%------------------------------------------------------------------------------ % File : Princess---230619 % Problem : SET090+1 : TPTP v8.1.2. Bugfixed v7.3.0. % Transfm : none % Format : tptp % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s % Computer : n015.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 15:23:43 EDT 2023 % Result : Theorem 28.77s 4.59s % Output : Proof 34.35s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.12 % Problem : SET090+1 : TPTP v8.1.2. Bugfixed v7.3.0. % 0.00/0.13 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s % 0.12/0.34 % Computer : n015.cluster.edu % 0.12/0.34 % Model : x86_64 x86_64 % 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.34 % Memory : 8042.1875MB % 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.34 % CPULimit : 300 % 0.12/0.34 % WCLimit : 300 % 0.12/0.34 % DateTime : Sat Aug 26 09:35:53 EDT 2023 % 0.12/0.34 % CPUTime : % 0.20/0.58 ________ _____ % 0.20/0.58 ___ __ \_________(_)________________________________ % 0.20/0.58 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/ % 0.20/0.58 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ ) % 0.20/0.58 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/ % 0.20/0.58 % 0.20/0.58 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic % 0.20/0.58 (2023-06-19) % 0.20/0.58 % 0.20/0.58 (c) Philipp Rümmer, 2009-2023 % 0.20/0.58 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen, % 0.20/0.58 Amanda Stjerna. % 0.20/0.58 Free software under BSD-3-Clause. % 0.20/0.58 % 0.20/0.58 For more information, visit http://www.philipp.ruemmer.org/princess.shtml % 0.20/0.58 % 0.20/0.58 Loading /export/starexec/sandbox2/benchmark/theBenchmark.p ... % 0.20/0.59 Running up to 7 provers in parallel. % 0.20/0.60 Prover 0: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1042961893 % 0.20/0.60 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1571432423 % 0.20/0.60 Prover 2: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMinimalAndEmpty -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1065072994 % 0.20/0.60 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=1868514696 % 0.20/0.60 Prover 3: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1922548996 % 0.20/0.60 Prover 6: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximalOutermost -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1399714365 % 0.20/0.60 Prover 5: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=complete -randomSeed=1259561288 % 3.28/1.16 Prover 1: Preprocessing ... % 3.28/1.17 Prover 4: Preprocessing ... % 3.28/1.21 Prover 6: Preprocessing ... % 3.28/1.21 Prover 5: Preprocessing ... % 3.28/1.21 Prover 3: Preprocessing ... % 3.28/1.21 Prover 0: Preprocessing ... % 3.28/1.21 Prover 2: Preprocessing ... % 8.66/1.96 Prover 1: Warning: ignoring some quantifiers % 9.49/2.04 Prover 3: Warning: ignoring some quantifiers % 9.49/2.04 Prover 6: Proving ... % 9.49/2.06 Prover 1: Constructing countermodel ... % 9.49/2.08 Prover 3: Constructing countermodel ... % 9.49/2.08 Prover 5: Proving ... % 9.49/2.08 Prover 4: Warning: ignoring some quantifiers % 9.49/2.13 Prover 4: Constructing countermodel ... % 9.49/2.17 Prover 2: Proving ... % 11.29/2.27 Prover 0: Proving ... % 28.77/4.59 Prover 2: proved (3989ms) % 28.77/4.59 % 28.77/4.59 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p % 28.77/4.59 % 28.77/4.59 Prover 3: stopped % 28.77/4.59 Prover 5: stopped % 28.77/4.59 Prover 0: stopped % 28.77/4.62 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470 % 28.77/4.62 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089 % 28.77/4.62 Prover 10: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=919308125 % 28.77/4.62 Prover 6: stopped % 28.77/4.62 Prover 11: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1509710984 % 28.77/4.62 Prover 13: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=1138197443 % 29.57/4.70 Prover 10: Preprocessing ... % 29.57/4.70 Prover 7: Preprocessing ... % 29.57/4.71 Prover 13: Preprocessing ... % 29.64/4.71 Prover 8: Preprocessing ... % 29.64/4.72 Prover 11: Preprocessing ... % 30.40/4.87 Prover 10: Warning: ignoring some quantifiers % 30.40/4.88 Prover 10: Constructing countermodel ... % 30.40/4.88 Prover 8: Warning: ignoring some quantifiers % 30.40/4.89 Prover 7: Warning: ignoring some quantifiers % 30.40/4.89 Prover 8: Constructing countermodel ... % 31.08/4.90 Prover 13: Warning: ignoring some quantifiers % 31.08/4.91 Prover 7: Constructing countermodel ... % 31.08/4.91 Prover 13: Constructing countermodel ... % 31.08/4.94 Prover 11: Warning: ignoring some quantifiers % 31.08/4.95 Prover 11: Constructing countermodel ... % 33.38/5.22 Prover 10: gave up % 33.38/5.23 Prover 7: Found proof (size 88) % 33.38/5.23 Prover 7: proved (627ms) % 33.38/5.23 Prover 16: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=completeFrugal -randomSeed=-2043353683 % 33.38/5.23 Prover 8: stopped % 33.38/5.24 Prover 11: stopped % 33.38/5.24 Prover 13: stopped % 33.38/5.24 Prover 1: stopped % 33.38/5.24 Prover 4: stopped % 33.74/5.26 Prover 16: Preprocessing ... % 33.74/5.31 Prover 16: stopped % 33.74/5.31 % 33.74/5.31 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p % 33.74/5.31 % 33.74/5.32 % SZS output start Proof for theBenchmark % 33.74/5.33 Assumptions after simplification: % 33.74/5.33 --------------------------------- % 33.74/5.33 % 33.74/5.33 (compose_defn1) % 33.74/5.35 $i(universal_class) & ? [v0: $i] : (cross_product(universal_class, % 33.74/5.35 universal_class) = v0 & $i(v0) & ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : % 33.74/5.35 ( ~ (compose(v2, v1) = v3) | ~ $i(v2) | ~ $i(v1) | subclass(v3, v0))) % 33.74/5.35 % 33.74/5.35 (element_relation) % 33.74/5.35 $i(element_relation) & $i(universal_class) & ? [v0: $i] : % 33.74/5.35 (cross_product(universal_class, universal_class) = v0 & $i(v0) & % 33.74/5.35 subclass(element_relation, v0)) % 33.74/5.35 % 33.74/5.35 (flip) % 33.74/5.35 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0, % 33.74/5.35 universal_class) = v1 & cross_product(universal_class, universal_class) = % 33.74/5.35 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (flip(v2) = v3) | ~ % 33.74/5.35 $i(v2) | subclass(v3, v1))) % 33.74/5.35 % 33.74/5.35 (flip_defn) % 34.17/5.36 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0, % 34.17/5.36 universal_class) = v1 & cross_product(universal_class, universal_class) = % 34.17/5.36 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] % 34.17/5.36 : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~ % 34.17/5.36 (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v3, v2) = v6) | ~ $i(v5) | % 34.17/5.36 ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v5) | ? [v9: $i] : ? % 34.17/5.36 [v10: $i] : (ordered_pair(v9, v4) = v10 & ordered_pair(v2, v3) = v9 & % 34.17/5.36 $i(v10) & $i(v9) & ( ~ member(v10, v1) | member(v10, v8)))) & ! [v2: % 34.17/5.36 $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] % 34.17/5.36 : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ % 34.17/5.36 (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ % 34.17/5.36 $i(v2) | ~ member(v7, v8) | member(v7, v1)) & ! [v2: $i] : ! [v3: $i] : % 34.17/5.36 ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ % 34.17/5.36 (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v2, % 34.17/5.36 v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ % 34.17/5.36 member(v7, v8) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v4) = v10 % 34.17/5.36 & ordered_pair(v3, v2) = v9 & $i(v10) & $i(v9) & member(v10, v5))) & ! % 34.17/5.36 [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: % 34.17/5.36 $i] : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | % 34.17/5.36 ~ (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ % 34.17/5.36 $i(v2) | ~ member(v7, v1) | member(v7, v8) | ? [v9: $i] : ? [v10: $i] : % 34.17/5.36 (ordered_pair(v9, v4) = v10 & ordered_pair(v3, v2) = v9 & $i(v10) & $i(v9) % 34.17/5.36 & ~ member(v10, v5))) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! % 34.17/5.36 [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (flip(v5) = v6) | % 34.17/5.36 ~ (ordered_pair(v7, v4) = v8) | ~ (ordered_pair(v3, v2) = v7) | ~ $i(v5) % 34.17/5.36 | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] : % 34.17/5.36 (ordered_pair(v9, v4) = v10 & ordered_pair(v2, v3) = v9 & $i(v10) & $i(v9) % 34.17/5.36 & ( ~ member(v10, v6) | (member(v10, v1) & member(v8, v5)))))) % 34.17/5.36 % 34.17/5.36 (function_defn) % 34.17/5.36 $i(identity_relation) & $i(universal_class) & ? [v0: $i] : % 34.17/5.36 (cross_product(universal_class, universal_class) = v0 & $i(v0) & ! [v1: $i] : % 34.17/5.36 ! [v2: $i] : ( ~ (inverse(v1) = v2) | ~ $i(v1) | ~ function(v1) | % 34.17/5.36 subclass(v1, v0)) & ! [v1: $i] : ! [v2: $i] : ( ~ (inverse(v1) = v2) | % 34.17/5.36 ~ $i(v1) | ~ function(v1) | ? [v3: $i] : (compose(v1, v2) = v3 & $i(v3) % 34.17/5.36 & subclass(v3, identity_relation))) & ! [v1: $i] : ! [v2: $i] : ( ~ % 34.17/5.36 (inverse(v1) = v2) | ~ $i(v1) | ~ subclass(v1, v0) | function(v1) | ? % 34.17/5.36 [v3: $i] : (compose(v1, v2) = v3 & $i(v3) & ~ subclass(v3, % 34.17/5.36 identity_relation)))) % 34.17/5.36 % 34.17/5.36 (member_of_singleton) % 34.17/5.36 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ( ~ (v2 = v1) % 34.17/5.36 & member_of(v0) = v2 & singleton(v1) = v0 & $i(v2) & $i(v1) & $i(v0) & % 34.17/5.36 member(v1, universal_class)) % 34.17/5.36 % 34.17/5.36 (member_singleton_singleton) % 34.17/5.36 $i(universal_class) & ! [v0: $i] : ! [v1: $i] : ( ~ (singleton(v0) = v1) | % 34.17/5.36 ~ $i(v0) | ~ member(v0, universal_class) | ? [v2: $i] : (member_of(v1) = % 34.17/5.36 v2 & singleton(v2) = v1 & $i(v2) & $i(v1))) % 34.17/5.36 % 34.17/5.36 (member_singleton_universal) % 34.17/5.36 $i(universal_class) & ! [v0: $i] : ! [v1: $i] : ( ~ (singleton(v0) = v1) | % 34.17/5.36 ~ $i(v0) | ~ member(v0, universal_class) | ? [v2: $i] : (member_of(v1) = % 34.17/5.36 v2 & $i(v2) & member(v2, universal_class))) % 34.17/5.36 % 34.17/5.36 (rotate) % 34.17/5.36 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0, % 34.17/5.36 universal_class) = v1 & cross_product(universal_class, universal_class) = % 34.17/5.36 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (rotate(v2) = v3) | % 34.17/5.36 ~ $i(v2) | subclass(v3, v1))) % 34.17/5.36 % 34.17/5.36 (rotate_defn) % 34.17/5.37 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0, % 34.17/5.37 universal_class) = v1 & cross_product(universal_class, universal_class) = % 34.17/5.37 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] % 34.17/5.37 : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~ % 34.17/5.37 (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = v6) | ~ $i(v5) | % 34.17/5.37 ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v8) | member(v7, v1)) & % 34.17/5.37 ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: % 34.17/5.37 $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~ (ordered_pair(v6, v5) = v7) % 34.17/5.37 | ~ (ordered_pair(v3, v4) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ % 34.17/5.37 $i(v2) | ~ member(v7, v8) | ? [v9: $i] : ? [v10: $i] : % 34.17/5.37 (ordered_pair(v9, v3) = v10 & ordered_pair(v4, v5) = v9 & $i(v10) & $i(v9) % 34.17/5.37 & member(v10, v2))) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: % 34.17/5.37 $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~ % 34.17/5.37 (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = v6) | ~ $i(v5) | % 34.17/5.37 ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v1) | member(v7, v8) | % 34.17/5.37 ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v3) = v10 & ordered_pair(v4, % 34.17/5.37 v5) = v9 & $i(v10) & $i(v9) & ~ member(v10, v2))) & ! [v2: $i] : ! % 34.17/5.37 [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: % 34.17/5.37 $i] : ( ~ (rotate(v2) = v8) | ~ (ordered_pair(v6, v3) = v7) | ~ % 34.17/5.37 (ordered_pair(v4, v5) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ % 34.17/5.37 $i(v2) | ~ member(v7, v2) | ? [v9: $i] : ? [v10: $i] : % 34.17/5.37 (ordered_pair(v9, v5) = v10 & ordered_pair(v3, v4) = v9 & $i(v10) & $i(v9) % 34.17/5.37 & ( ~ member(v10, v1) | member(v10, v8)))) & ! [v2: $i] : ! [v3: $i] : % 34.17/5.37 ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ % 34.17/5.37 (rotate(v2) = v6) | ~ (ordered_pair(v7, v3) = v8) | ~ (ordered_pair(v4, % 34.17/5.37 v5) = v7) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ? [v9: % 34.17/5.37 $i] : ? [v10: $i] : (ordered_pair(v9, v5) = v10 & ordered_pair(v3, v4) % 34.17/5.37 = v9 & $i(v10) & $i(v9) & ( ~ member(v10, v6) | (member(v10, v1) & % 34.17/5.37 member(v8, v2)))))) % 34.17/5.37 % 34.17/5.37 (singleton_set_defn) % 34.17/5.37 ! [v0: $i] : ! [v1: $i] : ( ~ (singleton(v0) = v1) | ~ $i(v0) | % 34.17/5.37 (unordered_pair(v0, v0) = v1 & $i(v1))) & ! [v0: $i] : ! [v1: $i] : ( ~ % 34.17/5.37 (unordered_pair(v0, v0) = v1) | ~ $i(v0) | (singleton(v0) = v1 & $i(v1))) % 34.17/5.37 % 34.17/5.37 (successor_relation_defn1) % 34.17/5.37 $i(successor_relation) & $i(universal_class) & ? [v0: $i] : % 34.17/5.37 (cross_product(universal_class, universal_class) = v0 & $i(v0) & % 34.17/5.37 subclass(successor_relation, v0)) % 34.17/5.37 % 34.17/5.37 (unordered_pair_defn) % 34.17/5.37 $i(universal_class) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : % 34.17/5.37 (v2 = v0 | v1 = v0 | ~ (unordered_pair(v1, v2) = v3) | ~ $i(v2) | ~ $i(v1) % 34.17/5.37 | ~ $i(v0) | ~ member(v0, v3)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : % 34.17/5.37 ! [v3: $i] : ( ~ (unordered_pair(v1, v2) = v3) | ~ $i(v2) | ~ $i(v1) | ~ % 34.17/5.37 $i(v0) | ~ member(v0, v3) | member(v0, universal_class)) & ! [v0: $i] : ! % 34.17/5.37 [v1: $i] : ! [v2: $i] : ( ~ (unordered_pair(v1, v0) = v2) | ~ $i(v1) | ~ % 34.17/5.37 $i(v0) | ~ member(v0, universal_class) | member(v0, v2)) & ! [v0: $i] : ! % 34.17/5.37 [v1: $i] : ! [v2: $i] : ( ~ (unordered_pair(v0, v1) = v2) | ~ $i(v1) | ~ % 34.17/5.37 $i(v0) | ~ member(v0, universal_class) | member(v0, v2)) % 34.17/5.37 % 34.17/5.37 (function-axioms) % 34.17/5.38 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : (v1 = v0 % 34.17/5.38 | ~ (restrict(v4, v3, v2) = v1) | ~ (restrict(v4, v3, v2) = v0)) & ! [v0: % 34.17/5.38 $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (apply(v3, v2) % 34.17/5.38 = v1) | ~ (apply(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] % 34.17/5.38 : ! [v3: $i] : (v1 = v0 | ~ (compose(v3, v2) = v1) | ~ (compose(v3, v2) = % 34.17/5.38 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | % 34.17/5.38 ~ (image(v3, v2) = v1) | ~ (image(v3, v2) = v0)) & ! [v0: $i] : ! [v1: % 34.17/5.38 $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (union(v3, v2) = v1) | ~ % 34.17/5.38 (union(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: % 34.17/5.38 $i] : (v1 = v0 | ~ (intersection(v3, v2) = v1) | ~ (intersection(v3, v2) = % 34.17/5.38 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | % 34.17/5.38 ~ (cross_product(v3, v2) = v1) | ~ (cross_product(v3, v2) = v0)) & ! [v0: % 34.17/5.38 $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ % 34.17/5.38 (ordered_pair(v3, v2) = v1) | ~ (ordered_pair(v3, v2) = v0)) & ! [v0: $i] % 34.17/5.38 : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (unordered_pair(v3, % 34.17/5.38 v2) = v1) | ~ (unordered_pair(v3, v2) = v0)) & ! [v0: $i] : ! [v1: % 34.17/5.38 $i] : ! [v2: $i] : (v1 = v0 | ~ (member_of(v2) = v1) | ~ (member_of(v2) = % 34.17/5.38 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ % 34.17/5.38 (power_class(v2) = v1) | ~ (power_class(v2) = v0)) & ! [v0: $i] : ! [v1: % 34.17/5.38 $i] : ! [v2: $i] : (v1 = v0 | ~ (sum_class(v2) = v1) | ~ (sum_class(v2) = % 34.17/5.38 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ % 34.17/5.38 (range_of(v2) = v1) | ~ (range_of(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : % 34.17/5.38 ! [v2: $i] : (v1 = v0 | ~ (inverse(v2) = v1) | ~ (inverse(v2) = v0)) & ! % 34.17/5.38 [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (successor(v2) = v1) | ~ % 34.17/5.38 (successor(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | % 34.17/5.38 ~ (flip(v2) = v1) | ~ (flip(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! % 34.17/5.38 [v2: $i] : (v1 = v0 | ~ (rotate(v2) = v1) | ~ (rotate(v2) = v0)) & ! [v0: % 34.17/5.38 $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (domain_of(v2) = v1) | ~ % 34.17/5.38 (domain_of(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | % 34.17/5.38 ~ (complement(v2) = v1) | ~ (complement(v2) = v0)) & ! [v0: $i] : ! [v1: % 34.17/5.38 $i] : ! [v2: $i] : (v1 = v0 | ~ (first(v2) = v1) | ~ (first(v2) = v0)) & % 34.17/5.38 