%------------------------------------------------------------------------------ % File : Princess---230619 % Problem : SET102+1 : TPTP v8.1.2. Bugfixed v5.4.0. % Transfm : none % Format : tptp % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s % Computer : n009.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 15:23:48 EDT 2023 % Result : Theorem 13.22s 2.45s % Output : Proof 19.38s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.12 % Problem : SET102+1 : TPTP v8.1.2. Bugfixed v5.4.0. % 0.07/0.13 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s % 0.12/0.34 % Computer : n009.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 08:53:05 EDT 2023 % 0.12/0.34 % CPUTime : % 0.49/0.62 ________ _____ % 0.49/0.62 ___ __ \_________(_)________________________________ % 0.49/0.62 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/ % 0.49/0.62 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ ) % 0.49/0.62 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/ % 0.49/0.62 % 0.49/0.62 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic % 0.49/0.62 (2023-06-19) % 0.49/0.62 % 0.49/0.62 (c) Philipp Rümmer, 2009-2023 % 0.49/0.62 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen, % 0.49/0.62 Amanda Stjerna. % 0.49/0.62 Free software under BSD-3-Clause. % 0.49/0.62 % 0.49/0.62 For more information, visit http://www.philipp.ruemmer.org/princess.shtml % 0.49/0.62 % 0.49/0.62 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ... % 0.49/0.63 Running up to 7 provers in parallel. % 0.68/0.65 Prover 0: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1042961893 % 0.68/0.65 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1571432423 % 0.68/0.65 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=1868514696 % 0.68/0.65 Prover 2: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMinimalAndEmpty -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1065072994 % 0.68/0.65 Prover 3: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1922548996 % 0.68/0.65 Prover 5: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=complete -randomSeed=1259561288 % 0.68/0.65 Prover 6: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximalOutermost -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1399714365 % 3.41/1.14 Prover 1: Preprocessing ... % 3.41/1.14 Prover 4: Preprocessing ... % 3.68/1.18 Prover 6: Preprocessing ... % 3.68/1.18 Prover 2: Preprocessing ... % 3.68/1.18 Prover 3: Preprocessing ... % 3.68/1.18 Prover 0: Preprocessing ... % 3.68/1.19 Prover 5: Preprocessing ... % 8.89/1.87 Prover 1: Warning: ignoring some quantifiers % 8.89/1.88 Prover 3: Warning: ignoring some quantifiers % 8.89/1.89 Prover 5: Proving ... % 8.89/1.90 Prover 6: Proving ... % 8.89/1.91 Prover 1: Constructing countermodel ... % 8.89/1.91 Prover 3: Constructing countermodel ... % 9.46/1.94 Prover 4: Warning: ignoring some quantifiers % 9.77/1.98 Prover 4: Constructing countermodel ... % 9.77/1.99 Prover 2: Proving ... % 10.30/2.06 Prover 0: Proving ... % 13.22/2.45 Prover 0: proved (1805ms) % 13.22/2.45 % 13.22/2.45 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % 13.22/2.45 % 13.22/2.45 Prover 2: stopped % 13.22/2.45 Prover 3: stopped % 13.55/2.46 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089 % 13.55/2.46 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470 % 13.55/2.46 Prover 5: stopped % 13.55/2.47 Prover 10: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=919308125 % 13.55/2.47 Prover 6: stopped % 13.55/2.48 Prover 11: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1509710984 % 13.55/2.49 Prover 13: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=1138197443 % 14.23/2.55 Prover 7: Preprocessing ... % 14.23/2.56 Prover 8: Preprocessing ... % 14.23/2.57 Prover 11: Preprocessing ... % 14.23/2.57 Prover 10: Preprocessing ... % 14.54/2.59 Prover 13: Preprocessing ... % 15.64/2.73 Prover 10: Warning: ignoring some quantifiers % 15.64/2.73 Prover 8: Warning: ignoring some quantifiers % 15.64/2.74 Prover 7: Warning: ignoring some quantifiers % 15.64/2.75 Prover 10: Constructing countermodel ... % 15.64/2.75 Prover 8: Constructing countermodel ... % 15.64/2.76 Prover 7: Constructing countermodel ... % 15.99/2.80 Prover 13: Warning: ignoring some quantifiers % 15.99/2.82 Prover 13: Constructing countermodel ... % 16.65/2.85 Prover 11: Warning: ignoring some quantifiers % 16.65/2.86 Prover 11: Constructing countermodel ... % 18.31/3.10 Prover 10: gave up % 18.31/3.11 Prover 16: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=completeFrugal -randomSeed=-2043353683 % 18.31/3.12 Prover 7: Found proof (size 74) % 18.31/3.12 Prover 7: proved (665ms) % 18.31/3.12 Prover 8: stopped % 18.31/3.12 Prover 13: stopped % 18.31/3.12 Prover 1: stopped % 18.31/3.12 Prover 4: stopped % 18.31/3.12 Prover 11: stopped % 18.31/3.14 Prover 16: Preprocessing ... % 18.90/3.18 Prover 