%------------------------------------------------------------------------------ % File : Princess---230619 % Problem : KLE150+2 : TPTP v8.1.2. Released v4.0.0. % Transfm : none % Format : tptp % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s % Computer : n017.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 05:34:42 EDT 2023 % Result : Theorem 24.17s 3.94s % Output : Proof 78.11s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.12 % Problem : KLE150+2 : TPTP v8.1.2. Released v4.0.0. % 0.07/0.13 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s % 0.13/0.34 % Computer : n017.cluster.edu % 0.13/0.34 % Model : x86_64 x86_64 % 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.13/0.34 % Memory : 8042.1875MB % 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.13/0.34 % CPULimit : 300 % 0.13/0.34 % WCLimit : 300 % 0.13/0.34 % DateTime : Tue Aug 29 10:39:58 EDT 2023 % 0.13/0.34 % CPUTime : % 0.20/0.62 ________ _____ % 0.20/0.62 ___ __ \_________(_)________________________________ % 0.20/0.62 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/ % 0.20/0.62 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ ) % 0.20/0.62 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/ % 0.20/0.62 % 0.20/0.62 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic % 0.20/0.62 (2023-06-19) % 0.20/0.62 % 0.20/0.62 (c) Philipp Rümmer, 2009-2023 % 0.20/0.62 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen, % 0.20/0.62 Amanda Stjerna. % 0.20/0.62 Free software under BSD-3-Clause. % 0.20/0.62 % 0.20/0.62 For more information, visit http://www.philipp.ruemmer.org/princess.shtml % 0.20/0.62 % 0.20/0.62 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ... % 0.20/0.63 Running up to 7 provers in parallel. % 0.20/0.65 Prover 0: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1042961893 % 0.20/0.65 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1571432423 % 0.20/0.65 Prover 3: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1922548996 % 0.20/0.65 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=1868514696 % 0.20/0.65 Prover 2: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMinimalAndEmpty -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1065072994 % 0.20/0.65 Prover 5: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=complete -randomSeed=1259561288 % 0.20/0.65 Prover 6: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximalOutermost -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1399714365 % 2.11/1.07 Prover 1: Preprocessing ... % 2.11/1.07 Prover 4: Preprocessing ... % 2.96/1.11 Prover 5: Preprocessing ... % 2.96/1.11 Prover 0: Preprocessing ... % 2.96/1.11 Prover 3: Preprocessing ... % 2.96/1.11 Prover 2: Preprocessing ... % 2.96/1.11 Prover 6: Preprocessing ... % 4.32/1.40 Prover 6: Constructing countermodel ... % 4.32/1.42 Prover 1: Constructing countermodel ... % 4.32/1.42 Prover 3: Constructing countermodel ... % 5.39/1.45 Prover 4: Constructing countermodel ... % 5.65/1.47 Prover 0: Proving ... % 5.65/1.48 Prover 5: Proving ... % 5.89/1.52 Prover 2: Proving ... % 6.73/1.64 Prover 3: gave up % 6.73/1.64 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470 % 6.73/1.68 Prover 7: Preprocessing ... % 7.88/1.77 Prover 7: Constructing countermodel ... % 8.34/1.93 Prover 6: gave up % 8.34/1.93 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089 % 8.34/1.97 Prover 8: Preprocessing ... % 9.36/2.05 Prover 8: Warning: ignoring some quantifiers % 9.92/2.06 Prover 8: Constructing countermodel ... % 12.27/2.38 Prover 8: gave up % 12.27/2.38 Prover 9: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allMinimal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1423531889 % 12.27/2.40 Prover 9: Preprocessing ... % 13.25/2.49 Prover 9: Constructing countermodel ... % 16.30/2.94 Prover 1: gave up % 17.02/2.97 Prover 10: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=919308125 % 17.16/3.01 Prover 10: Preprocessing ... % 17.45/3.06 Prover 10: Constructing countermodel ... % 17.45/3.09 Prover 10: gave up % 17.45/3.09 Prover 11: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1509710984 % 18.06/3.11 Prover 11: Preprocessing ... % 18.06/3.18 Prover 11: Constructing countermodel ... % 24.17/3.93 Prover 9: proved (1552ms) % 24.17/3.94 % 24.17/3.94 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % 24.17/3.94 % 24.17/3.94 Prover 5: stopped % 24.50/3.96 Prover 13: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=1138197443 % 24.50/3.96 Prover 2: stopped % 24.50/3.97 Prover 13: Preprocessing ... % 24.50/3.97 Prover 16: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=completeFrugal -randomSeed=-2043353683 % 24.50/3.97 Prover 0: stopped % 24.50/3.97 Prover 19: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=-1780594085 % 24.50/3.97 Prover 16: Preprocessing ... % 24.50/3.99 Prover 19: Preprocessing ... % 24.95/4.01 Prover 13: Warning: ignoring some quantifiers % 24.95/4.02 Prover 13: Constructing countermodel ... % 24.95/4.02 Prover 16: Warning: ignoring some quantifiers % 24.95/4.02 Prover 19: Warning: ignoring some quantifiers % 24.95/4.02 Prover 16: Constructing countermodel ... % 24.95/4.03 Prover 19: Constructing countermodel ... % 24.95/4.04 Prover 13: gave up % 25.37/4.09 Prover 19: gave up % 26.74/4.28 Prover 16: gave up % 77.11/14.04 Prover 4: Found proof (size 140) % 77.11/14.04 Prover 4: proved (13393ms) % 77.11/14.04 Prover 7: stopped % 77.11/14.04 Prover 11: stopped % 77.11/14.04 % 77.11/14.04 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % 77.11/14.04 % 77.48/14.09 % SZS output start Proof for theBenchmark % 77.48/14.10 Assumptions after simplification: % 77.48/14.10 --------------------------------- % 77.48/14.10 % 77.48/14.10 (additive_associativity) % 