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (second(v2) = v1) | ~ % 34.17/5.38 (second(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ % 34.17/5.38 (singleton(v2) = v1) | ~ (singleton(v2) = v0)) % 34.17/5.38 % 34.17/5.38 Further assumptions not needed in the proof: % 34.17/5.38 -------------------------------------------- % 34.17/5.38 apply_defn, choice, class_elements_are_sets, complement, compose_defn2, % 34.17/5.38 cross_product, cross_product_defn, disjoint_defn, domain_of, % 34.17/5.38 element_relation_defn, extensionality, first_second, identity_relation, % 34.17/5.38 image_defn, inductive_defn, infinity, intersection, inverse_defn, % 34.17/5.38 member_universal_self, null_class_defn, ordered_pair_defn, power_class, % 34.17/5.38 power_class_defn, range_of_defn, regularity, replacement, restrict_defn, % 34.17/5.38 singleton_self, subclass_defn, successor_defn, successor_relation_defn2, % 34.17/5.38 sum_class, sum_class_defn, union_defn, unordered_pair % 34.17/5.38 % 34.17/5.38 Those formulas are unsatisfiable: % 34.17/5.38 --------------------------------- % 34.17/5.38 % 34.17/5.38 Begin of proof % 34.17/5.38 | % 34.17/5.38 | ALPHA: (unordered_pair_defn) implies: % 34.17/5.38 | (1) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (unordered_pair(v1, v0) = % 34.17/5.38 | v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v0, universal_class) | % 34.17/5.38 | member(v0, v2)) % 34.17/5.38 | (2) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v2 = v0 | v1 = % 34.17/5.38 | v0 | ~ (unordered_pair(v1, v2) = v3) | ~ $i(v2) | ~ $i(v1) | ~ % 34.17/5.38 | $i(v0) | ~ member(v0, v3)) % 34.17/5.38 | % 34.17/5.38 | ALPHA: (singleton_set_defn) implies: % 34.17/5.38 | (3) ! [v0: $i] : ! [v1: $i] : ( ~ (singleton(v0) = v1) | ~ $i(v0) | % 34.17/5.38 | (unordered_pair(v0, v0) = v1 & $i(v1))) % 34.17/5.38 | % 34.17/5.38 | ALPHA: (element_relation) implies: % 34.17/5.38 | (4) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 & % 34.17/5.38 | $i(v0) & subclass(element_relation, v0)) % 34.17/5.38 | % 34.17/5.38 | ALPHA: (rotate_defn) implies: % 34.17/5.38 | (5) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 & % 34.17/5.38 | cross_product(universal_class, universal_class) = v0 & $i(v1) & % 34.17/5.38 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! % 34.17/5.38 | [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~ % 34.17/5.38 | (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = v6) | ~ % 34.17/5.38 | $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v8) | % 34.17/5.38 | member(v7, v1)) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: % 34.17/5.38 | $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = % 34.17/5.38 | v8) | ~ (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = % 34.17/5.38 | v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ % 34.17/5.38 | member(v7, v8) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v3) % 34.17/5.39 | = v10 & ordered_pair(v4, v5) = v9 & $i(v10) & $i(v9) & % 34.17/5.39 | member(v10, v2))) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! % 34.17/5.39 | [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) % 34.17/5.39 | = v8) | ~ (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) % 34.17/5.39 | = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ % 34.17/5.39 | member(v7, v1) | member(v7, v8) | ? [v9: $i] : ? [v10: $i] : % 34.17/5.39 | (ordered_pair(v9, v3) = v10 & ordered_pair(v4, v5) = v9 & $i(v10) & % 34.17/5.39 | $i(v9) & ~ member(v10, v2))) & ! [v2: $i] : ! [v3: $i] : ! % 34.17/5.39 | [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( % 34.17/5.39 | ~ (rotate(v2) = v8) | ~ (ordered_pair(v6, v3) = v7) | ~ % 34.17/5.39 | (ordered_pair(v4, v5) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 34.17/5.39 | ~ $i(v2) | ~ member(v7, v2) | ? [v9: $i] : ? [v10: $i] : % 34.17/5.39 | (ordered_pair(v9, v5) = v10 & ordered_pair(v3, v4) = v9 & $i(v10) & % 34.17/5.39 | $i(v9) & ( ~ member(v10, v1) | member(v10, v8)))) & ! [v2: $i] : % 34.17/5.39 | ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : % 34.17/5.39 | ! [v8: $i] : ( ~ (rotate(v2) = v6) | ~ (ordered_pair(v7, v3) = v8) % 34.17/5.39 | | ~ (ordered_pair(v4, v5) = v7) | ~ $i(v5) | ~ $i(v4) | ~ % 34.17/5.39 | $i(v3) | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, % 34.17/5.39 | v5) = v10 & ordered_pair(v3, v4) = v9 & $i(v10) & $i(v9) & ( ~ % 34.17/5.39 | member(v10, v6) | (member(v10, v1) & member(v8, v2)))))) % 34.17/5.39 | % 34.17/5.39 | ALPHA: (rotate) implies: % 34.17/5.39 | (6) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 & % 34.17/5.39 | cross_product(universal_class, universal_class) = v0 & $i(v1) & % 34.17/5.39 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (rotate(v2) = v3) | ~ % 34.17/5.39 | $i(v2) | subclass(v3, v1))) % 34.17/5.39 | % 34.17/5.39 | ALPHA: (flip_defn) implies: % 34.35/5.39 | (7) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 & % 34.35/5.39 | cross_product(universal_class, universal_class) = v0 & $i(v1) & % 34.35/5.39 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! % 34.35/5.39 | [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~ % 34.35/5.39 | (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v3, v2) = v6) | ~ % 34.35/5.39 | $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v5) | ? % 34.35/5.39 | [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v4) = v10 & % 34.35/5.39 | ordered_pair(v2, v3) = v9 & $i(v10) & $i(v9) & ( ~ member(v10, % 34.35/5.39 | v1) | member(v10, v8)))) & ! [v2: $i] : ! [v3: $i] : ! % 34.35/5.39 | [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( % 34.35/5.39 | ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ % 34.35/5.39 | (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 34.35/5.39 | ~ $i(v2) | ~ member(v7, v8) | member(v7, v1)) & ! [v2: $i] : ! % 34.35/5.39 | [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! % 34.35/5.39 | [v8: $i] : ( ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ % 34.35/5.39 | (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 34.35/5.39 | ~ $i(v2) | ~ member(v7, v8) | ? [v9: $i] : ? [v10: $i] : % 34.35/5.39 | (ordered_pair(v9, v4) = v10 & ordered_pair(v3, v2) = v9 & $i(v10) & % 34.35/5.39 | $i(v9) & member(v10, v5))) & ! [v2: $i] : ! [v3: $i] : ! [v4: % 34.35/5.39 | $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ % 34.35/5.39 | (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ % 34.35/5.39 | (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 34.35/5.39 | ~ $i(v2) | ~ member(v7, v1) | member(v7, v8) | ? [v9: $i] : ? % 34.35/5.39 | [v10: $i] : (ordered_pair(v9, v4) = v10 & ordered_pair(v3, v2) = v9 % 34.35/5.39 | & $i(v10) & $i(v9) & ~ member(v10, v5))) & ! [v2: $i] : ! [v3: % 34.35/5.39 | $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! % 34.35/5.39 | [v8: $i] : ( ~ (flip(v5) = v6) | ~ (ordered_pair(v7, v4) = v8) | ~ % 34.35/5.39 | (ordered_pair(v3, v2) = v7) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 34.35/5.39 | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v4) = v10 % 34.35/5.39 | & ordered_pair(v2, v3) = v9 & $i(v10) & $i(v9) & ( ~ member(v10, % 34.35/5.39 | v6) | (member(v10, v1) & member(v8, v5)))))) % 34.35/5.39 | % 34.35/5.39 | ALPHA: (flip) implies: % 34.35/5.39 | (8) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 & % 34.35/5.39 | cross_product(universal_class, universal_class) = v0 & $i(v1) & % 34.35/5.39 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (flip(v2) = v3) | ~ $i(v2) % 34.35/5.39 | | subclass(v3, v1))) % 34.35/5.39 | % 34.35/5.39 | ALPHA: (successor_relation_defn1) implies: % 34.35/5.39 | (9) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 & % 34.35/5.39 | $i(v0) & subclass(successor_relation, v0)) % 34.35/5.39 | % 34.35/5.39 | ALPHA: (compose_defn1) implies: % 34.35/5.39 | (10) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 & % 34.35/5.39 | $i(v0) & ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~ (compose(v2, % 34.35/5.39 | v1) = v3) | ~ $i(v2) | ~ $i(v1) | subclass(v3, v0))) % 34.35/5.39 | % 34.35/5.39 | ALPHA: (function_defn) implies: % 34.35/5.39 | (11) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 & % 34.35/5.39 | $i(v0) & ! [v1: $i] : ! [v2: $i] : ( ~ (inverse(v1) = v2) | ~ % 34.35/5.39 | $i(v1) | ~ function(v1) | subclass(v1, v0)) & ! [v1: $i] : ! % 34.35/5.39 | [v2: $i] : ( ~ (inverse(v1) = v2) | ~ $i(v1) | ~ function(v1) | ? % 34.35/5.39 | [v3: $i] : (compose(v1, v2) = v3 & $i(v3) & subclass(v3, % 34.35/5.39 | identity_relation))) & ! [v1: $i] : ! [v2: $i] : ( ~ % 34.35/5.39 | (inverse(v1) = v2) | ~ $i(v1) | ~ subclass(v1, v0) | % 34.35/5.39 | function(v1) | ? [v3: $i] : (compose(v1, v2) = v3 & $i(v3) & ~ % 34.35/5.39 | subclass(v3, identity_relation)))) % 34.35/5.39 | % 34.35/5.39 | ALPHA: (member_singleton_universal) implies: % 34.35/5.39 | (12) ! [v0: $i] : ! [v1: $i] : ( ~ (singleton(v0) = v1) | ~ $i(v0) | ~ % 34.35/5.39 | member(v0, universal_class) | ? [v2: $i] : (member_of(v1) = v2 & % 34.35/5.39 | $i(v2) & member(v2, universal_class))) % 34.35/5.39 | % 34.35/5.39 | ALPHA: (member_singleton_singleton) implies: % 34.35/5.40 | (13) ! [v0: $i] : ! [v1: $i] : ( ~ (singleton(v0) = v1) | ~ $i(v0) | ~ % 34.35/5.40 | member(v0, universal_class) | ? [v2: $i] : (member_of(v1) = v2 & % 34.35/5.40 | singleton(v2) = v1 & $i(v2) & $i(v1))) % 34.35/5.40 | % 34.35/5.40 | ALPHA: (member_of_singleton) implies: % 34.35/5.40 | (14) ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ( ~ (v2 = v1) & % 34.35/5.40 | member_of(v0) = v2 & singleton(v1) = v0 & $i(v2) & $i(v1) & $i(v0) & % 34.35/5.40 | member(v1, universal_class)) % 34.35/5.40 | % 34.35/5.40 | ALPHA: (function-axioms) implies: % 34.35/5.40 | (15) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (member_of(v2) % 34.35/5.40 | = v1) | ~ (member_of(v2) = v0)) % 34.35/5.40 | (16) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ % 34.35/5.40 | (cross_product(v3, v2) = v1) | ~ (cross_product(v3, v2) = v0)) % 34.35/5.40 | % 34.35/5.40 | DELTA: instantiating (9) with fresh symbol all_41_0 gives: % 34.35/5.40 | (17) cross_product(universal_class, universal_class) = all_41_0 & % 34.35/5.40 | $i(all_41_0) & subclass(successor_relation, all_41_0) % 34.35/5.40 | % 34.35/5.40 | ALPHA: (17) implies: % 34.35/5.40 | (18) cross_product(universal_class, universal_class) = all_41_0 % 34.35/5.40 | % 34.35/5.40 | DELTA: instantiating (4) with fresh symbol all_43_0 gives: % 34.35/5.40 | (19) cross_product(universal_class, universal_class) = all_43_0 & % 34.35/5.40 | $i(all_43_0) & subclass(element_relation, all_43_0) % 34.35/5.40 | % 34.35/5.40 | ALPHA: (19) implies: % 34.35/5.40 | (20) cross_product(universal_class, universal_class) = all_43_0 % 34.35/5.40 | % 34.35/5.40 | DELTA: instantiating (10) with fresh symbol all_47_0 gives: % 34.35/5.40 | (21) cross_product(universal_class, universal_class) = all_47_0 & % 34.35/5.40 | $i(all_47_0) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ % 34.35/5.40 | (compose(v1, v0) = v2) | ~ $i(v1) | ~ $i(v0) | subclass(v2, % 34.35/5.40 | all_47_0)) % 34.35/5.40 | % 34.35/5.40 | ALPHA: (21) implies: % 34.35/5.40 | (22) cross_product(universal_class, universal_class) = all_47_0 % 34.35/5.40 | % 34.35/5.40 | DELTA: instantiating (14) with fresh symbols all_54_0, all_54_1, all_54_2 % 34.35/5.40 | gives: % 34.35/5.40 | (23) ~ (all_54_0 = all_54_1) & member_of(all_54_2) = all_54_0 & % 34.35/5.40 | singleton(all_54_1) = all_54_2 & $i(all_54_0) & $i(all_54_1) & % 34.35/5.40 | $i(all_54_2) & member(all_54_1, universal_class) % 34.35/5.40 | % 34.35/5.40 | ALPHA: (23) implies: % 34.35/5.40 | (24) ~ (all_54_0 = all_54_1) % 34.35/5.40 | (25) member(all_54_1, universal_class) % 34.35/5.40 | (26) $i(all_54_1) % 34.35/5.40 | (27) singleton(all_54_1) = all_54_2 % 34.35/5.40 | (28) member_of(all_54_2) = all_54_0 % 34.35/5.40 | % 34.35/5.40 | DELTA: instantiating (6) with fresh symbols all_56_0, all_56_1 gives: % 34.35/5.40 | (29) cross_product(all_56_1, universal_class) = all_56_0 & % 34.35/5.40 | cross_product(universal_class, universal_class) = all_56_1 & % 34.35/5.40 | $i(all_56_0) & $i(all_56_1) & ! [v0: $i] : ! [v1: $i] : ( ~ % 34.35/5.40 | (rotate(v0) = v1) | ~ $i(v0) | subclass(v1, all_56_0)) % 34.35/5.40 | % 34.35/5.40 | ALPHA: (29) implies: % 34.35/5.40 | (30) cross_product(universal_class, universal_class) = all_56_1 % 34.35/5.40 | (31) cross_product(all_56_1, universal_class) = all_56_0 % 34.35/5.40 | % 34.35/5.40 | DELTA: instantiating (8) with fresh symbols all_62_0, all_62_1 gives: % 34.35/5.40 | (32) cross_product(all_62_1, universal_class) = all_62_0 & % 34.35/5.40 | cross_product(universal_class, universal_class) = all_62_1 & % 34.35/5.40 | $i(all_62_0) & $i(all_62_1) & ! [v0: $i] : ! [v1: $i] : ( ~ % 34.35/5.40 | (flip(v0) = v1) | ~ $i(v0) | subclass(v1, all_62_0)) % 34.35/5.40 | % 34.35/5.40 | ALPHA: (32) implies: % 34.35/5.40 | (33) cross_product(universal_class, universal_class) = all_62_1 % 34.35/5.40 | % 34.35/5.40 | DELTA: instantiating (11) with fresh symbol all_65_0 gives: % 34.35/5.40 | (34) cross_product(universal_class, universal_class) = all_65_0 & % 34.35/5.40 | $i(all_65_0) & ! [v0: $i] : ! [v1: $i] : ( ~ (inverse(v0) = v1) | ~ % 34.35/5.40 | $i(v0) | ~ function(v0) | subclass(v0, all_65_0)) & ! [v0: $i] : % 34.35/5.40 | ! [v1: $i] : ( ~ (inverse(v0) = v1) | ~ $i(v0) | ~ function(v0) | ? % 34.35/5.40 | [v2: $i] : (compose(v0, v1) = v2 & $i(v2) & subclass(v2, % 34.35/5.40 | identity_relation))) & ! [v0: $i] : ! [v1: $i] : ( ~ % 34.35/5.40 | (inverse(v0) = v1) | ~ $i(v0) | ~ subclass(v0, all_65_0) | % 34.35/5.40 | function(v0) | ? [v2: $i] : (compose(v0, v1) = v2 & $i(v2) & ~ % 34.35/5.40 | subclass(v2, identity_relation))) % 34.35/5.40 | % 34.35/5.40 | ALPHA: (34) implies: % 34.35/5.40 | (35) cross_product(universal_class, universal_class) = all_65_0 % 34.35/5.40 | % 34.35/5.40 | DELTA: instantiating (7) with fresh symbols all_68_0, all_68_1 gives: % 34.35/5.41 | (36) cross_product(all_68_1, universal_class) = all_68_0 & % 34.35/5.41 | cross_product(universal_class, universal_class) = all_68_1 & % 34.35/5.41 | $i(all_68_0) & $i(all_68_1) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] % 34.35/5.41 | : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ % 34.35/5.41 | (flip(v3) = v6) | ~ (ordered_pair(v4, v2) = v5) | ~ % 34.35/5.41 | (ordered_pair(v1, v0) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ % 34.35/5.41 | $i(v0) | ~ member(v5, v3) | ? [v7: $i] : ? [v8: $i] : % 34.35/5.41 | (ordered_pair(v7, v2) = v8 & ordered_pair(v0, v1) = v7 & $i(v8) & % 34.35/5.41 | $i(v7) & ( ~ member(v8, all_68_0) | member(v8, v6)))) & ! [v0: % 34.35/5.41 | $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! % 34.35/5.41 | [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v6) | ~ (ordered_pair(v4, % 34.35/5.41 | v2) = v5) | ~ (ordered_pair(v0, v1) = v4) | ~ $i(v3) | ~ % 34.35/5.41 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v5, v6) | member(v5, % 34.35/5.41 | all_68_0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] % 34.35/5.41 | : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v6) | ~ % 34.35/5.41 | (ordered_pair(v4, v2) = v5) | ~ (ordered_pair(v0, v1) = v4) | ~ % 34.35/5.41 | $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v5, v6) | ? % 34.35/5.41 | [v7: $i] : ? [v8: $i] : (ordered_pair(v7, v2) = v8 & % 34.35/5.41 | ordered_pair(v1, v0) = v7 & $i(v8) & $i(v7) & member(v8, v3))) & % 34.35/5.41 | ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : % 34.35/5.41 | ! [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v6) | ~ (ordered_pair(v4, % 34.35/5.41 | v2) = v5) | ~ (ordered_pair(v0, v1) = v4) | ~ $i(v3) | ~ % 34.35/5.41 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v5, all_68_0) | % 34.35/5.41 | member(v5, v6) | ? [v7: $i] : ? [v8: $i] : (ordered_pair(v7, v2) = % 34.35/5.41 | v8 & ordered_pair(v1, v0) = v7 & $i(v8) & $i(v7) & ~ member(v8, % 34.35/5.41 | v3))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : % 34.35/5.41 | ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v4) | ~ % 34.35/5.41 | (ordered_pair(v5, v2) = v6) | ~ (ordered_pair(v1, v0) = v5) | ~ % 34.35/5.41 | $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v7: $i] : ? [v8: % 34.35/5.41 | $i] : (ordered_pair(v7, v2) = v8 & ordered_pair(v0, v1) = v7 & % 34.35/5.41 | $i(v8) & $i(v7) & ( ~ member(v8, v4) | (member(v8, all_68_0) & % 34.35/5.41 | member(v6, v3))))) % 34.35/5.41 | % 34.35/5.41 | ALPHA: (36) implies: % 34.35/5.41 | (37) cross_product(universal_class, universal_class) = all_68_1 % 34.35/5.41 | (38) cross_product(all_68_1, universal_class) = all_68_0 % 34.35/5.41 | % 34.35/5.41 | DELTA: instantiating (5) with fresh symbols all_71_0, all_71_1 