16: stopped % 18.90/3.18 % 18.90/3.18 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % 18.90/3.18 % 19.05/3.19 % SZS output start Proof for theBenchmark % 19.05/3.19 Assumptions after simplification: % 19.05/3.19 --------------------------------- % 19.05/3.19 % 19.05/3.19 (compose_defn1) % 19.05/3.22 $i(universal_class) & ? [v0: $i] : (cross_product(universal_class, % 19.05/3.22 universal_class) = v0 & $i(v0) & ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : % 19.05/3.22 ( ~ (compose(v2, v1) = v3) | ~ $i(v2) | ~ $i(v1) | subclass(v3, v0))) % 19.05/3.22 % 19.05/3.22 (element_relation) % 19.05/3.22 $i(element_relation) & $i(universal_class) & ? [v0: $i] : % 19.05/3.22 (cross_product(universal_class, universal_class) = v0 & $i(v0) & % 19.05/3.22 subclass(element_relation, v0)) % 19.05/3.22 % 19.05/3.22 (flip) % 19.05/3.22 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0, % 19.05/3.22 universal_class) = v1 & cross_product(universal_class, universal_class) = % 19.05/3.22 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (flip(v2) = v3) | ~ % 19.05/3.22 $i(v2) | subclass(v3, v1))) % 19.05/3.22 % 19.05/3.22 (flip_defn) % 19.26/3.23 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0, % 19.26/3.23 universal_class) = v1 & cross_product(universal_class, universal_class) = % 19.26/3.23 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] % 19.26/3.23 : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~ % 19.26/3.23 (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v3, v2) = v6) | ~ $i(v5) | % 19.26/3.23 ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v5) | ? [v9: $i] : ? % 19.26/3.23 [v10: $i] : (ordered_pair(v9, v4) = v10 & ordered_pair(v2, v3) = v9 & % 19.26/3.23 $i(v10) & $i(v9) & ( ~ member(v10, v1) | member(v10, v8)))) & ! [v2: % 19.26/3.23 $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] % 19.26/3.23 : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ % 19.26/3.23 (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ % 19.26/3.23 $i(v2) | ~ member(v7, v8) | member(v7, v1)) & ! [v2: $i] : ! [v3: $i] : % 19.26/3.23 ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ % 19.26/3.23 (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v2, % 19.26/3.23 v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ % 19.26/3.23 member(v7, v8) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v4) = v10 % 19.26/3.23 & ordered_pair(v3, v2) = v9 & $i(v10) & $i(v9) & member(v10, v5))) & ! % 19.26/3.23 [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: % 19.26/3.23 $i] : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | % 19.26/3.23 ~ (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ % 19.26/3.23 $i(v2) | ~ member(v7, v1) | member(v7, v8) | ? [v9: $i] : ? [v10: $i] : % 19.26/3.23 (ordered_pair(v9, v4) = v10 & ordered_pair(v3, v2) = v9 & $i(v10) & $i(v9) % 19.26/3.23 & ~ member(v10, v5))) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! % 19.26/3.23 [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (flip(v5) = v6) | % 19.26/3.23 ~ (ordered_pair(v7, v4) = v8) | ~ (ordered_pair(v3, v2) = v7) | ~ $i(v5) % 19.26/3.23 | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] : % 19.26/3.23 (ordered_pair(v9, v4) = v10 & ordered_pair(v2, v3) = v9 & $i(v10) & $i(v9) % 19.26/3.23 & ( ~ member(v10, v6) | (member(v10, v1) & member(v8, v5)))))) % 19.26/3.23 % 19.26/3.23 (function_defn) % 19.26/3.23 $i(identity_relation) & $i(universal_class) & ? [v0: $i] : % 19.26/3.23 (cross_product(universal_class, universal_class) = v0 & $i(v0) & ! [v1: $i] : % 19.26/3.23 ! [v2: $i] : ( ~ (inverse(v1) = v2) | ~ $i(v1) | ~ function(v1) | % 19.26/3.23 subclass(v1, v0)) & ! [v1: $i] : ! [v2: $i] : ( ~ (inverse(v1) = v2) | % 19.26/3.23 ~ $i(v1) | ~ function(v1) | ? [v3: $i] : (compose(v1, v2) = v3 & $i(v3) % 19.26/3.23 & subclass(v3, identity_relation))) & ! [v1: $i] : ! [v2: $i] : ( ~ % 19.26/3.23 (inverse(v1) = v2) | ~ $i(v1) | ~ subclass(v1, v0) | function(v1) | ? % 19.26/3.23 [v3: $i] : (compose(v1, v2) = v3 & $i(v3) & ~ subclass(v3, % 19.26/3.23 identity_relation)))) % 19.26/3.23 % 19.26/3.23 (ordered_pair_defn) % 19.26/3.24 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~ (singleton(v1) = % 19.26/3.24 v2) | ~ (unordered_pair(v0, v2) = v3) | ~ $i(v1) | ~ $i(v0) | ? [v4: % 19.26/3.24 $i] : ? [v5: $i] : (ordered_pair(v0, v1) = v4 & singleton(v0) = v5 & % 19.26/3.24 unordered_pair(v5, v3) = v4 & $i(v5) & $i(v4))) & ! [v0: $i] : ! [v1: % 19.26/3.24 $i] : ! [v2: $i] : ( ~ (ordered_pair(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) % 19.26/3.24 | ? [v3: $i] : ? [v4: $i] : ? [v5: $i] : (singleton(v1) = v4 & % 19.26/3.24 singleton(v0) = v3 & unordered_pair(v3, v5) = v2 & unordered_pair(v0, v4) % 19.26/3.24 = v5 & $i(v5) & $i(v4) & $i(v3) & $i(v2))) % 19.26/3.24 % 19.26/3.24 (rotate) % 19.26/3.24 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0, % 19.26/3.24 