77.48/14.14 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( ~ % 77.48/14.15 (addition(v3, v0) = v4) | ~ (addition(v2, v1) = v3) | ~ $i(v2) | ~ $i(v1) % 77.48/14.15 | ~ $i(v0) | ? [v5: $i] : (addition(v2, v5) = v4 & addition(v1, v0) = v5 & % 77.48/14.15 $i(v5) & $i(v4))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] % 77.48/14.15 : ! [v4: $i] : ( ~ (addition(v2, v3) = v4) | ~ (addition(v1, v0) = v3) | ~ % 77.48/14.15 $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v5: $i] : (addition(v5, v0) = v4 & % 77.48/14.15 addition(v2, v1) = v5 & $i(v5) & $i(v4))) % 77.48/14.15 % 77.48/14.15 (additive_commutativity) % 77.48/14.15 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (addition(v1, v0) = v2) | ~ % 77.48/14.15 $i(v1) | ~ $i(v0) | (addition(v0, v1) = v2 & $i(v2))) & ! [v0: $i] : ! % 77.48/14.15 [v1: $i] : ! [v2: $i] : ( ~ (addition(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | % 77.48/14.15 (addition(v1, v0) = v2 & $i(v2))) % 77.48/14.15 % 77.48/14.15 (distributivity1) % 77.48/14.15 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: % 77.48/14.15 $i] : ( ~ (multiplication(v0, v2) = v4) | ~ (multiplication(v0, v1) = v3) | % 77.48/14.15 ~ (addition(v3, v4) = v5) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v6: $i] % 77.48/14.15 : (multiplication(v0, v6) = v5 & addition(v1, v2) = v6 & $i(v6) & $i(v5))) & % 77.48/14.15 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( ~ % 77.48/14.15 (multiplication(v0, v3) = v4) | ~ (addition(v1, v2) = v3) | ~ $i(v2) | ~ % 77.48/14.15 $i(v1) | ~ $i(v0) | ? [v5: $i] : ? [v6: $i] : (multiplication(v0, v2) = % 77.48/14.15 v6 & multiplication(v0, v1) = v5 & addition(v5, v6) = v4 & $i(v6) & $i(v5) % 77.48/14.15 & $i(v4))) % 77.48/14.15 % 77.48/14.15 (distributivity2) % 77.48/14.16 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: % 77.48/14.16 $i] : ( ~ (multiplication(v1, v2) = v4) | ~ (multiplication(v0, v2) = v3) | % 77.48/14.16 ~ (addition(v3, v4) = v5) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v6: $i] % 77.48/14.16 : (multiplication(v6, v2) = v5 & addition(v0, v1) = v6 & $i(v6) & $i(v5))) & % 77.48/14.16 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( ~ % 77.48/14.16 (multiplication(v3, v2) = v4) | ~ (addition(v0, v1) = v3) | ~ $i(v2) | ~ % 77.48/14.16 $i(v1) | ~ $i(v0) | ? [v5: $i] : ? [v6: $i] : (multiplication(v1, v2) = % 77.48/14.16 v6 & multiplication(v0, v2) = v5 & addition(v5, v6) = v4 & $i(v6) & $i(v5) % 77.48/14.16 & $i(v4))) % 77.48/14.16 % 77.48/14.16 (goals) % 77.48/14.16 $i(one) & $i(zero) & ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : % 77.48/14.16 ? [v4: any] : ? [v5: any] : (strong_iteration(v1) = v2 & leq(v3, v2) = v5 & % 77.48/14.16 leq(v2, v3) = v4 & multiplication(v0, zero) = v1 & addition(one, v1) = v3 & % 77.48/14.16 $i(v3) & $i(v2) & $i(v1) & $i(v0) & ( ~ (v5 = 0) | ~ (v4 = 0))) % 77.48/14.16 % 77.48/14.16 (idempotence) % 77.48/14.16 ! [v0: $i] : ! [v1: $i] : (v1 = v0 | ~ (addition(v0, v0) = v1) | ~ $i(v0)) % 77.48/14.16 % 77.48/14.16 (infty_coinduction) % 77.48/14.17 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: % 77.48/14.17 int] : (v5 = 0 | ~ (strong_iteration(v0) = v3) | ~ (leq(v2, v4) = v5) | ~ % 77.48/14.17 (multiplication(v3, v1) = v4) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v6: % 77.48/14.17 $i] : ? [v7: $i] : ? [v8: int] : ( ~ (v8 = 0) & leq(v2, v7) = v8 & % 77.48/14.17 multiplication(v0, v2) = v6 & addition(v6, v1) = v7 & $i(v7) & $i(v6))) & % 77.48/14.17 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( ~ % 77.48/14.17 (multiplication(v0, v2) = v3) | ~ (addition(v3, v1) = v4) | ~ $i(v2) | ~ % 77.48/14.17 $i(v1) | ~ $i(v0) | ? [v5: any] : ? [v6: $i] : ? [v7: $i] : ? [v8: any] % 77.48/14.17 : (strong_iteration(v0) = v6 & leq(v2, v7) = v8 & leq(v2, v4) = v5 & % 77.48/14.17 multiplication(v6, v1) = v7 & $i(v7) & $i(v6) & ( ~ (v5 = 0) | v8 = 0))) % 77.48/14.17 % 77.48/14.17 (infty_unfold1) % 77.48/14.17 $i(one) & ! [v0: $i] : ! [v1: $i] : ( ~ (strong_iteration(v0) = v1) | ~ % 77.48/14.17 $i(v0) | ? [v2: $i] : (multiplication(v0, v1) = v2 & addition(v2, one) = v1 % 77.48/14.17 & $i(v2) & $i(v1))) % 77.48/14.17 % 77.48/14.17 (isolation) % 77.48/14.17 $i(zero) & ! [v0: $i] : ! [v1: $i] : ( ~ (strong_iteration(v0) = v1) | ~ % 77.48/14.17 $i(v0) | ? [v2: $i] : ? [v3: $i] : (star(v0) = v2 & multiplication(v1, % 77.48/14.17 zero) = v3 & addition(v2, v3) = v1 & $i(v3) & $i(v2) & $i(v1))) & ! % 77.48/14.17 [v0: $i] : ! [v1: $i] : ( ~ (star(v0) = v1) | ~ $i(v0) | ? [v2: $i] : ? % 77.48/14.17 [v3: $i] : (strong_iteration(v0) = v2 & multiplication(v2, zero) = v3 & % 77.48/14.17 addition(v1, v3) = v2 & $i(v3) & $i(v2))) % 77.48/14.17 % 77.48/14.17 (left_annihilation) % 77.48/14.17 $i(zero) & ! [v0: $i] : ! [v1: $i] : (v1 = zero | ~ (multiplication(zero, % 77.48/14.17 v0) = v1) | ~ $i(v0)) % 77.48/14.17 % 77.48/14.17 (multiplicative_associativity) % 77.48/14.17 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( ~ % 77.48/14.17 (multiplication(v3, v2) = v4) | ~ (multiplication(v0, v1) = v3) | ~ $i(v2) % 77.48/14.17 | ~ $i(v1) | ~ $i(v0) | ? [v5: $i] : (multiplication(v1, v2) = v5 & % 77.48/14.17 multiplication(v0, v5) = v4 & $i(v5) & $i(v4))) & ! [v0: $i] : ! [v1: % 77.48/14.18 $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( ~ (multiplication(v1, v2) % 77.48/14.18 = v3) | ~ (multiplication(v0, v3) = v4) | ~ $i(v2) | ~ $i(v1) | ~ % 77.48/14.18 $i(v0) | ? [v5: $i] : (multiplication(v5, v2) = v4 & multiplication(v0, v1) % 77.48/14.18 = v5 & $i(v5) & $i(v4))) % 77.48/14.18 % 77.48/14.18 (multiplicative_right_identity) % 77.90/14.18 $i(one) & ! [v0: $i] : ! [v1: $i] : (v1 = v0 | ~ (multiplication(v0, one) = % 77.90/14.18 v1) | ~ $i(v0)) % 77.90/14.18 % 77.90/14.18 (order) % 77.90/14.18 