.07 | ~ (implies(v0, v2) = v3) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | % 40.81/6.07 | is_a_theorem(v8)))) % 40.81/6.07 | % 40.81/6.07 | ALPHA: (9) implies: % 40.81/6.07 | (10) (or(all_94_8, all_94_7) = all_94_3 & implies(all_94_3, all_94_6) = % 40.81/6.07 | all_94_2 & implies(all_94_4, all_94_2) = all_94_1 & % 40.81/6.07 | implies(all_94_5, all_94_1) = all_94_0 & implies(all_94_7, all_94_6) % 40.81/6.07 | = all_94_4 & implies(all_94_8, all_94_6) = all_94_5 & $i(all_94_0) & % 40.81/6.07 | $i(all_94_1) & $i(all_94_2) & $i(all_94_3) & $i(all_94_4) & % 40.81/6.07 | $i(all_94_5) & ~ or_3 & ~ is_a_theorem(all_94_0)) | (or_3 & ! % 40.81/6.07 | [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : % 40.81/6.07 | ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (or(v0, % 40.81/6.07 | v1) = v5) | ~ (implies(v5, v2) = v6) | ~ (implies(v4, v6) = % 40.81/6.07 | v7) | ~ (implies(v3, v7) = v8) | ~ (implies(v1, v2) = v4) | ~ % 40.81/6.07 | (implies(v0, v2) = v3) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | % 40.81/6.07 | is_a_theorem(v8))) % 40.81/6.07 | % 40.81/6.07 | BETA: splitting (6) gives: % 40.81/6.07 | % 40.81/6.07 | Case 1: % 40.81/6.07 | | % 40.81/6.08 | | (11) strict_implies(all_22_1, all_22_3) = all_22_0 & and(all_22_3, % 40.81/6.08 | | all_22_2) = all_22_1 & $i(all_22_0) & $i(all_22_1) & ~ axiom_m2 & % 40.81/6.08 | | ~ is_a_theorem(all_22_0) % 40.81/6.08 | | % 40.81/6.08 | | ALPHA: (11) implies: % 40.81/6.08 | | (12) ~ is_a_theorem(all_22_0) % 40.81/6.08 | | (13) $i(all_22_1) % 40.81/6.08 | | (14) and(all_22_3, all_22_2) = all_22_1 % 40.81/6.08 | | (15) strict_implies(all_22_1, all_22_3) = all_22_0 % 40.81/6.08 | | % 40.81/6.08 | | BETA: splitting (2) gives: % 40.81/6.08 | | % 40.81/6.08 | | Case 1: % 40.81/6.08 | | | % 40.81/6.08 | | | (16) necessarily(all_4_1) = all_4_0 & $i(all_4_0) & % 40.81/6.08 | | | is_a_theorem(all_4_1) & ~ necessitation & ~ % 40.81/6.08 | | | is_a_theorem(all_4_0) % 40.81/6.08 | | | % 40.81/6.08 | | | ALPHA: (16) implies: % 40.81/6.08 | | | (17) ~ necessitation % 40.81/6.08 | | | % 40.81/6.08 | | | PRED_UNIFY: (17), (km5_necessitation) imply: % 40.81/6.08 | | | (18) $false % 40.81/6.08 | | | % 40.81/6.08 | | | CLOSE: (18) is inconsistent. % 40.81/6.08 | | | % 40.81/6.08 | | Case 2: % 40.81/6.08 | | | % 40.81/6.08 | | | (19) necessitation & ! [v0: $i] : ! [v1: $i] : ( ~ (necessarily(v0) = % 40.81/6.08 | | | v1) | ~ $i(v0) | ~ is_a_theorem(v0) | is_a_theorem(v1)) % 40.81/6.08 | | | % 40.81/6.08 | | | ALPHA: (19) implies: % 40.81/6.08 | | | (20) ! [v0: $i] : ! [v1: $i] : ( ~ (necessarily(v0) = v1) | ~ $i(v0) % 40.81/6.08 | | | | ~ is_a_theorem(v0) | is_a_theorem(v1)) % 40.81/6.08 | | | % 40.81/6.08 | | | BETA: splitting (op_strict_implies) gives: % 40.81/6.08 | | | % 40.81/6.08 | | | Case 1: % 40.81/6.08 | | | | % 40.81/6.08 | | | | (21) ~ op_strict_implies % 40.81/6.08 | | | | % 40.81/6.08 | | | | PRED_UNIFY: (21), (s1_0_op_strict_implies) imply: % 40.81/6.08 | | | | (22) $false % 40.81/6.08 | | | | % 40.81/6.08 | | | | CLOSE: (22) is inconsistent. % 40.81/6.08 | | | | % 40.81/6.08 | | | Case 2: % 40.81/6.08 | | | | % 40.81/6.08 | | | | (23) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ % 40.81/6.08 | | | | (strict_implies(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? % 40.81/6.08 | | | | [v3: $i] : (necessarily(v3) = v2 & implies(v0, v1) = v3 & % 40.81/6.08 | | | | $i(v3) & $i(v2))) % 40.81/6.08 | | | | % 40.81/6.08 | | | | BETA: splitting (8) gives: % 40.81/6.08 | | | | % 40.81/6.09 | | | | Case 1: % 40.81/6.09 | | | | | % 40.81/6.09 | | | | | (24) and(all_26_3, all_26_2) = all_26_1 & implies(all_26_1, % 40.81/6.09 | | | | | all_26_3) = all_26_0 & $i(all_26_0) & $i(all_26_1) & ~ % 40.81/6.09 | | | | | and_1 & ~ is_a_theorem(all_26_0) % 40.81/6.09 | | | | | % 40.81/6.09 | | | | | ALPHA: (24) implies: % 40.81/6.09 | | | | | (25) ~ and_1 % 40.81/6.09 | | | | | % 40.81/6.09 | | | | | PRED_UNIFY: (25), (hilbert_and_1) imply: % 40.81/6.09 | | | | | (26) $false % 40.81/6.09 | | | | | % 40.81/6.09 | | | | | CLOSE: (26) is inconsistent. % 40.81/6.09 | | | | | % 40.81/6.09 | | | | Case 2: % 40.81/6.09 | | | | | % 40.81/6.09 | | | | | (27) and_1 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] % 40.81/6.09 | | | | | : ( ~ (and(v0, v1) = v2) | ~ (implies(v2, v0) = v3) | ~ % 40.81/6.09 | | | | | $i(v1) | ~ $i(v0) | is_a_theorem(v3)) % 40.81/6.09 | | | | | % 40.81/6.09 | | | | | ALPHA: (27) implies: % 40.81/6.09 | | | | | (28) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~ % 40.81/6.09 | | | | | (and(v0, v1) = v2) | ~ (implies(v2, v0) = v3) | ~ $i(v1) | % 40.81/6.09 | | | | | ~ $i(v0) | is_a_theorem(v3)) % 40.81/6.09 | | | | | % 40.81/6.09 | | | | | BETA: splitting (10) gives: % 40.81/6.09 | | | | | % 40.81/6.09 | | | | | Case 1: % 40.81/6.09 | | | | | | % 40.81/6.09 | | | | | | (29) or(all_94_8, all_94_7) = all_94_3 & implies(all_94_3, % 40.81/6.09 | | | | | | all_94_6) = all_94_2 & implies(all_94_4, all_94_2) = % 40.81/6.09 | | | | | | all_94_1 & implies(all_94_5, all_94_1) = all_94_0 & % 40.81/6.09 | | | | | | implies(all_94_7, all_94_6) = all_94_4 & implies(all_94_8, % 40.81/6.09 | | | | | | all_94_6) = all_94_5 & $i(all_94_0) & $i(all_94_1) & % 40.81/6.09 | | | | | | $i(all_94_2) & $i(all_94_3) & $i(all_94_4) & $i(all_94_5) & % 40.81/6.09 | | | | | | ~ or_3 & ~ is_a_theorem(all_94_0) % 40.81/6.09 | | | | | | % 40.81/6.09 | | | | | | ALPHA: (29) implies: % 40.81/6.09 | | | | | | (30) ~ or_3 % 40.81/6.09 | | | | | | % 40.81/6.09 | | | | | | PRED_UNIFY: (30), (hilbert_or_3) imply: % 40.81/6.09 | | | | | | (31) $false % 40.81/6.09 | | | | | | % 40.81/6.09 | | | | | | CLOSE: (31) is inconsistent. % 40.81/6.09 | | | | | | % 40.81/6.09 | | | | | Case 2: % 40.81/6.09 | | | | | | % 40.81/6.09 | | | | | | % 40.81/6.09 | | | | | | GROUND_INST: instantiating (23) with all_22_1, all_22_3, all_22_0, % 40.81/6.09 | | | | | | simplifying with (4), (13), (15) gives: % 40.81/6.10 | | | | | | (32) ? [v0: $i] : (necessarily(v0) = all_22_0 & % 40.81/6.10 | | | | | | implies(all_22_1, all_22_3) = v0 & $i(v0) & $i(all_22_0)) % 40.81/6.10 | | | | | | % 40.81/6.10 | | | | | | DELTA: instantiating (32) with fresh symbol all_212_0 gives: % 40.81/6.10 | | | | | | (33) necessarily(all_212_0) = all_22_0 & implies(all_22_1, % 40.81/6.10 | | | | | | all_22_3) = all_212_0 & $i(all_212_0) & $i(all_22_0) % 40.81/6.10 | | | | | | % 40.81/6.10 | | | | | | ALPHA: (33) implies: % 40.81/6.10 | | | | | | (34) $i(all_212_0) % 40.81/6.10 | | | | | | (35) implies(all_22_1, all_22_3) = all_212_0 % 40.81/6.10 | | | | | | (36) necessarily(all_212_0) = all_22_0 % 40.81/6.10 | | | | | | % 40.81/6.10 | | | | | | GROUND_INST: instantiating (28) with all_22_3, all_22_2, all_22_1, % 40.81/6.10 | | | | | | all_212_0, simplifying with (4), (5), (14), (35) gives: % 40.81/6.10 | | | | | | (37) is_a_theorem(all_212_0) % 40.81/6.10 | | | | | | % 40.81/6.10 | | | | | | GROUND_INST: instantiating (20) with all_212_0, all_22_0, % 40.81/6.10 | | | | | | simplifying with (12), (34), (36), (37) gives: % 40.81/6.10 | | | | | | (38) $false % 40.81/6.10 | | | | | | % 40.81/6.10 | | | | | | CLOSE: (38) is inconsistent. % 40.81/6.10 | | | | | | % 40.81/6.10 | | | | | End of split % 40.81/6.10 | | | | | % 40.81/6.10 | | | | End of split % 40.81/6.10 | | | | % 40.81/6.10 | | | End of split % 40.81/6.10 | | | % 40.81/6.10 | | End of split % 40.81/6.10 | | % 40.81/6.10 | Case 2: % 40.81/6.10 | | % 40.81/6.10 | | (39) axiom_m2 & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( % 40.81/6.10 | | ~ (strict_implies(v2, v0) = v3) | ~ (and(v0, v1) = v2) | ~ % 40.81/6.10 | | $i(v1) | ~ $i(v0) | is_a_theorem(v3)) % 40.81/6.10 | | % 40.81/6.10 | | ALPHA: (39) implies: % 40.81/6.10 | | (40) axiom_m2 % 40.81/6.10 | | % 40.81/6.10 | | PRED_UNIFY: (40), (s1_0_axiom_m2) imply: % 40.81/6.10 | | (41) $false % 40.81/6.10 | | % 40.81/6.10 | | CLOSE: (41) is inconsistent. % 40.81/6.10 | | % 40.81/6.10 | End of split % 40.81/6.10 | % 40.81/6.10 End of proof % 40.81/6.10 % SZS output end Proof for theBenchmark % 40.81/6.10 % 40.81/6.10 5539ms %------------------------------------------------------------------------------