universal_class) = v1 & cross_product(universal_class, universal_class) = % 19.26/3.24 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (rotate(v2) = v3) | % 19.26/3.24 ~ $i(v2) | subclass(v3, v1))) % 19.26/3.24 % 19.26/3.24 (rotate_defn) % 19.26/3.24 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0, % 19.26/3.24 universal_class) = v1 & cross_product(universal_class, universal_class) = % 19.26/3.24 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] % 19.26/3.24 : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~ % 19.26/3.24 (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = v6) | ~ $i(v5) | % 19.26/3.24 ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v8) | member(v7, v1)) & % 19.26/3.24 ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: % 19.26/3.24 $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~ (ordered_pair(v6, v5) = v7) % 19.26/3.24 | ~ (ordered_pair(v3, v4) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ % 19.26/3.24 $i(v2) | ~ member(v7, v8) | ? [v9: $i] : ? [v10: $i] : % 19.26/3.24 (ordered_pair(v9, v3) = v10 & ordered_pair(v4, v5) = v9 & $i(v10) & $i(v9) % 19.26/3.24 & member(v10, v2))) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: % 19.26/3.24 $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~ % 19.26/3.24 (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = v6) | ~ $i(v5) | % 19.26/3.24 ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v1) | member(v7, v8) | % 19.26/3.24 ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v3) = v10 & ordered_pair(v4, % 19.26/3.24 v5) = v9 & $i(v10) & $i(v9) & ~ member(v10, v2))) & ! [v2: $i] : ! % 19.26/3.24 [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: % 19.26/3.24 $i] : ( ~ (rotate(v2) = v8) | ~ (ordered_pair(v6, v3) = v7) | ~ % 19.26/3.24 (ordered_pair(v4, v5) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ % 19.26/3.24 $i(v2) | ~ member(v7, v2) | ? [v9: $i] : ? [v10: $i] : % 19.26/3.24 (ordered_pair(v9, v5) = v10 & ordered_pair(v3, v4) = v9 & $i(v10) & $i(v9) % 19.26/3.24 & ( ~ member(v10, v1) | member(v10, v8)))) & ! [v2: $i] : ! [v3: $i] : % 19.26/3.24 ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ % 19.26/3.24 (rotate(v2) = v6) | ~ (ordered_pair(v7, v3) = v8) | ~ (ordered_pair(v4, % 19.26/3.24 v5) = v7) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ? [v9: % 19.26/3.24 $i] : ? [v10: $i] : (ordered_pair(v9, v5) = v10 & ordered_pair(v3, v4) % 19.26/3.24 = v9 & $i(v10) & $i(v9) & ( ~ member(v10, v6) | (member(v10, v1) & % 19.26/3.24 member(v8, v2)))))) % 19.26/3.24 % 19.26/3.24 (successor_relation_defn1) % 19.26/3.24 $i(successor_relation) & $i(universal_class) & ? [v0: $i] : % 19.26/3.24 (cross_product(universal_class, universal_class) = v0 & $i(v0) & % 19.26/3.24 subclass(successor_relation, v0)) % 19.26/3.24 % 19.26/3.24 (unordered_pair) % 19.26/3.24 $i(universal_class) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ % 19.26/3.24 (unordered_pair(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | member(v2, % 19.26/3.24 universal_class)) % 19.26/3.24 % 19.26/3.24 (unordered_pair_defn) % 19.26/3.25 $i(universal_class) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : % 19.26/3.25 (v2 = v0 | v1 = v0 | ~ (unordered_pair(v1, v2) = v3) | ~ $i(v2) | ~ $i(v1) % 19.26/3.25 | ~ $i(v0) | ~ member(v0, v3)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : % 19.26/3.25 ! [v3: $i] : ( ~ (unordered_pair(v1, v2) = v3) | ~ $i(v2) | ~ $i(v1) | ~ % 19.26/3.25 $i(v0) | ~ member(v0, v3) | member(v0, universal_class)) & ! [v0: $i] : ! % 19.26/3.25 [v1: $i] : ! [v2: $i] : ( ~ (unordered_pair(v1, v0) = v2) | ~ $i(v1) | ~ % 19.26/3.25 $i(v0) | ~ member(v0, universal_class) | member(v0, v2)) & ! [v0: $i] : ! % 19.26/3.25 [v1: $i] : ! [v2: $i] : ( ~ (unordered_pair(v0, v1) = v2) | ~ $i(v1) | ~ % 19.26/3.25 $i(v0) | ~ member(v0, universal_class) | member(v0, v2)) % 19.26/3.25 % 19.26/3.25 (unordered_pair_member_of_ordered_pair) % 19.26/3.25 ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ? [v3: $i] : ? [v4: $i] : % 19.26/3.25 (ordered_pair(v0, v1) = v4 & singleton(v1) = v2 & unordered_pair(v0, v2) = v3 % 19.26/3.25 & $i(v4) & $i(v3) & $i(v2) & $i(v1) & $i(v0) & ~ member(v3, v4)) % 19.26/3.25 % 19.26/3.25 (function-axioms) % 19.26/3.25 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : (v1 = v0 % 19.26/3.25 | ~ (restrict(v4, v3, v2) = v1) | ~ (restrict(v4, v3, v2) = v0)) & ! [v0: % 19.26/3.25 $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (apply(v3, v2) % 19.26/3.25 = v1) | ~ (apply(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] % 19.26/3.25 : ! [v3: $i] : (v1 = v0 | ~ (compose(v3, v2) = v1) | ~ (compose(v3, v2) = % 19.26/3.25 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | % 19.26/3.25 ~ (image(v3, v2) = v1) | ~ (image(v3, v2) = v0)) & ! [v0: $i] : ! [v1: % 19.26/3.25 $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (union(v3, v2) = v1) | ~ % 19.26/3.25 (union(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: % 19.26/3.25 $i] : (v1 = v0 | ~ (intersection(v3, v2) = v1) | ~ (intersection(v3, v2) = % 19.26/3.25 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | % 19.26/3.25 ~ (cross_product(v3, v2) = v1) | ~ (cross_product(v3, v2) = v0)) & ! [v0: % 19.26/3.25 $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ % 19.26/3.25 (ordered_pair(v3, v2) = v1) | ~ (ordered_pair(v3, v2) = v0)) & ! [v0: $i] % 19.26/3.25 : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (unordered_pair(v3, % 19.26/3.25 v2) = v1) | ~ (unordered_pair(v3, v2) = v0)) & ! [v0: $i] : ! [v1: % 19.26/3.25 $i] : ! [v2: $i] : (v1 = v0 | ~ (power_class(v2) = v1) | ~ % 19.26/3.25 (power_class(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 % 19.26/3.25 | ~ (sum_class(v2) = v1) | ~ (sum_class(v2) = v0)) & ! [v0: $i] : ! [v1: % 19.26/3.25 $i] : ! [v2: $i] : (v1 = v0 | ~ (range_of(v2) = v1) | ~ (range_of(v2) = % 19.26/3.25 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ % 19.26/3.25 (inverse(v2) = v1) | ~ (inverse(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! % 19.26/3.25 [v2: $i] : (v1 = v0 | ~ (successor(v2) = v1) | ~ (successor(v2) = v0)) & ! % 19.26/3.25 [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (flip(v2) = v1) | ~ % 19.26/3.25 (flip(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ % 19.26/3.25 (rotate(v2) = v1) | ~ (rotate(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! % 19.26/3.25 [v2: $i] : (v1 = v0 | ~ (domain_of(v2) = v1) | ~ (domain_of(v2) = v0)) & ! % 19.26/3.25 [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (complement(v2) = v1) | % 19.26/3.25 ~ (complement(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = % 19.26/3.25 v0 | ~ (first(v2) = v1) | ~ (first(v2) = v0)) & ! [v0: $i] : ! [v1: $i] % 19.26/3.25 : ! [v2: $i] : (v1 = v0 | ~ (second(v2) = v1) | ~ (second(v2) = v0)) & ! % 19.26/3.25 [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (singleton(v2) = v1) | ~ % 19.26/3.25 (singleton(v2) = v0)) % 19.26/3.25 % 19.26/3.25 Further assumptions not needed in the proof: % 19.26/3.25 -------------------------------------------- % 19.38/3.25 apply_defn, choice, class_elements_are_sets, complement, compose_defn2, % 19.38/3.25 cross_product, cross_product_defn, disjoint_defn, domain_of, % 19.38/3.25 element_relation_defn, extensionality, first_second, identity_relation, % 19.38/3.25 image_defn, inductive_defn, infinity, intersection, inverse_defn, % 19.38/3.25 null_class_defn, power_class, power_class_defn, range_of_defn, regularity, % 19.38/3.25 replacement, restrict_defn, singleton_set_defn, subclass_defn, successor_defn, % 19.38/3.25 successor_relation_defn2, sum_class, sum_class_defn, union_defn % 19.38/3.25 % 19.38/3.25 Those formulas are unsatisfiable: % 19.38/3.25 --------------------------------- % 19.38/3.25 % 19.38/3.25 Begin of proof % 19.38/3.25 | % 19.38/3.25 | ALPHA: (unordered_pair_defn) implies: % 19.38/3.25 | (1) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (unordered_pair(v1, v0) = % 19.38/3.26 | v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v0, universal_class) | % 19.38/3.26 | member(v0, v2)) % 19.38/3.26 | % 19.38/3.26 | ALPHA: (unordered_pair) implies: % 19.38/3.26 | (2) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (unordered_pair(v0, v1) = % 19.38/3.26 | v2) | ~ $i(v1) | ~ $i(v0) | member(v2, universal_class)) % 19.38/3.26 | % 19.38/3.26 | ALPHA: (ordered_pair_defn) implies: % 19.38/3.26 | (3) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (ordered_pair(v0, v1) = % 19.38/3.26 | v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: $i] : ? [v5: % 19.38/3.26 | $i] : (singleton(v1) = v4 & singleton(v0) = v3 & unordered_pair(v3, % 19.38/3.26 | v5) = v2 & unordered_pair(v0, v4) = v5 & $i(v5) & $i(v4) & $i(v3) % 19.38/3.26 | & $i(v2))) % 19.38/3.26 | (4) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~ % 19.38/3.26 | (singleton(v1) = v2) | ~ (unordered_pair(v0, v2) = v3) | ~ $i(v1) | % 19.38/3.26 | ~ $i(v0) | ? [v4: $i] : ? [v5: $i] : (ordered_pair(v0, v1) = v4 & % 19.38/3.26 | singleton(v0) = v5 & unordered_pair(v5, v3) = v4 & $i(v5) & % 19.38/3.26 | $i(v4))) % 19.38/3.26 | % 19.38/3.26 | ALPHA: (element_relation) implies: % 19.38/3.26 | (5) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 & % 19.38/3.26 | $i(v0) & subclass(element_relation, v0)) % 19.38/3.26 | % 19.38/3.26 | ALPHA: (rotate_defn) implies: % 19.38/3.26 | (6) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 & % 19.38/3.26 | cross_product(universal_class, universal_class) = v0 & $i(v1) & % 19.38/3.26 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! % 19.38/3.26 | [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~ % 19.38/3.26 | (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = v6) | ~ % 19.38/3.26 | $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v8) | % 