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v2 = v1 | ~ (addition(v0, v1) = % 77.90/14.18 v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: int] : ( ~ (v3 = 0) & leq(v0, v1) = % 77.90/14.18 v3)) & ! [v0: $i] : ! [v1: $i] : ! [v2: int] : (v2 = 0 | ~ (leq(v0, % 77.90/14.18 v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ( ~ (v3 = v1) & % 77.90/14.18 addition(v0, v1) = v3 & $i(v3))) & ! [v0: $i] : ! [v1: $i] : ( ~ % 77.90/14.18 (leq(v0, v1) = 0) | ~ $i(v1) | ~ $i(v0) | addition(v0, v1) = v1) & ! [v0: % 77.90/14.18 $i] : ! [v1: $i] : ( ~ (addition(v0, v1) = v1) | ~ $i(v1) | ~ $i(v0) | % 77.90/14.18 leq(v0, v1) = 0) % 77.90/14.18 % 77.90/14.18 (star_induction1) % 77.90/14.18 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: % 77.90/14.18 int] : (v5 = 0 | ~ (leq(v4, v2) = v5) | ~ (star(v0) = v3) | ~ % 77.90/14.18 (multiplication(v3, v1) = v4) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v6: % 77.90/14.18 $i] : ? [v7: $i] : ? [v8: int] : ( ~ (v8 = 0) & leq(v7, v2) = v8 & % 77.90/14.18 multiplication(v0, v2) = v6 & addition(v6, v1) = v7 & $i(v7) & $i(v6))) & % 77.90/14.18 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( ~ % 77.90/14.18 (multiplication(v0, v2) = v3) | ~ (addition(v3, v1) = v4) | ~ $i(v2) | ~ % 77.90/14.18 $i(v1) | ~ $i(v0) | ? [v5: any] : ? [v6: $i] : ? [v7: $i] : ? [v8: any] % 77.90/14.18 : (leq(v7, v2) = v8 & leq(v4, v2) = v5 & star(v0) = v6 & multiplication(v6, % 77.90/14.18 v1) = v7 & $i(v7) & $i(v6) & ( ~ (v5 = 0) | v8 = 0))) % 77.90/14.18 % 77.90/14.18 (star_induction2) % 77.90/14.19 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: % 77.90/14.19 int] : (v5 = 0 | ~ (leq(v4, v2) = v5) | ~ (star(v0) = v3) | ~ % 77.90/14.19 (multiplication(v1, v3) = v4) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v6: % 77.90/14.19 $i] : ? [v7: $i] : ? [v8: int] : ( ~ (v8 = 0) & leq(v7, v2) = v8 & % 77.90/14.19 multiplication(v2, v0) = v6 & addition(v6, v1) = v7 & $i(v7) & $i(v6))) & % 77.90/14.19 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( ~ % 77.90/14.19 (multiplication(v2, v0) = v3) | ~ (addition(v3, v1) = v4) | ~ $i(v2) | ~ % 77.90/14.19 $i(v1) | ~ $i(v0) | ? [v5: any] : ? [v6: $i] : ? [v7: $i] : ? [v8: any] % 77.90/14.19 : (leq(v7, v2) = v8 & leq(v4, v2) = v5 & star(v0) = v6 & multiplication(v1, % 77.90/14.19 v6) = v7 & $i(v7) & $i(v6) & ( ~ (v5 = 0) | v8 = 0))) % 77.90/14.19 % 77.90/14.19 (star_unfold2) % 77.90/14.19 $i(one) & ! [v0: $i] : ! [v1: $i] : ( ~ (star(v0) = v1) | ~ $i(v0) | ? % 77.90/14.19 [v2: $i] : (multiplication(v1, v0) = v2 & addition(one, v2) = v1 & $i(v2) & % 77.90/14.19 $i(v1))) % 77.90/14.19 % 77.90/14.19 (function-axioms) % 77.90/14.19 ! [v0: MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: $i] : ! % 77.90/14.19 [v3: $i] : (v1 = v0 | ~ (leq(v3, v2) = v1) | ~ (leq(v3, v2) = v0)) & ! [v0: % 77.90/14.19 $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ % 77.90/14.19 (multiplication(v3, v2) = v1) | ~ (multiplication(v3, v2) = v0)) & ! [v0: % 77.90/14.19 $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (addition(v3, % 77.90/14.19 v2) = v1) | ~ (addition(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! % 77.90/14.19 [v2: $i] : (v1 = v0 | ~ (strong_iteration(v2) = v1) | ~ % 77.90/14.19 (strong_iteration(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 % 77.90/14.19 = v0 | ~ (star(v2) = v1) | ~ (star(v2) = v0)) % 77.90/14.19 % 77.90/14.19 Further assumptions not needed in the proof: % 77.90/14.19 -------------------------------------------- % 77.90/14.19 additive_identity, multiplicative_left_identity, star_unfold1 % 77.90/14.19 % 77.90/14.19 Those formulas are unsatisfiable: % 77.90/14.19 --------------------------------- % 77.90/14.19 % 77.90/14.19 Begin of proof % 77.97/14.19 | % 77.97/14.19 | ALPHA: (additive_commutativity) implies: % 77.97/14.19 | (1) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (addition(v1, v0) = v2) | % 77.97/14.19 | ~ $i(v1) | ~ $i(v0) | (addition(v0, v1) = v2 & $i(v2))) % 77.97/14.19 | % 77.97/14.19 | ALPHA: (additive_associativity) implies: % 77.97/14.19 | (2) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( % 77.97/14.19 | ~ (addition(v3, v0) = v4) | ~ (addition(v2, v1) = v3) | ~ $i(v2) | % 77.97/14.19 | ~ $i(v1) | ~ $i(v0) | ? [v5: $i] : (addition(v2, v5) = v4 & % 77.97/14.19 | addition(v1, v0) = v5 & $i(v5) & $i(v4))) % 77.97/14.19 | % 77.97/14.19 | ALPHA: (multiplicative_associativity) implies: % 77.97/14.19 | (3) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( % 77.97/14.19 | ~ (multiplication(v3, v2) = v4) | ~ (multiplication(v0, v1) = v3) | % 77.97/14.19 | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v5: $i] : (multiplication(v1, % 77.97/14.19 | v2) = v5 & multiplication(v0, v5) = v4 & $i(v5) & $i(v4))) % 77.97/14.19 | % 77.97/14.19 | ALPHA: (multiplicative_right_identity) implies: % 77.97/14.19 | (4) ! [v0: $i] : ! [v1: $i] : (v1 = v0 | ~ (multiplication(v0, one) = % 77.97/14.19 | v1) | ~ $i(v0)) % 77.97/14.19 | % 77.97/14.19 | ALPHA: (distributivity1) implies: % 77.97/14.20 | (5) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( % 77.97/14.20 | ~ (multiplication(v0, v3) = v4) | ~ (addition(v1, v2) = v3) | ~ % 77.97/14.20 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v5: $i] : ? [v6: $i] : % 77.97/14.20 | (multiplication(v0, v2) = v6 & multiplication(v0, v1) = v5 & % 77.97/14.20 | addition(v5, v6) = v4 & $i(v6) & $i(v5) & $i(v4))) % 77.97/14.20 | % 77.97/14.20 | ALPHA: (distributivity2) implies: % 77.97/14.20 | (6) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( % 77.97/14.20 | ~ (multiplication(v3, v2) = v4) | ~ (addition(v0, v1) = v3) | ~ % 77.97/14.20 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v5: $i] : ? [v6: $i] : % 77.97/14.20 | (multiplication(v1, v2) = v6 & multiplication(v0, v2) = v5 & % 77.97/14.20 | addition(v5, v6) = v4 & $i(v6) & $i(v5) & $i(v4))) % 77.97/14.20 | % 77.97/14.20 | ALPHA: (left_annihilation) implies: % 77.97/14.20 | (7) ! [v0: $i] : ! [v1: $i] : (v1 = zero | ~ (multiplication(zero, v0) = % 77.97/14.20 | v1) | ~ $i(v0)) % 77.97/14.20 | % 77.97/14.20 | ALPHA: (star_unfold2) implies: % 77.97/14.20 | (8) ! [v0: $i] : ! [v1: $i] : ( ~ (star(v0) = v1) | ~ $i(v0) | ? [v2: % 77.97/14.20 | $i] : (multiplication(v1, v0) = v2 & addition(one, v2) = v1 & % 77.97/14.20 | $i(v2) & $i(v1))) % 77.97/14.20 | % 77.97/14.20 | ALPHA: (star_induction1) implies: % 77.97/14.20 | (9) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ( % 77.97/14.20 | ~ (multiplication(v0, v2) = v3) | ~ (addition(v3, v1) = v4) | ~ % 77.97/14.20 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v5: any] : ? [v6: $i] : ? [v7: % 77.97/14.20 | $i] : ? [v8: any] : (leq(v7, v2) = v8 & leq(v4, v2) = v5 & % 77.97/14.20 | star(v0) = v6 & multiplication(v6, v1) = v7 & $i(v7) & $i(v6) & ( ~ % 77.97/14.20 | (v5 = 0) | v8 = 0))) % 77.97/14.20 | % 77.97/14.20 | ALPHA: (star_induction2) implies: % 77.97/14.20 | (10) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : % 77.97/14.20 | ( ~ (multiplication(v2, v0) = v3) | ~ (addition(v3, v1) = v4) | ~ % 77.97/14.20 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v5: any] : ? [v6: $i] : ? % 77.97/14.20 | [v7: $i] : ? [v8: any] : (leq(v7, v2) = v8 & leq(v4, v2) = v5 & % 77.97/14.20 | star(v0) = v6 & multiplication(v1, v6) = v7 & $i(v7) & $i(v6) & ( % 77.97/14.20 | ~ (v5 = 0) | v8 = 0))) % 77.97/14.20 | % 77.97/14.20 | ALPHA: (infty_unfold1) implies: % 77.97/14.20 | (11) ! [v0: $i] : ! [v1: $i] : ( ~ (strong_iteration(v0) = v1) | ~ % 77.97/14.20 | $i(v0) | ? [v2: $i] : (multiplication(v0, v1) = v2 & addition(v2, % 77.97/14.20 | one) = v1 & $i(v2) & $i(v1))) % 77.97/14.20 | % 77.97/14.20 | ALPHA: (infty_coinduction) implies: % 77.97/14.20 | (12) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : % 77.97/14.21 | ( ~ (multiplication(v0, v2) = v3) | ~ (addition(v3, v1) = v4) | ~ % 77.97/14.21 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v5: any] : ? [v6: $i] : ? % 77.97/14.21 | [v7: $i] : ? [v8: any] : (strong_iteration(v0) = v6 & leq(v2, v7) = % 77.97/14.21 | v8 & leq(v2, v4) = v5 & multiplication(v6, v1) = v7 & $i(v7) & % 77.97/14.21 | $i(v6) & ( ~ (v5 = 0) | v8 = 0))) % 77.97/14.21 | % 77.97/14.21 | ALPHA: (isolation) implies: % 77.97/14.21 | (13) ! [v0: $i] : ! [v1: $i] : ( ~ (strong_iteration(v0) = v1) | ~ % 77.97/14.21 | $i(v0) | ? [v2: $i] : ? [v3: $i] : (star(v0) = v2 & % 77.97/14.21 | multiplication(v1, zero) = v3 & addition(v2, v3) = v1 & $i(v3) & % 77.97/14.21 | $i(v2) & $i(v1))) % 77.97/14.21 | % 77.97/14.21 | ALPHA: (order) implies: % 77.97/14.21 | (14) ! [v0: $i] : ! [v1: $i] : ! [v2: int] : (v2 = 0 | ~ (leq(v0, v1) = % 77.97/14.21 | v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ( ~ (v3 = v1) & % 77.97/14.21 | addition(v0, v1) = v3 & $i(v3))) % 77.97/14.21 | % 77.97/14.21 | ALPHA: (goals) implies: % 77.97/14.21 | (15) $i(zero) % 77.97/14.21 | (16) $i(one) % 77.97/14.21 | (17) ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ? [v4: any] : % 77.97/14.21 | ? [v5: any] : (strong_iteration(v1) = v2 & leq(v3, v2) = v5 & leq(v2, % 77.97/14.21 | v3) = v4 & multiplication(v0, zero) = v1 & addition(one, v1) = v3 % 77.97/14.21 | & $i(v3) & $i(v2) & $i(v1) & $i(v0) & ( ~ (v5 = 0) | ~ (v4 = 0))) % 77.97/14.21 | % 77.97/14.21 | ALPHA: (function-axioms) implies: % 77.97/14.21 | (18) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ % 77.97/14.21 | (strong_iteration(v2) = v1) | ~ (strong_iteration(v2) = v0)) % 78.08/14.21 | (19) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ % 78.08/14.21 | (addition(v3, v2) = v1) | ~ (addition(v3, v2) = v0)) % 78.08/14.21 | (20) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ % 78.08/14.21 | (multiplication(v3, v2) = v1) | ~ (multiplication(v3, v2) = v0)) % 78.08/14.21 | (21) ! [v0: MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: $i] % 78.08/14.21 | : ! [v3: $i] : (v1 = v0 | ~ (leq(v3, v2) = v1) | ~ (leq(v3, v2) = % 78.08/14.21 | v0)) % 78.08/14.21 | % 78.08/14.21 | DELTA: instantiating (17) with fresh symbols all_22_0, all_22_1, all_22_2, % 78.08/14.21 | all_22_3, all_22_4, all_22_5 gives: % 78.08/14.21 | (22) strong_iteration(all_22_4) = all_22_3 & leq(all_22_2, all_22_3) = % 78.08/14.21 | all_22_0 & leq(all_22_3, all_22_2) = all_22_1 & % 78.08/14.21 | multiplication(all_22_5, zero) = all_22_4 & addition(one, all_22_4) = % 78.08/14.21 | all_22_2 & $i(all_22_2) & $i(all_22_3) & $i(all_22_4) & $i(all_22_5) & % 78.08/14.21 | ( ~ (all_22_0 = 0) | ~ (all_22_1 = 0)) % 78.08/14.21 | % 78.08/14.21 | ALPHA: (22) implies: % 78.08/14.21 | (23) $i(all_22_5) % 78.08/14.21 | (24) $i(all_22_4) % 78.08/14.21 | (25) $i(all_22_3) % 78.08/14.21 | (26) addition(one, all_22_4) = all_22_2 % 78.08/14.21 | (27) multiplication(all_22_5, zero) = all_22_4 % 78.08/14.21 | (28) leq(all_22_3, all_22_2) = all_22_1 % 78.08/14.22 | (29) leq(all_22_2, all_22_3) = all_22_0 % 78.08/14.22 | (30) strong_iteration(all_22_4) = all_22_3 % 78.08/14.22 | (31) ~ (all_22_0 = 0) | ~ (all_22_1 = 0) % 78.08/14.22 | % 78.08/14.22 | GROUND_INST: instantiating (1) with all_22_4, one, all_22_2, simplifying with % 78.08/14.22 | (16), (24), (26) gives: % 78.11/14.22 | (32) addition(all_22_4, one) = all_22_2 & $i(all_22_2) % 78.11/14.22 | % 78.11/14.22 | ALPHA: (32) implies: % 78.11/14.22 | (33) $i(all_22_2) % 78.11/14.22 | (34) addition(all_22_4, one) = all_22_2 % 78.11/14.22 | % 78.11/14.22 | GROUND_INST: instantiating (14) with all_22_3, all_22_2, all_22_1, simplifying % 78.11/14.22 | with (25), (28), (33) gives: % 78.11/14.22 | (35) all_22_1 = 0 | ? [v0: any] : ( ~ (v0 = all_22_2) & addition(all_22_3, % 78.11/14.22 | all_22_2) = v0 & $i(v0)) % 78.11/14.22 | % 78.11/14.22 | GROUND_INST: instantiating (14) with all_22_2, all_22_3, all_22_0, simplifying % 78.11/14.22 | with (25), (29), (33) gives: % 78.11/14.22 | (36) all_22_0 = 0 | ? [v0: any] : ( ~ (v0 = all_22_3) & addition(all_22_2, % 78.11/14.22 | all_22_3) = v0 & $i(v0)) % 78.11/14.22 | % 78.11/14.22 | GROUND_INST: instantiating (13) with all_22_4, all_22_3, simplifying with % 78.11/14.22 | (24), (30) gives: % 78.11/14.22 | (37) ? [v0: $i] : ? [v1: $i] : (star(all_22_4) = v0 & % 78.11/14.22 | multiplication(all_22_3, zero) = v1 & addition(v0, v1) = all_22_3 & % 78.11/14.22 | $i(v1) & $i(v0) & $i(all_22_3)) % 78.11/14.22 | % 78.11/14.22 | GROUND_INST: instantiating (11) with all_22_4, all_22_3, simplifying with % 78.11/14.22 | (24), (30) gives: % 78.11/14.22 | (38) ? [v0: $i] : (multiplication(all_22_4, all_22_3) = v0 & addition(v0, % 78.11/14.22 | one) = all_22_3 & $i(v0) & $i(all_22_3)) % 78.11/14.22 | % 78.11/14.22 | DELTA: instantiating (38) with fresh symbol all_30_0 gives: % 78.11/14.22 | (39) multiplication(all_22_4, all_22_3) = all_30_0 & addition(all_30_0, % 78.11/14.22 | one) = all_22_3 & $i(all_30_0) & $i(all_22_3) % 78.11/14.22 | % 78.11/14.22 | ALPHA: (39) implies: % 78.11/14.22 | (40) $i(all_30_0) % 78.11/14.22 | (41) addition(all_30_0, one) = all_22_3 % 78.11/14.22 | (42) multiplication(all_22_4, all_22_3) = all_30_0 % 78.11/14.22 | % 78.11/14.22 | DELTA: instantiating (37) with fresh symbols all_32_0, all_32_1 gives: % 78.11/14.22 | (43) star(all_22_4) = all_32_1 & multiplication(all_22_3, zero) = all_32_0 % 78.11/14.22 | & addition(all_32_1, all_32_0) = all_22_3 & $i(all_32_0) & % 78.11/14.22 | $i(all_32_1) & $i(all_22_3) % 78.11/14.22 | % 78.11/14.22 | ALPHA: (43) implies: % 78.11/14.22 | (44) $i(all_32_1) % 78.11/14.22 | (45) $i(all_32_0) % 78.11/14.22 | (46) addition(all_32_1, all_32_0) = all_22_3 % 78.11/14.22 | (47) multiplication(all_22_3, zero) = all_32_0 % 78.11/14.22 | (48) star(all_22_4) = all_32_1 % 78.11/14.22 | % 78.11/14.22 | GROUND_INST: instantiating (1) with all_32_0, all_32_1, all_22_3, simplifying % 78.11/14.22 | with (44), (45), (46) gives: % 78.11/14.22 | (49) addition(all_32_0, all_32_1) = all_22_3 & $i(all_22_3) % 78.11/14.22 | % 78.11/14.22 | ALPHA: (49) implies: % 78.11/14.22 | (50) addition(all_32_0, all_32_1) = all_22_3 % 78.11/14.22 | % 78.11/14.22 | GROUND_INST: instantiating (3) with all_22_5, zero, all_22_3, all_22_4, % 78.11/14.22 | all_30_0, simplifying with (15), (23), (25), (27), (42) gives: % 78.11/14.23 | (51) ? [v0: $i] : (multiplication(all_22_5, v0) = all_30_0 & % 78.11/14.23 | multiplication(zero, all_22_3) = v0 & $i(v0) & $i(all_30_0)) % 78.11/14.23 | % 78.11/14.23 | GROUND_INST: instantiating (12) with all_22_4, one, all_22_3, all_30_0, % 78.11/14.23 | all_22_3, simplifying with (16), (24), (25), (41), (42) gives: % 78.11/14.23 | (52) ? [v0: any] : ? [v1: $i] : ? [v2: $i] : ? [v3: any] : % 78.11/14.23 | (strong_iteration(all_22_4) = v1 & leq(all_22_3, v2) = v3 & % 78.11/14.23 | leq(all_22_3, all_22_3) = v0 & multiplication(v1, one) = v2 & $i(v2) % 78.11/14.23 | & $i(v1) & ( ~ (v0 = 0) | v3 = 0)) % 78.11/14.23 | % 78.11/14.23 | GROUND_INST: instantiating (9) with all_22_4, one, all_22_3, all_30_0, % 78.11/14.23 | all_22_3, simplifying with (16), (24), (25), (41), (42) gives: % 78.11/14.23 | (53) ? [v0: any] : ? [v1: $i] : ? [v2: $i] : ? [v3: any] : (leq(v2, % 78.11/14.23 | all_22_3) = v3 & leq(all_22_3, all_22_3) = v0 & star(all_22_4) = % 78.11/14.23 | v1 & multiplication(v1, one) = v2 & $i(v2) & $i(v1) & ( ~ (v0 = 0) | % 78.11/14.23 | v3 = 0)) % 78.11/14.23 | % 78.11/14.23 | GROUND_INST: instantiating (5) with all_22_4, all_30_0, one, all_22_3, % 78.11/14.23 | all_30_0, simplifying with (16), (24), (40), (41), (42) gives: % 78.11/14.23 | (54) ? [v0: $i] : ? [v1: $i] : (multiplication(all_22_4, all_30_0) = v0 & % 78.11/14.23 | multiplication(all_22_4, one) = v1 & addition(v0, v1) = all_30_0 & % 78.11/14.23 | $i(v1) & $i(v0)) % 78.11/14.23 | % 78.11/14.23 | GROUND_INST: instantiating (6) with all_30_0, one, zero, all_22_3, all_32_0, % 78.11/14.23 | simplifying with (15), (16), (40), (41), (47) gives: % 78.11/14.23 | (55) ? [v0: $i] : ? [v1: $i] : (multiplication(all_30_0, zero) = v0 & % 78.11/14.23 | multiplication(one, zero) = v1 & addition(v0, v1) = all_32_0 & % 78.11/14.23 | $i(v1) & $i(v0) & $i(all_32_0)) % 78.11/14.23 | % 78.11/14.23 | GROUND_INST: instantiating (8) with all_22_4, all_32_1, simplifying with (24), % 78.11/14.23 | (48) gives: % 78.11/14.23 | (56) ? [v0: $i] : (multiplication(all_32_1, all_22_4) = v0 & addition(one, % 78.11/14.23 | v0) = all_32_1 & $i(v0) & $i(all_32_1)) % 78.11/14.23 | % 78.11/14.23 | DELTA: instantiating (56) with fresh symbol all_42_0 gives: % 78.11/14.23 | (57) multiplication(all_32_1, all_22_4) = all_42_0 & addition(one, % 78.11/14.23 | all_42_0) = all_32_1 & $i(all_42_0) & $i(all_32_1) % 78.11/14.23 | % 78.11/14.23 | ALPHA: (57) implies: % 78.11/14.23 | (58) $i(all_42_0) % 78.11/14.23 | (59) addition(one, all_42_0) = all_32_1 % 78.11/14.23 | % 78.11/14.23 | DELTA: instantiating (51) with fresh symbol all_44_0 gives: % 78.11/14.23 | (60) multiplication(all_22_5, all_44_0) = all_30_0 & multiplication(zero, % 78.11/14.23 | all_22_3) = all_44_0 & $i(all_44_0) & $i(all_30_0) % 78.11/14.23 | % 78.11/14.23 | ALPHA: (60) implies: % 78.11/14.23 | (61) multiplication(zero, all_22_3) = all_44_0 % 78.11/14.23 | (62) multiplication(all_22_5, all_44_0) = all_30_0 % 78.11/14.23 | % 78.11/14.23 | DELTA: instantiating (54) with fresh symbols all_48_0, all_48_1 gives: % 78.11/14.23 | (63) multiplication(all_22_4, all_30_0) = all_48_1 & % 78.11/14.23 | multiplication(all_22_4, one) = all_48_0 & addition(all_48_1, % 78.11/14.23 | all_48_0) = all_30_0 & $i(all_48_0) & $i(all_48_1) % 78.11/14.23 | % 78.11/14.23 | ALPHA: (63) implies: % 78.11/14.23 | (64) $i(all_48_1) % 78.11/14.23 | (65) $i(all_48_0) % 78.11/14.23 | (66) addition(all_48_1, all_48_0) = all_30_0 % 78.11/14.23 | (67) multiplication(all_22_4, one) = all_48_0 % 78.11/14.23 | % 78.11/14.23 | DELTA: instantiating (55) with fresh symbols all_50_0, all_50_1 gives: % 78.11/14.24 | (68) multiplication(all_30_0, zero) = all_50_1 & multiplication(one, zero) % 78.11/14.24 | = all_50_0 & addition(all_50_1, all_50_0) = all_32_0 & $i(all_50_0) & % 78.11/14.24 | $i(all_50_1) & $i(all_32_0) % 78.11/14.24 | % 78.11/14.24 | DELTA: instantiating (53) with fresh symbols all_56_0, all_56_1, all_56_2, % 78.11/14.24 | all_56_3 gives: % 78.11/14.24 | (69) leq(all_56_1, all_22_3) = all_56_0 & leq(all_22_3, all_22_3) = % 78.11/14.24 | all_56_3 & star(all_22_4) = all_56_2 & multiplication(all_56_2, one) = % 78.11/14.24 | all_56_1 & $i(all_56_1) & $i(all_56_2) & ( ~ (all_56_3 = 0) | all_56_0 % 78.11/14.24 | = 0) % 78.11/14.24 | % 78.11/14.24 | ALPHA: (69) implies: % 78.11/14.24 | (70) leq(all_22_3, all_22_3) = all_56_3 % 78.11/14.24 | % 78.11/14.24 | DELTA: instantiating (52) with fresh symbols all_58_0, all_58_1, all_58_2, % 78.11/14.24 | all_58_3 gives: % 78.11/14.24 | (71) strong_iteration(all_22_4) = all_58_2 & leq(all_22_3, all_58_1) = % 78.11/14.24 | all_58_0 & leq(all_22_3, all_22_3) = all_58_3 & % 78.11/14.24 | multiplication(all_58_2, one) = all_58_1 & $i(all_58_1) & $i(all_58_2) % 78.11/14.24 | & ( ~ (all_58_3 = 0) | all_58_0 = 0) % 78.11/14.24 | % 78.11/14.24 | ALPHA: (71) implies: % 78.11/14.24 | (72) $i(all_58_2) % 78.11/14.24 | (73) $i(all_58_1) % 78.11/14.24 | (74) multiplication(all_58_2, one) = all_58_1 % 78.11/14.24 | (75) leq(all_22_3, all_22_3) = all_58_3 % 78.11/14.24 | (76) leq(all_22_3, all_58_1) = all_58_0 % 78.11/14.24 | (77) strong_iteration(all_22_4) = all_58_2 % 78.11/14.24 | % 78.11/14.24 | GROUND_INST: instantiating (21) with all_56_3, all_58_3, all_22_3, all_22_3, % 78.11/14.24 | simplifying with (70), (75) gives: % 78.11/14.24 | (78) all_58_3 = all_56_3 % 78.11/14.24 | % 78.11/14.24 | GROUND_INST: instantiating (18) with all_22_3, all_58_2, all_22_4, simplifying % 78.11/14.24 | with (30), (77) gives: % 78.11/14.24 | (79) all_58_2 = all_22_3 % 78.11/14.24 | % 78.11/14.24 | REDUCE: (74), (79) imply: % 78.11/14.24 | (80) multiplication(all_22_3, one) = all_58_1 % 78.11/14.24 | % 78.11/14.24 | GROUND_INST: instantiating (2) with all_32_0, all_42_0, one, all_32_1, % 78.11/14.24 | all_22_3, simplifying with (16), (45), (46), (58), (59) gives: % 78.11/14.24 | (81) ? [v0: $i] : (addition(all_42_0, all_32_0) = v0 & addition(one, v0) = % 78.11/14.24 | all_22_3 & $i(v0) & $i(all_22_3)) % 78.11/14.24 | % 78.11/14.24 | GROUND_INST: instantiating (1) with all_42_0, one, all_32_1, simplifying with % 78.11/14.24 | (16), (58), (59) gives: % 78.11/14.24 | (82) addition(all_42_0, one) = all_32_1 & $i(all_32_1) % 78.11/14.24 | % 78.11/14.24 | GROUND_INST: instantiating (10) with zero, all_32_1, all_22_3, all_32_0, % 78.11/14.24 | all_22_3, simplifying with (15), (25), (44), (47), (50) gives: % 78.11/14.24 | (83) ? [v0: any] : ? [v1: $i] : ? [v2: $i] : ? [v3: any] : (leq(v2, % 78.11/14.24 | all_22_3) = v3 & leq(all_22_3, all_22_3) = v0 & star(zero) = v1 & % 78.11/14.24 | multiplication(all_32_1, v1) = v2 & $i(v2) & $i(v1) & ( ~ (v0 = 0) | % 78.11/14.24 | v3 = 0)) % 78.11/14.24 | % 78.11/14.24 | GROUND_INST: instantiating (2) with one, all_48_0, all_48_1, all_30_0, % 78.11/14.24 | all_22_3, simplifying with (16), (41), (64), (65), (66) gives: % 78.11/14.24 | (84) ? [v0: $i] : (addition(all_48_0, one) = v0 & addition(all_48_1, v0) = % 78.11/14.24 | all_22_3 & $i(v0) & $i(all_22_3)) % 78.11/14.24 | % 78.11/14.24 | GROUND_INST: instantiating (7) with all_22_3, all_44_0, simplifying with (25), % 78.11/14.24 | (61) gives: % 78.11/14.24 | (85) all_44_0 = zero % 78.11/14.24 | % 78.11/14.24 | GROUND_INST: instantiating (4) with all_22_4, all_48_0, simplifying with (24), % 78.11/14.24 | (67) gives: % 78.11/14.24 | (86) all_48_0 = all_22_4 % 78.11/14.24 | % 78.11/14.25 | GROUND_INST: instantiating (4) with all_22_3, all_58_1, simplifying with (25), % 78.11/14.25 | (80) gives: % 78.11/14.25 | (87) all_58_1 = all_22_3 % 78.11/14.25 | % 78.11/14.25 | GROUND_INST: instantiating (14) with all_22_3, all_22_3, all_56_3, simplifying % 78.11/14.25 | with (25), (70) gives: % 78.11/14.25 | (88) all_56_3 = 0 | ? [v0: any] : ( ~ (v0 = all_22_3) & addition(all_22_3, % 78.11/14.25 | all_22_3) = v0 & $i(v0)) % 78.11/14.25 | % 78.11/14.25 | GROUND_INST: instantiating (14) with all_22_3, all_58_1, all_58_0, simplifying % 78.11/14.25 | with (25), (73), (76) gives: % 78.11/14.25 | (89) all_58_0 = 0 | ? [v0: any] : ( ~ (v0 = all_58_1) & addition(all_22_3, % 78.11/14.25 | all_58_1) = v0 & $i(v0)) % 78.11/14.25 | % 78.11/14.25 | DELTA: instantiating (81) with fresh symbol all_94_0 gives: % 78.11/14.25 | (90) addition(all_42_0, all_32_0) = all_94_0 & addition(one, all_94_0) = % 78.11/14.25 | all_22_3 & $i(all_94_0) & $i(all_22_3) % 78.11/14.25 | % 78.11/14.25 | ALPHA: (90) implies: % 78.11/14.25 | (91) $i(all_94_0) % 78.11/14.25 | (92) addition(one, all_94_0) = all_22_3 % 78.11/14.25 | % 78.11/14.25 | DELTA: instantiating (84) with fresh symbol all_124_0 gives: % 78.11/14.25 | (93) addition(all_48_0, one) = all_124_0 & addition(all_48_1, all_124_0) = % 78.11/14.25 | all_22_3 & $i(all_124_0) & $i(all_22_3) % 78.11/14.25 | % 78.11/14.25 | ALPHA: (93) implies: % 78.11/14.25 | (94) addition(all_48_0, one) = all_124_0 % 78.11/14.25 | % 78.11/14.25 | DELTA: instantiating (83) with fresh symbols