gives: % 34.35/5.41 | (39) cross_product(all_71_1, universal_class) = all_71_0 & % 34.35/5.41 | cross_product(universal_class, universal_class) = all_71_1 & % 34.35/5.41 | $i(all_71_0) & $i(all_71_1) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] % 34.35/5.41 | : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ % 34.35/5.41 | (rotate(v0) = v6) | ~ (ordered_pair(v4, v3) = v5) | ~ % 34.35/5.41 | (ordered_pair(v1, v2) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ % 34.35/5.41 | $i(v0) | ~ member(v5, v6) | member(v5, all_71_0)) & ! [v0: $i] : % 34.35/5.41 | ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : % 34.35/5.41 | ! [v6: $i] : ( ~ (rotate(v0) = v6) | ~ (ordered_pair(v4, v3) = v5) | % 34.35/5.41 | ~ (ordered_pair(v1, v2) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | % 34.35/5.41 | ~ $i(v0) | ~ member(v5, v6) | ? [v7: $i] : ? [v8: $i] : % 34.35/5.41 | (ordered_pair(v7, v1) = v8 & ordered_pair(v2, v3) = v7 & $i(v8) & % 34.35/5.41 | $i(v7) & member(v8, v0))) & ! [v0: $i] : ! [v1: $i] : ! [v2: % 34.35/5.41 | $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ % 34.35/5.41 | (rotate(v0) = v6) | ~ (ordered_pair(v4, v3) = v5) | ~ % 34.35/5.41 | (ordered_pair(v1, v2) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ % 34.35/5.41 | $i(v0) | ~ member(v5, all_71_0) | member(v5, v6) | ? [v7: $i] : ? % 34.35/5.41 | [v8: $i] : (ordered_pair(v7, v1) = v8 & ordered_pair(v2, v3) = v7 & % 34.35/5.41 | $i(v8) & $i(v7) & ~ member(v8, v0))) & ! [v0: $i] : ! [v1: $i] % 34.35/5.41 | : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] % 34.35/5.41 | : ( ~ (rotate(v0) = v6) | ~ (ordered_pair(v4, v1) = v5) | ~ % 34.35/5.41 | (ordered_pair(v2, v3) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ % 34.35/5.41 | $i(v0) | ~ member(v5, v0) | ? [v7: $i] : ? [v8: $i] : % 34.35/5.41 | (ordered_pair(v7, v3) = v8 & ordered_pair(v1, v2) = v7 & $i(v8) & % 34.35/5.41 | $i(v7) & ( ~ member(v8, all_71_0) | member(v8, v6)))) & ! [v0: % 34.35/5.41 | $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! % 34.35/5.41 | [v5: $i] : ! [v6: $i] : ( ~ (rotate(v0) = v4) | ~ (ordered_pair(v5, % 34.35/5.41 | v1) = v6) | ~ (ordered_pair(v2, v3) = v5) | ~ $i(v3) | ~ % 34.35/5.41 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v7: $i] : ? [v8: $i] : % 34.35/5.41 | (ordered_pair(v7, v3) = v8 & ordered_pair(v1, v2) = v7 & $i(v8) & % 34.35/5.41 | $i(v7) & ( ~ member(v8, v4) | (member(v8, all_71_0) & member(v6, % 34.35/5.41 | v0))))) % 34.35/5.41 | % 34.35/5.41 | ALPHA: (39) implies: % 34.35/5.41 | (40) cross_product(universal_class, universal_class) = all_71_1 % 34.35/5.41 | (41) cross_product(all_71_1, universal_class) = all_71_0 % 34.35/5.41 | % 34.35/5.41 | GROUND_INST: instantiating (16) with all_62_1, all_65_0, universal_class, % 34.35/5.41 | universal_class, simplifying with (33), (35) gives: % 34.35/5.41 | (42) all_65_0 = all_62_1 % 34.35/5.41 | % 34.35/5.42 | GROUND_INST: instantiating (16) with all_43_0, all_65_0, universal_class, % 34.35/5.42 | universal_class, simplifying with (20), (35) gives: % 34.35/5.42 | (43) all_65_0 = all_43_0 % 34.35/5.42 | % 34.35/5.42 | GROUND_INST: instantiating (16) with all_62_1, all_68_1, universal_class, % 34.35/5.42 | universal_class, simplifying with (33), (37) gives: % 34.35/5.42 | (44) all_68_1 = all_62_1 % 34.35/5.42 | % 34.35/5.42 | GROUND_INST: instantiating (16) with all_56_1, all_68_1, universal_class, % 34.35/5.42 | universal_class, simplifying with (30), (37) gives: % 34.35/5.42 | (45) all_68_1 = all_56_1 % 34.35/5.42 | % 34.35/5.42 | GROUND_INST: instantiating (16) with all_47_0, all_68_1, universal_class, % 34.35/5.42 | universal_class, simplifying with (22), (37) gives: % 34.35/5.42 | (46) all_68_1 = all_47_0 % 34.35/5.42 | % 34.35/5.42 | GROUND_INST: instantiating (16) with all_68_1, all_71_1, universal_class, % 34.35/5.42 | universal_class, simplifying with (37), (40) gives: % 34.35/5.42 | (47) all_71_1 = all_68_1 % 34.35/5.42 | % 34.35/5.42 | GROUND_INST: instantiating (16) with all_41_0, all_71_1, universal_class, % 34.35/5.42 | universal_class, simplifying with (18), (40) gives: % 34.35/5.42 | (48) all_71_1 = all_41_0 % 34.35/5.42 | % 34.35/5.42 | GROUND_INST: instantiating (16) with all_68_0, all_71_0, universal_class, % 34.35/5.42 | all_68_1, simplifying with (38) gives: % 34.35/5.42 | (49) all_71_0 = all_68_0 | ~ (cross_product(all_68_1, universal_class) = % 34.35/5.42 | all_71_0) % 34.35/5.42 | % 34.35/5.42 | GROUND_INST: instantiating (16) with all_56_0, all_71_0, universal_class, % 34.35/5.42 | all_56_1, simplifying with (31) gives: % 34.35/5.42 | (50) all_71_0 = all_56_0 | ~ (cross_product(all_56_1, universal_class) = % 34.35/5.42 | all_71_0) % 34.35/5.42 | % 34.35/5.42 | COMBINE_EQS: (47), (48) imply: % 34.35/5.42 | (51) all_68_1 = all_41_0 % 34.35/5.42 | % 34.35/5.42 | SIMP: (51) implies: % 34.35/5.42 | (52) all_68_1 = all_41_0 % 34.35/5.42 | % 34.35/5.42 | COMBINE_EQS: (45), (46) imply: % 34.35/5.42 | (53) all_56_1 = all_47_0 % 34.35/5.42 | % 34.35/5.42 | COMBINE_EQS: (45), (52) imply: % 34.35/5.42 | (54) all_56_1 = all_41_0 % 34.35/5.42 | % 34.35/5.42 | COMBINE_EQS: (44), (45) imply: % 34.35/5.42 | (55) all_62_1 = all_56_1 % 34.35/5.42 | % 34.35/5.42 | SIMP: (55) implies: % 34.35/5.42 | (56) all_62_1 = all_56_1 % 34.35/5.42 | % 34.35/5.42 | COMBINE_EQS: (42), (43) imply: % 34.35/5.42 | (57) all_62_1 = all_43_0 % 34.35/5.42 | % 34.35/5.42 | SIMP: (57) implies: % 34.35/5.42 | (58) all_62_1 = all_43_0 % 34.35/5.42 | % 34.35/5.42 | COMBINE_EQS: (56), (58) imply: % 34.35/5.42 | (59) all_56_1 = all_43_0 % 34.35/5.42 | % 34.35/5.42 | SIMP: (59) implies: % 34.35/5.42 | (60) all_56_1 = all_43_0 % 34.35/5.42 | % 34.35/5.42 | COMBINE_EQS: (53), (54) imply: % 34.35/5.42 | (61) all_47_0 = all_41_0 % 34.35/5.42 | % 34.35/5.42 | COMBINE_EQS: (53), (60) imply: % 34.35/5.42 | (62) all_47_0 = all_43_0 % 34.35/5.42 | % 34.35/5.42 | COMBINE_EQS: (61), (62) imply: % 34.35/5.42 | (63) all_43_0 = all_41_0 % 34.35/5.42 | % 34.35/5.42 | REDUCE: (41), (48) imply: % 34.35/5.42 | (64) cross_product(all_41_0, universal_class) = all_71_0 % 34.35/5.42 | % 34.35/5.42 | BETA: splitting (49) gives: % 34.35/5.42 | % 34.35/5.42 | Case 1: % 34.35/5.42 | | % 34.35/5.42 | | (65) ~ (cross_product(all_68_1, universal_class) = all_71_0) % 34.35/5.42 | | % 34.35/5.42 | | REDUCE: (52), (65) imply: % 34.35/5.42 | | (66) ~ (cross_product(all_41_0, universal_class) = all_71_0) % 34.35/5.42 | | % 34.35/5.42 | | PRED_UNIFY: (64), (66) imply: % 34.35/5.42 | | (67) $false % 34.35/5.42 | | % 34.35/5.42 | | CLOSE: (67) is inconsistent. % 34.35/5.42 | | % 34.35/5.42 | Case 2: % 34.35/5.42 | | % 34.35/5.42 | | (68) all_71_0 = all_68_0 % 34.35/5.42 | | % 34.35/5.42 | | REDUCE: (64), (68) imply: % 34.35/5.42 | | (69) cross_product(all_41_0, universal_class) = all_68_0 % 34.35/5.42 | | % 34.35/5.42 | | BETA: splitting (50) gives: % 34.35/5.42 | | % 34.35/5.42 | | Case 1: % 34.35/5.42 | | | % 34.35/5.42 | | | (70) ~ (cross_product(all_56_1, universal_class) = all_71_0) % 34.35/5.42 | | | % 34.35/5.42 | | | REDUCE: (54), (68), (70) imply: % 34.35/5.42 | | | (71) ~ (cross_product(all_41_0, universal_class) = all_68_0) % 34.35/5.42 | | | % 34.35/5.42 | | | PRED_UNIFY: (69), (71) imply: % 34.35/5.42 | | | (72) $false % 34.35/5.42 | | | % 34.35/5.42 | | | CLOSE: (72) is inconsistent. % 34.35/5.42 | | | % 34.35/5.42 | | Case 2: % 34.35/5.42 | | | % 34.35/5.42 | | | % 34.35/5.42 | | | GROUND_INST: instantiating (13) with all_54_1, all_54_2, simplifying with % 34.35/5.42 | | | (25), (26), (27) gives: % 34.35/5.42 | | | (73) ? [v0: $i] : (member_of(all_54_2) = v0 & singleton(v0) = all_54_2 % 34.35/5.42 | | | & $i(v0) & $i(all_54_2)) % 34.35/5.42 | | | % 34.35/5.42 | | | GROUND_INST: instantiating (12) with all_54_1, all_54_2, simplifying with % 34.35/5.42 | | | (25), (26), (27) gives: % 34.35/5.42 | | | (74) ? [v0: $i] : (member_of(all_54_2) = v0 & $i(v0) & member(v0, % 34.35/5.42 | | | universal_class)) % 34.35/5.42 | | | % 34.35/5.42 | | | GROUND_INST: instantiating (3) with all_54_1, all_54_2, simplifying with % 34.35/5.42 | | | (26), (27) gives: % 34.35/5.42 | | | (75) unordered_pair(all_54_1, all_54_1) = all_54_2 & $i(all_54_2) % 34.35/5.42 | | | % 34.35/5.42 | | | ALPHA: (75) implies: % 34.35/5.42 | | | (76) unordered_pair(all_54_1, all_54_1) = all_54_2 % 34.35/5.42 | | | % 34.35/5.42 | | | DELTA: instantiating (74) with fresh symbol all_98_0 gives: % 34.35/5.42 | | | (77) member_of(all_54_2) = all_98_0 & $i(all_98_0) & member(all_98_0, % 34.35/5.42 | | | universal_class) % 34.35/5.42 | | | % 34.35/5.42 | | | ALPHA: (77) implies: % 34.35/5.42 | | | (78) $i(all_98_0) % 34.35/5.42 | | | (79) member_of(all_54_2) = all_98_0 % 34.35/5.42 | | | % 34.35/5.42 | | | DELTA: instantiating (73) with fresh symbol all_100_0 gives: % 34.35/5.43 | | | (80) member_of(all_54_2) = all_100_0 & singleton(all_100_0) = all_54_2 % 34.35/5.43 | | | & $i(all_100_0) & $i(all_54_2) % 34.35/5.43 | | | % 34.35/5.43 | | | ALPHA: (80) implies: % 34.35/5.43 | | | (81) singleton(all_100_0) = all_54_2 % 34.35/5.43 | | | (82) member_of(all_54_2) = all_100_0 % 34.35/5.43 | | | % 34.35/5.43 | | | GROUND_INST: instantiating (15) with all_54_0, all_100_0, all_54_2, % 34.35/5.43 | | | simplifying with (28), (82) gives: % 34.35/5.43 | | | (83) all_100_0 = all_54_0 % 34.35/5.43 | | | % 34.35/5.43 | | | GROUND_INST: instantiating (15) with all_98_0, all_100_0, all_54_2, % 34.35/5.43 | | | simplifying with (79), (82) gives: % 34.35/5.43 | | | (84) all_100_0 = all_98_0 % 34.35/5.43 | | | % 34.35/5.43 | | | COMBINE_EQS: (83), (84) imply: % 34.35/5.43 | | | (85) all_98_0 = all_54_0 % 34.35/5.43 | | | % 34.35/5.43 | | | REDUCE: (81), (83) imply: % 34.35/5.43 | | | (86) singleton(all_54_0) = all_54_2 % 34.35/5.43 | | | % 34.35/5.43 | | | REDUCE: (78), (85) imply: % 34.35/5.43 | | | (87) $i(all_54_0) % 34.35/5.43 | | | % 34.35/5.43 | | | GROUND_INST: instantiating (1) with all_54_1, all_54_1, all_54_2, % 34.35/5.43 | | | simplifying with (25), (26), (76) gives: % 34.35/5.43 | | | (88) member(all_54_1, all_54_2) % 34.35/5.43 | | | % 34.35/5.43 | | | GROUND_INST: instantiating (3) with all_54_0, all_54_2, simplifying with % 34.35/5.43 | | | (86), (87) gives: % 34.35/5.43 | | | (89) unordered_pair(all_54_0, all_54_0) = all_54_2 & $i(all_54_2) % 34.35/5.43 | | | % 34.35/5.43 | | | ALPHA: (89) implies: % 34.35/5.43 | | | (90) unordered_pair(all_54_0, all_54_0) = all_54_2 % 34.35/5.43 | | | % 34.35/5.43 | | | GROUND_INST: instantiating (2) with all_54_1, all_54_0, all_54_0, % 34.35/5.43 | | | all_54_2, simplifying with (26), (87), (88), (90) gives: % 34.35/5.43 | | | (91) all_54_0 = all_54_1 % 34.35/5.43 | | | % 34.35/5.43 | | | REDUCE: (24), (91) imply: % 34.35/5.43 | | | (92) $false % 34.35/5.43 | | | % 34.35/5.43 | | | CLOSE: (92) is inconsistent. % 34.35/5.43 | | | % 34.35/5.43 | | End of split % 34.35/5.43 | | % 34.35/5.43 | End of split % 34.35/5.43 | % 34.35/5.43 End of proof % 34.35/5.43 % SZS output end Proof for theBenchmark % 34.35/5.43 % 34.35/5.43 4850ms %------------------------------------------------------------------------------