19.38/3.26 | member(v7, v1)) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: % 19.38/3.26 | $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = % 19.38/3.26 | v8) | ~ (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = % 19.38/3.26 | v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ % 19.38/3.26 | member(v7, v8) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v3) % 19.38/3.26 | = v10 & ordered_pair(v4, v5) = v9 & $i(v10) & $i(v9) & % 19.38/3.26 | member(v10, v2))) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! % 19.38/3.26 | [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) % 19.38/3.26 | = v8) | ~ (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) % 19.38/3.26 | = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ % 19.38/3.26 | member(v7, v1) | member(v7, v8) | ? [v9: $i] : ? [v10: $i] : % 19.38/3.26 | (ordered_pair(v9, v3) = v10 & ordered_pair(v4, v5) = v9 & $i(v10) & % 19.38/3.26 | $i(v9) & ~ member(v10, v2))) & ! [v2: $i] : ! [v3: $i] : ! % 19.38/3.26 | [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( % 19.38/3.26 | ~ (rotate(v2) = v8) | ~ (ordered_pair(v6, v3) = v7) | ~ % 19.38/3.26 | (ordered_pair(v4, v5) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 19.38/3.26 | ~ $i(v2) | ~ member(v7, v2) | ? [v9: $i] : ? [v10: $i] : % 19.38/3.26 | (ordered_pair(v9, v5) = v10 & ordered_pair(v3, v4) = v9 & $i(v10) & % 19.38/3.26 | $i(v9) & ( ~ member(v10, v1) | member(v10, v8)))) & ! [v2: $i] : % 19.38/3.26 | ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : % 19.38/3.26 | ! [v8: $i] : ( ~ (rotate(v2) = v6) | ~ (ordered_pair(v7, v3) = v8) % 19.38/3.26 | | ~ (ordered_pair(v4, v5) = v7) | ~ $i(v5) | ~ $i(v4) | ~ % 19.38/3.26 | $i(v3) | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, % 19.38/3.26 | v5) = v10 & ordered_pair(v3, v4) = v9 & $i(v10) & $i(v9) & ( ~ % 19.38/3.26 | member(v10, v6) | (member(v10, v1) & member(v8, v2)))))) % 19.38/3.26 | % 19.38/3.26 | ALPHA: (rotate) implies: % 19.38/3.26 | (7) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 & % 19.38/3.26 | cross_product(universal_class, universal_class) = v0 & $i(v1) & % 19.38/3.26 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (rotate(v2) = v3) | ~ % 19.38/3.26 | $i(v2) | subclass(v3, v1))) % 19.38/3.26 | % 19.38/3.26 | ALPHA: (flip_defn) implies: % 19.38/3.27 | (8) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 & % 19.38/3.27 | cross_product(universal_class, universal_class) = v0 & $i(v1) & % 19.38/3.27 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! % 19.38/3.27 | [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~ % 19.38/3.27 | (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v3, v2) = v6) | ~ % 19.38/3.27 | $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v5) | ? % 19.38/3.27 | [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v4) = v10 & % 19.38/3.27 | ordered_pair(v2, v3) = v9 & $i(v10) & $i(v9) & ( ~ member(v10, % 19.38/3.27 | v1) | member(v10, v8)))) & ! [v2: $i] : ! [v3: $i] : ! % 19.38/3.27 | [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( % 19.38/3.27 | ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ % 19.38/3.27 | (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 19.38/3.27 | ~ $i(v2) | ~ member(v7, v8) | member(v7, v1)) & ! [v2: $i] : ! % 19.38/3.27 | [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! % 19.38/3.27 | [v8: $i] : ( ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ % 19.38/3.27 | (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 19.38/3.27 | ~ $i(v2) | ~ member(v7, v8) | ? [v9: $i] : ? [v10: $i] : % 19.38/3.27 | (ordered_pair(v9, v4) = v10 & ordered_pair(v3, v2) = v9 & $i(v10) & % 19.38/3.27 | $i(v9) & member(v10, v5))) & ! [v2: $i] : ! [v3: $i] : ! [v4: % 19.38/3.27 | $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ % 19.38/3.27 | (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ % 19.38/3.27 | (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 19.38/3.27 | ~ $i(v2) | ~ member(v7, v1) | member(v7, v8) | ? [v9: $i] : ? % 19.38/3.27 | [v10: $i] : (ordered_pair(v9, v4) = v10 & ordered_pair(v3, v2) = v9 % 19.38/3.27 | & $i(v10) & $i(v9) & ~ member(v10, v5))) & ! [v2: $i] : ! [v3: % 19.38/3.27 | $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! % 19.38/3.27 | [v8: $i] : ( ~ (flip(v5) = v6) | ~ (ordered_pair(v7, v4) = v8) | ~ % 19.38/3.27 | (ordered_pair(v3, v2) = v7) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 19.38/3.27 | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v4) = v10 % 19.38/3.27 | & ordered_pair(v2, v3) = v9 & $i(v10) & $i(v9) & ( ~ member(v10, % 19.38/3.27 | v6) | (member(v10, v1) & member(v8, v5)))))) % 19.38/3.27 | % 19.38/3.27 | ALPHA: (flip) implies: % 19.38/3.27 | (9) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 & % 19.38/3.27 | cross_product(universal_class, universal_class) = v0 & $i(v1) & % 