all_216_0, all_216_1, all_216_2, % 78.11/14.25 | all_216_3 gives: % 78.11/14.25 | (95) leq(all_216_1, all_22_3) = all_216_0 & leq(all_22_3, all_22_3) = % 78.11/14.25 | all_216_3 & star(zero) = all_216_2 & multiplication(all_32_1, % 78.11/14.25 | all_216_2) = all_216_1 & $i(all_216_1) & $i(all_216_2) & ( ~ % 78.11/14.25 | (all_216_3 = 0) | all_216_0 = 0) % 78.11/14.25 | % 78.11/14.25 | ALPHA: (95) implies: % 78.11/14.25 | (96) leq(all_22_3, all_22_3) = all_216_3 % 78.11/14.25 | % 78.11/14.25 | REDUCE: (76), (87) imply: % 78.11/14.25 | (97) leq(all_22_3, all_22_3) = all_58_0 % 78.11/14.25 | % 78.11/14.25 | REDUCE: (62), (85) imply: % 78.11/14.25 | (98) multiplication(all_22_5, zero) = all_30_0 % 78.11/14.25 | % 78.11/14.25 | REDUCE: (86), (94) imply: % 78.11/14.25 | (99) addition(all_22_4, one) = all_124_0 % 78.11/14.25 | % 78.11/14.25 | GROUND_INST: instantiating (19) with all_22_2, all_124_0, one, all_22_4, % 78.11/14.25 | simplifying with (34), (99) gives: % 78.11/14.25 | (100) all_124_0 = all_22_2 % 78.11/14.25 | % 78.11/14.25 | GROUND_INST: instantiating (20) with all_22_4, all_30_0, zero, all_22_5, % 78.11/14.25 | simplifying with (27), (98) gives: % 78.11/14.25 | (101) all_30_0 = all_22_4 % 78.11/14.25 | % 78.11/14.25 | GROUND_INST: instantiating (21) with all_56_3, all_216_3, all_22_3, all_22_3, % 78.11/14.25 | simplifying with (70), (96) gives: % 78.11/14.25 | (102) all_216_3 = all_56_3 % 78.11/14.25 | % 78.11/14.25 | GROUND_INST: instantiating (21) with all_58_0, all_216_3, all_22_3, all_22_3, % 78.11/14.25 | simplifying with (96), (97) gives: % 78.11/14.25 | (103) all_216_3 = all_58_0 % 78.11/14.25 | % 78.11/14.25 | COMBINE_EQS: (102), (103) imply: % 78.11/14.25 | (104) all_58_0 = all_56_3 % 78.11/14.25 | % 78.11/14.25 | SIMP: (104) implies: % 78.11/14.25 | (105) all_58_0 = all_56_3 % 78.11/14.25 | % 78.11/14.25 | REDUCE: (41), (101) imply: % 78.11/14.25 | (106) addition(all_22_4, one) = all_22_3 % 78.11/14.25 | % 78.11/14.25 | GROUND_INST: instantiating (19) with all_22_2, all_22_3, one, all_22_4, % 78.11/14.25 | simplifying with (34), (106) gives: % 78.11/14.25 | (107) all_22_2 = all_22_3 % 78.11/14.25 | % 78.11/14.25 | REDUCE: (29), (107) imply: % 78.11/14.25 | (108) leq(all_22_3, all_22_3) = all_22_0 % 78.11/14.25 | % 78.11/14.25 | REDUCE: (28), (107) imply: % 78.11/14.25 | (109) leq(all_22_3, all_22_3) = all_22_1 % 78.11/14.25 | % 78.11/14.25 | GROUND_INST: instantiating (21) with all_56_3, all_22_1, all_22_3, all_22_3, % 78.11/14.25 | simplifying with (70), (109) gives: % 78.11/14.25 | (110) all_56_3 = all_22_1 % 78.11/14.25 | % 78.11/14.25 | GROUND_INST: instantiating (21) with all_56_3, all_22_0, all_22_3, all_22_3, % 78.11/14.25 | simplifying with (70), (108) gives: % 78.11/14.25 | (111) all_56_3 = all_22_0 % 78.11/14.25 | % 78.11/14.25 | COMBINE_EQS: (110), (111) imply: % 78.11/14.25 | (112) all_22_0 = all_22_1 % 78.11/14.25 | % 78.11/14.25 | COMBINE_EQS: (105), (110) imply: % 78.11/14.26 | (113) all_58_0 = all_22_1 % 78.11/14.26 | % 78.11/14.26 | BETA: splitting (31) gives: % 78.11/14.26 | % 78.11/14.26 | Case 1: % 78.11/14.26 | | % 78.11/14.26 | | (114) ~ (all_22_0 = 0) % 78.11/14.26 | | % 78.11/14.26 | | REDUCE: (112), (114) imply: % 78.11/14.26 | | (115) ~ (all_22_1 = 0) % 78.11/14.26 | | % 78.11/14.26 | | BETA: splitting (35) gives: % 78.11/14.26 | | % 78.11/14.26 | | Case 1: % 78.11/14.26 | | | % 78.11/14.26 | | | (116) all_22_1 = 0 % 78.11/14.26 | | | % 78.11/14.26 | | | REDUCE: (115), (116) imply: % 78.11/14.26 | | | (117) $false % 78.11/14.26 | | | % 78.11/14.26 | | | CLOSE: (117) is inconsistent. % 78.11/14.26 | | | % 78.11/14.26 | | Case 2: % 78.11/14.26 | | | % 78.11/14.26 | | | (118) ? [v0: any] : ( ~ (v0 = all_22_2) & addition(all_22_3, all_22_2) % 78.11/14.26 | | | = v0 & $i(v0)) % 78.11/14.26 | | | % 78.11/14.26 | | | DELTA: instantiating (118) with fresh symbol all_256_0 gives: % 78.11/14.26 | | | (119) ~ (all_256_0 = all_22_2) & addition(all_22_3, all_22_2) = % 78.11/14.26 | | | all_256_0 & $i(all_256_0) % 78.11/14.26 | | | % 78.11/14.26 | | | ALPHA: (119) implies: % 78.11/14.26 | | | (120) addition(all_22_3, all_22_2) = all_256_0 % 78.11/14.26 | | | % 78.11/14.26 | | | REDUCE: (107), (120) imply: % 78.11/14.26 | | | (121) addition(all_22_3, all_22_3) = all_256_0 % 78.11/14.26 | | | % 78.11/14.26 | | | BETA: splitting (36) gives: % 78.11/14.26 | | | % 78.11/14.26 | | | Case 1: % 78.11/14.26 | | | | % 78.11/14.26 | | | | (122) all_22_0 = 0 % 78.11/14.26 | | | | % 78.11/14.26 | | | | COMBINE_EQS: (112), (122) imply: % 78.11/14.26 | | | | (123) all_22_1 = 0 % 78.11/14.26 | | | | % 78.11/14.26 | | | | SIMP: (123) implies: % 78.11/14.26 | | | | (124) all_22_1 = 0 % 78.11/14.26 | | | | % 78.11/14.26 | | | | REDUCE: (115), (124) imply: % 78.11/14.26 | | | | (125) $false % 78.11/14.26 | | | | % 78.11/14.26 | | | | CLOSE: (125) is inconsistent. % 78.11/14.26 | | | | % 78.11/14.26 | | | Case 2: % 78.11/14.26 | | | | % 78.11/14.26 | | | | (126) ? [v0: any] : ( ~ (v0 = all_22_3) & addition(all_22_2, % 78.11/14.26 | | | | all_22_3) = v0 & $i(v0)) % 78.11/14.26 | | | | % 78.11/14.26 | | | | DELTA: instantiating (126) with fresh symbol all_261_0 gives: % 78.11/14.26 | | | | (127) ~ (all_261_0 = all_22_3) & addition(all_22_2, all_22_3) = % 78.11/14.26 | | | | all_261_0 & $i(all_261_0) % 78.11/14.26 | | | | % 78.11/14.26 | | | | ALPHA: (127) implies: % 78.11/14.26 | | | | (128) ~ (all_261_0 = all_22_3) % 78.11/14.26 | | | | (129) addition(all_22_2, all_22_3) = all_261_0 % 78.11/14.26 | | | | % 78.11/14.26 | | | | REDUCE: (107), (129) imply: % 78.11/14.26 | | | | (130) addition(all_22_3, all_22_3) = all_261_0 % 78.11/14.26 | | | | % 78.11/14.26 | | | | BETA: splitting (89) gives: % 78.11/14.26 | | | | % 78.11/14.26 | | | | Case 