19.38/3.27 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (flip(v2) = v3) | ~ $i(v2) % 19.38/3.27 | | subclass(v3, v1))) % 19.38/3.27 | % 19.38/3.27 | ALPHA: (successor_relation_defn1) implies: % 19.38/3.27 | (10) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 & % 19.38/3.27 | $i(v0) & subclass(successor_relation, v0)) % 19.38/3.27 | % 19.38/3.27 | ALPHA: (compose_defn1) implies: % 19.38/3.27 | (11) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 & % 19.38/3.27 | $i(v0) & ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~ (compose(v2, % 19.38/3.27 | v1) = v3) | ~ $i(v2) | ~ $i(v1) | subclass(v3, v0))) % 19.38/3.27 | % 19.38/3.27 | ALPHA: (function_defn) implies: % 19.38/3.27 | (12) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 & % 19.38/3.27 | $i(v0) & ! [v1: $i] : ! [v2: $i] : ( ~ (inverse(v1) = v2) | ~ % 19.38/3.27 | $i(v1) | ~ function(v1) | subclass(v1, v0)) & ! [v1: $i] : ! % 19.38/3.27 | [v2: $i] : ( ~ (inverse(v1) = v2) | ~ $i(v1) | ~ function(v1) | ? % 19.38/3.27 | [v3: $i] : (compose(v1, v2) = v3 & $i(v3) & subclass(v3, % 19.38/3.27 | identity_relation))) & ! [v1: $i] : ! [v2: $i] : ( ~ % 19.38/3.27 | (inverse(v1) = v2) | ~ $i(v1) | ~ subclass(v1, v0) | % 19.38/3.27 | function(v1) | ? [v3: $i] : (compose(v1, v2) = v3 & $i(v3) & ~ % 19.38/3.27 | subclass(v3, identity_relation)))) % 19.38/3.27 | % 19.38/3.27 | ALPHA: (function-axioms) implies: % 19.38/3.27 | (13) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (singleton(v2) % 19.38/3.27 | = v1) | ~ (singleton(v2) = v0)) % 19.38/3.27 | (14) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ % 19.38/3.27 | (ordered_pair(v3, v2) = v1) | ~ (ordered_pair(v3, v2) = v0)) % 19.38/3.27 | (15) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ % 19.38/3.27 | (cross_product(v3, v2) = v1) | ~ (cross_product(v3, v2) = v0)) % 19.38/3.27 | % 19.38/3.27 | DELTA: instantiating (10) with fresh symbol all_37_0 gives: % 19.38/3.27 | (16) cross_product(universal_class, universal_class) = all_37_0 & % 19.38/3.27 | $i(all_37_0) & subclass(successor_relation, all_37_0) % 19.38/3.27 | % 19.38/3.27 | ALPHA: (16) implies: % 19.38/3.27 | (17) cross_product(universal_class, universal_class) = all_37_0 % 19.38/3.27 | % 19.38/3.27 | DELTA: instantiating (5) with fresh symbol all_39_0 gives: % 19.38/3.28 | (18) cross_product(universal_class, universal_class) = all_39_0 & % 19.38/3.28 | $i(all_39_0) & subclass(element_relation, all_39_0) % 19.38/3.28 | % 19.38/3.28 | ALPHA: (18) implies: % 19.38/3.28 | (19) cross_product(universal_class, universal_class) = all_39_0 % 19.38/3.28 | % 19.38/3.28 | DELTA: instantiating (11) with fresh symbol all_45_0 gives: % 19.38/3.28 | (20) cross_product(universal_class, universal_class) = all_45_0 & % 19.38/3.28 | $i(all_45_0) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ % 19.38/3.28 | (compose(v1, v0) = v2) | ~ $i(v1) | ~ $i(v0) | subclass(v2, % 19.38/3.28 | all_45_0)) % 19.38/3.28 | % 19.38/3.28 | ALPHA: (20) implies: % 19.38/3.28 | (21) cross_product(universal_class, universal_class) = all_45_0 % 19.38/3.28 | % 19.38/3.28 | DELTA: instantiating (7) with fresh symbols all_53_0, all_53_1 gives: % 19.38/3.28 | (22) cross_product(all_53_1, universal_class) = all_53_0 & % 19.38/3.28 | cross_product(universal_class, universal_class) = all_53_1 & % 19.38/3.28 | $i(all_53_0) & $i(all_53_1) & ! [v0: $i] : ! [v1: $i] : ( ~ % 19.38/3.28 | (rotate(v0) = v1) | ~ $i(v0) | subclass(v1, all_53_0)) % 19.38/3.28 | % 19.38/3.28 | ALPHA: (22) implies: % 19.38/3.28 | (23) cross_product(universal_class, universal_class) = all_53_1 % 19.38/3.28 | (24) cross_product(all_53_1, universal_class) = all_53_0 % 19.38/3.28 | % 19.38/3.28 | DELTA: instantiating (9) with fresh symbols all_56_0, all_56_1 gives: % 19.38/3.28 | (25) cross_product(all_56_1, universal_class) = all_56_0 & % 19.38/3.28 | cross_product(universal_class, universal_class) = all_56_1 & % 19.38/3.28 | $i(all_56_0) & $i(all_56_1) & ! [v0: $i] : ! [v1: $i] : ( ~ % 19.38/3.28 | (flip(v0) = v1) | ~ $i(v0) | subclass(v1, all_56_0)) % 19.38/3.28 | % 19.38/3.28 | ALPHA: (25) implies: % 19.38/3.28 | (26) cross_product(universal_class, universal_class) = all_56_1 % 19.38/3.28 | % 19.38/3.28 | DELTA: instantiating (unordered_pair_member_of_ordered_pair) with fresh % 19.38/3.28 | symbols all_59_0, all_59_1, all_59_2, all_59_3, all_59_4 gives: % 19.38/3.28 | (27) ordered_pair(all_59_4, all_59_3) = all_59_0 & singleton(all_59_3) = % 19.38/3.28 | all_59_2 & unordered_pair(all_59_4, all_59_2) = all_59_1 & % 19.38/3.28 | $i(all_59_0) & $i(all_59_1) & $i(all_59_2) & $i(all_59_3) & % 19.38/3.28 | $i(all_59_4) & ~ member(all_59_1, all_59_0) % 19.38/3.28 | % 19.38/3.28 | ALPHA: (27) implies: % 19.38/3.28 | (28) ~ member(all_59_1, all_59_0) % 19.38/3.28 | (29) $i(all_59_4) % 19.38/3.28 | (30) $i(all_59_3) % 19.38/3.28 | (31) $i(all_59_2) % 19.38/3.28 | (32) $i(all_59_1) % 