1: % 78.11/14.26 | | | | | % 78.11/14.26 | | | | | (131) all_58_0 = 0 % 78.11/14.26 | | | | | % 78.11/14.26 | | | | | COMBINE_EQS: (113), (131) imply: % 78.11/14.27 | | | | | (132) all_22_1 = 0 % 78.11/14.27 | | | | | % 78.11/14.27 | | | | | SIMP: (132) implies: % 78.11/14.27 | | | | | (133) all_22_1 = 0 % 78.11/14.27 | | | | | % 78.11/14.27 | | | | | REDUCE: (115), (133) imply: % 78.11/14.27 | | | | | (134) $false % 78.11/14.27 | | | | | % 78.11/14.27 | | | | | CLOSE: (134) is inconsistent. % 78.11/14.27 | | | | | % 78.11/14.27 | | | | Case 2: % 78.11/14.27 | | | | | % 78.11/14.27 | | | | | (135) ~ (all_58_0 = 0) % 78.11/14.27 | | | | | (136) ? [v0: any] : ( ~ (v0 = all_58_1) & addition(all_22_3, % 78.11/14.27 | | | | | all_58_1) = v0 & $i(v0)) % 78.11/14.27 | | | | | % 78.11/14.27 | | | | | DELTA: instantiating (136) with fresh symbol all_266_0 gives: % 78.11/14.27 | | | | | (137) ~ (all_266_0 = all_58_1) & addition(all_22_3, all_58_1) = % 78.11/14.27 | | | | | all_266_0 & $i(all_266_0) % 78.11/14.27 | | | | | % 78.11/14.27 | | | | | ALPHA: (137) implies: % 78.11/14.27 | | | | | (138) addition(all_22_3, all_58_1) = all_266_0 % 78.11/14.27 | | | | | % 78.11/14.27 | | | | | REDUCE: (87), (138) imply: % 78.11/14.27 | | | | | (139) addition(all_22_3, all_22_3) = all_266_0 % 78.11/14.27 | | | | | % 78.11/14.27 | | | | | BETA: splitting (88) gives: % 78.11/14.27 | | | | | % 78.11/14.27 | | | | | Case 1: % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | (140) all_56_3 = 0 % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | COMBINE_EQS: (110), (140) imply: % 78.11/14.27 | | | | | | (141) all_22_1 = 0 % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | SIMP: (141) implies: % 78.11/14.27 | | | | | | (142) all_22_1 = 0 % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | REDUCE: (115), (142) imply: % 78.11/14.27 | | | | | | (143) $false % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | CLOSE: (143) is inconsistent. % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | Case 2: % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | (144) ? [v0: any] : ( ~ (v0 = all_22_3) & addition(all_22_3, % 78.11/14.27 | | | | | | all_22_3) = v0 & $i(v0)) % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | DELTA: instantiating (144) with fresh symbol all_271_0 gives: % 78.11/14.27 | | | | | | (145) ~ (all_271_0 = all_22_3) & addition(all_22_3, all_22_3) = % 78.11/14.27 | | | | | | all_271_0 & $i(all_271_0) % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | ALPHA: (145) implies: % 78.11/14.27 | | | | | | (146) addition(all_22_3, all_22_3) = all_271_0 % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | GROUND_INST: instantiating (19) with all_261_0, all_266_0, all_22_3, % 78.11/14.27 | | | | | | all_22_3, simplifying with (130), (139) gives: % 78.11/14.27 | | | | | | (147) all_266_0 = all_261_0 % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | GROUND_INST: instantiating (19) with all_266_0, all_271_0, all_22_3, % 78.11/14.27 | | | | | | all_22_3, simplifying with (139), (146) gives: % 78.11/14.27 | | | | | | (148) all_271_0 = all_266_0 % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | GROUND_INST: instantiating (19) with all_256_0, all_271_0, all_22_3, % 78.11/14.27 | | | | | | all_22_3, simplifying with (121), (146) gives: % 78.11/14.27 | | | | | | (149) all_271_0 = all_256_0 % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | COMBINE_EQS: (148), (149) imply: % 78.11/14.27 | | | | | | (150) all_266_0 = all_256_0 % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | SIMP: (150) implies: % 78.11/14.27 | | | | | | (151) all_266_0 = all_256_0 % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | COMBINE_EQS: (147), (151) imply: % 78.11/14.27 | | | | | | (152) all_261_0 = all_256_0 % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | SIMP: (152) implies: % 78.11/14.27 | | | | | | (153) all_261_0 = all_256_0 % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | REDUCE: (128), (153) imply: % 78.11/14.27 | | | | | | (154) ~ (all_256_0 = all_22_3) % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | GROUND_INST: instantiating (1) with all_94_0, one, all_22_3, % 78.11/14.27 | | | | | | simplifying with (16), (91), (92) gives: % 78.11/14.27 | | | | | | (155) addition(all_94_0, one) = all_22_3 & $i(all_22_3) % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | GROUND_INST: instantiating (idempotence) with all_22_3, all_256_0, % 78.11/14.27 | | | | | | simplifying with (25), (121) gives: % 78.11/14.27 | | | | | | (156) all_256_0 = all_22_3 % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | REDUCE: (154), (156) imply: % 78.11/14.27 | | | | | | (157) $false % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | | CLOSE: (157) is inconsistent. % 78.11/14.27 | | | | | | % 78.11/14.27 | | | | | End of split % 78.11/14.27 | | | | | % 78.11/14.27 | | | | End of split % 78.11/14.27 | | | | % 78.11/14.27 | | | End of split % 78.11/14.27 | | | % 78.11/14.27 | | End of split % 78.11/14.27 | | % 78.11/14.27 | Case 2: % 78.11/14.27 | | % 78.11/14.27 | | (158) all_22_0 = 0 % 78.11/14.27 | | (159) ~ (all_22_1 = 0) % 78.11/14.27 | | % 78.11/14.27 | | COMBINE_EQS: (112), (158) imply: % 78.11/14.27 | | (160) all_22_1 = 0 % 78.11/14.27 | | % 78.11/14.27 | | SIMP: (160) implies: % 78.11/14.27 | | (161) all_22_1 = 0 % 78.11/14.27 | | % 78.11/14.27 | | REDUCE: (159), (161) imply: % 78.11/14.27 | | (162) $false % 78.11/14.27 | | % 78.11/14.27 | | CLOSE: (162) is inconsistent. % 78.11/14.27 | | % 78.11/14.27 | End of split % 78.11/14.27 | % 78.11/14.27 End of proof % 78.11/14.27 % SZS output end Proof for theBenchmark % 78.11/14.27 % 78.11/14.27 13650ms %------------------------------------------------------------------------------