19.38/3.28 | (33) unordered_pair(all_59_4, all_59_2) = all_59_1 % 19.38/3.28 | (34) singleton(all_59_3) = all_59_2 % 19.38/3.28 | (35) ordered_pair(all_59_4, all_59_3) = all_59_0 % 19.38/3.28 | % 19.38/3.28 | DELTA: instantiating (12) with fresh symbol all_61_0 gives: % 19.38/3.28 | (36) cross_product(universal_class, universal_class) = all_61_0 & % 19.38/3.28 | $i(all_61_0) & ! [v0: $i] : ! [v1: $i] : ( ~ (inverse(v0) = v1) | ~ % 19.38/3.28 | $i(v0) | ~ function(v0) | subclass(v0, all_61_0)) & ! [v0: $i] : % 19.38/3.28 | ! [v1: $i] : ( ~ (inverse(v0) = v1) | ~ $i(v0) | ~ function(v0) | ? % 19.38/3.28 | [v2: $i] : (compose(v0, v1) = v2 & $i(v2) & subclass(v2, % 19.38/3.28 | identity_relation))) & ! [v0: $i] : ! [v1: $i] : ( ~ % 19.38/3.28 | (inverse(v0) = v1) | ~ $i(v0) | ~ subclass(v0, all_61_0) | % 19.38/3.28 | function(v0) | ? [v2: $i] : (compose(v0, v1) = v2 & $i(v2) & ~ % 19.38/3.28 | subclass(v2, identity_relation))) % 19.38/3.28 | % 19.38/3.28 | ALPHA: (36) implies: % 19.38/3.28 | (37) cross_product(universal_class, universal_class) = all_61_0 % 19.38/3.28 | % 19.38/3.28 | DELTA: instantiating (8) with fresh symbols all_64_0, all_64_1 gives: % 19.38/3.28 | (38) cross_product(all_64_1, universal_class) = all_64_0 & % 19.38/3.28 | cross_product(universal_class, universal_class) = all_64_1 & % 19.38/3.28 | $i(all_64_0) & $i(all_64_1) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] % 19.38/3.29 | : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ % 19.38/3.29 | (flip(v3) = v6) | ~ (ordered_pair(v4, v2) = v5) | ~ % 19.38/3.29 | (ordered_pair(v1, v0) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ % 19.38/3.29 | $i(v0) | ~ member(v5, v3) | ? [v7: $i] : ? [v8: $i] : % 19.38/3.29 | (ordered_pair(v7, v2) = v8 & ordered_pair(v0, v1) = v7 & $i(v8) & % 19.38/3.29 | $i(v7) & ( ~ member(v8, all_64_0) | member(v8, v6)))) & ! [v0: % 19.38/3.29 | $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! % 19.38/3.29 | [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v6) | ~ (ordered_pair(v4, % 19.38/3.29 | v2) = v5) | ~ (ordered_pair(v0, v1) = v4) | ~ $i(v3) | ~ % 19.38/3.29 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v5, v6) | member(v5, % 19.38/3.29 | all_64_0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] % 19.38/3.29 | : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v6) | ~ % 19.38/3.29 | (ordered_pair(v4, v2) = v5) | ~ (ordered_pair(v0, v1) = v4) | ~ % 19.38/3.29 | $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v5, v6) | ? % 19.38/3.29 | [v7: $i] : ? [v8: $i] : (ordered_pair(v7, v2) = v8 & % 19.38/3.29 | ordered_pair(v1, v0) = v7 & $i(v8) & $i(v7) & member(v8, v3))) & % 19.38/3.29 | ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : % 19.38/3.29 | ! [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v6) | ~ (ordered_pair(v4, % 19.38/3.29 | v2) = v5) | ~ (ordered_pair(v0, v1) = v4) | ~ $i(v3) | ~ % 19.38/3.29 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v5, all_64_0) | % 19.38/3.29 | member(v5, v6) | ? [v7: $i] : ? [v8: $i] : (ordered_pair(v7, v2) = % 19.38/3.29 | v8 & ordered_pair(v1, v0) = v7 & $i(v8) & $i(v7) & ~ member(v8, % 19.38/3.29 | v3))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : % 19.38/3.29 | ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v4) | ~ % 19.38/3.29 | (ordered_pair(v5, v2) = v6) | ~ (ordered_pair(v1, v0) = v5) | ~ % 19.38/3.29 | $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v7: $i] : ? [v8: % 19.38/3.29 | $i] : (ordered_pair(v7, v2) = v8 & ordered_pair(v0, v1) = v7 & % 19.38/3.29 | $i(v8) & $i(v7) & ( ~ member(v8, v4) | (member(v8, all_64_0) & % 19.38/3.29 | member(v6, v3))))) % 19.38/3.29 | % 19.38/3.29 | ALPHA: (38) implies: % 19.38/3.29 | (39) cross_product(universal_class, universal_class) = all_64_1 % 19.38/3.29 | (40) cross_product(all_64_1, universal_class) = all_64_0 % 19.38/3.29 | % 19.38/3.29 | DELTA: instantiating (6) with fresh symbols all_67_0, all_67_1 gives: % 19.38/3.29 | (41) cross_product(all_67_1, universal_class) = all_67_0 & % 19.38/3.29 | cross_product(universal_class, universal_class) = all_67_1 & % 19.38/3.29 | $i(all_67_0) & $i(all_67_1) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] % 19.38/3.29 | : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ % 19.38/3.29 | (rotate(v0) = v6) | ~ (ordered_pair(v4, v3) = v5) | ~ % 19.38/3.29 | (ordered_pair(v1, v2) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ % 19.38/3.29 | $i(v0) | ~ member(v5, v6) | member(v5, all_67_0)) & ! [v0: $i] : % 19.38/3.29 | ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : % 19.38/3.29 | ! [v6: $i] : ( ~ (rotate(v0) = v6) | ~ (ordered_pair(v4, v3) = v5) | % 19.38/3.29 | ~ (ordered_pair(v1, v2) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | % 19.38/3.29 | ~ $i(v0) | ~ member(v5, v6) | ? [v7: $i] : ? [v8: $i] : % 19.38/3.29 | (ordered_pair(v7, v1) = v8 & ordered_pair(v2, v3) = v7 & $i(v8) & % 19.38/3.29 | $i(v7) & member(v8, v0))) & ! [v0: $i] : ! [v1: $i] : ! [v2: % 19.38/3.29 | $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ % 19.38/3.29 | (rotate(v0) = v6) | ~ (ordered_pair(v4, v3) = v5) | ~ % 19.38/3.29 | (ordered_pair(v1, v2) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ % 19.38/3.29 | $i(v0) | ~ member(v5, all_67_0) | member(v5, v6) | ? [v7: $i] : ? % 19.38/3.29 | [v8: $i] : (ordered_pair(v7, v1) = v8 & ordered_pair(v2, v3) = v7 & % 19.38/3.29 | $i(v8) & $i(v7) & ~ member(v8, v0))) & ! [v0: $i] : ! [v1: $i] % 19.38/3.29 | : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] % 19.38/3.29 | : ( ~ (rotate(v0) = v6) | ~ (ordered_pair(v4, v1) = v5) | ~ % 19.38/3.29 | (ordered_pair(v2, v3) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ % 19.38/3.29 | $i(v0) | ~ member(v5, v0) | ? [v7: $i] : ? [v8: $i] : % 19.38/3.29 | (ordered_pair(v7, v3) = v8 & ordered_pair(v1, v2) = v7 & $i(v8) & % 19.38/3.29 | $i(v7) & ( ~ member(v8, all_67_0) | member(v8, v6)))) & ! [v0: % 19.38/3.29 | $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! % 19.38/3.29 | [v5: $i] : ! [v6: $i] : ( ~ (rotate(v0) = v4) | ~ (ordered_pair(v5, % 19.38/3.29 | v1) = v6) | ~ (ordered_pair(v2, v3) = v5) | ~ $i(v3) | ~ % 19.38/3.29 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v7: $i] : ? [v8: $i] : % 19.38/3.29 | (ordered_pair(v7, v3) = v8 & ordered_pair(v1, v2) = v7 & $i(v8) & % 19.38/3.29 | $i(v7) & ( ~ member(v8, v4) | (member(v8, all_67_0) & member(v6, % 19.38/3.29 | v0))))) % 19.38/3.29 | % 19.38/3.29 | ALPHA: (41) implies: % 19.38/3.29 | (42) cross_product(universal_class, universal_class) = all_67_1 % 19.38/3.29 | % 19.38/3.29 | GROUND_INST: instantiating (15) with all_56_1, all_61_0, universal_class, % 19.38/3.29 | universal_class, simplifying with (26), (37) gives: % 19.38/3.29 | (43) all_61_0 = all_56_1 % 19.38/3.29 | % 19.38/3.29 | GROUND_INST: instantiating (15) with all_39_0, all_61_0, universal_class, % 19.38/3.29 | universal_class, simplifying with (19), (37) gives: % 19.38/3.29 | (44) all_61_0 = all_39_0 % 19.38/3.29 | % 19.38/3.29 | GROUND_INST: instantiating (15) with all_56_1, all_64_1, universal_class, % 19.38/3.29 | universal_class, simplifying with (26), (39) gives: % 19.38/3.29 | (45) all_64_1 = all_56_1 % 19.38/3.29 | % 19.38/3.29 | GROUND_INST: instantiating (15) with all_53_1, all_64_1, universal_class, % 19.38/3.29 | universal_class, simplifying with (23), (39) gives: % 19.38/3.29 | (46) all_64_1 = all_53_1 % 19.38/3.29 | % 19.38/3.29 | GROUND_INST: instantiating (15) with all_45_0, all_64_1, universal_class, % 19.38/3.29 | universal_class, simplifying with (21), (39) gives: % 19.38/3.29 | (47) all_64_1 = all_45_0 % 19.38/3.29 | % 19.38/3.29 | GROUND_INST: instantiating (15) with all_64_1, all_67_1, universal_class, % 19.38/3.29 | universal_class, simplifying with (39), (42) gives: % 19.38/3.29 | (48) all_67_1 = all_64_1 % 19.38/3.29 | % 19.38/3.29 | GROUND_INST: instantiating (15) with all_37_0, all_67_1, universal_class, % 19.38/3.29 | universal_class, simplifying with (17), (42) gives: % 19.38/3.29 | (49) all_67_1 = all_37_0 % 19.38/3.29 | % 19.38/3.29 | GROUND_INST: instantiating (15) with all_53_0, all_64_0, universal_class, % 19.38/3.29 | all_53_1, simplifying with (24) gives: % 19.38/3.29 | (50) all_64_0 = all_53_0 | ~ (cross_product(all_53_1, universal_class) = % 19.38/3.29 | all_64_0) % 19.38/3.29 | % 19.38/3.29 | COMBINE_EQS: (48), (49) imply: % 19.38/3.29 | (51) all_64_1 = all_37_0 % 19.38/3.29 | % 19.38/3.29 | SIMP: (51) implies: % 19.38/3.29 | (52) all_64_1 = all_37_0 % 19.38/3.29 | % 19.38/3.29 | COMBINE_EQS: (46), (47) imply: % 19.38/3.29 | (53) all_53_1 = all_45_0 % 19.38/3.29 | % 19.38/3.29 | COMBINE_EQS: (46), (52) imply: % 19.38/3.29 | (54) all_53_1 = all_37_0 % 19.38/3.29 | % 19.38/3.29 | COMBINE_EQS: (45), (46) imply: % 19.38/3.29 | (55) all_56_1 = all_53_1 % 19.38/3.29 | % 19.38/3.29 | SIMP: (55) implies: % 19.38/3.29 | (56) all_56_1 = all_53_1 % 19.38/3.29 | % 19.38/3.29 | COMBINE_EQS: (43), (44) imply: % 19.38/3.29 | (57) all_56_1 = all_39_0 % 19.38/3.29 | % 19.38/3.29 | SIMP: (57) implies: % 19.38/3.29 | (58) all_56_1 = all_39_0 % 19.38/3.29 | % 19.38/3.29 | COMBINE_EQS: (56), (58) imply: % 19.38/3.29 | (59) all_53_1 = all_39_0 % 19.38/3.29 | % 19.38/3.29 | SIMP: (59) implies: % 19.38/3.29 | (60) all_53_1 = all_39_0 % 19.38/3.29 | % 19.38/3.29 | COMBINE_EQS: (53), (54) imply: % 19.38/3.30 | (61) all_45_0 = all_37_0 % 19.38/3.30 | % 19.38/3.30 | COMBINE_EQS: (53), (60) imply: % 19.38/3.30 | (62) all_45_0 = all_39_0 % 19.38/3.30 | % 19.38/3.30 | COMBINE_EQS: (61), (62) imply: % 19.38/3.30 | (63) all_39_0 = all_37_0 % 19.38/3.30 | % 19.38/3.30 | REDUCE: (40), (52) imply: % 19.38/3.30 | (64) cross_product(all_37_0, universal_class) = all_64_0 % 19.38/3.30 | % 19.38/3.30 | BETA: splitting (50) gives: % 19.38/3.30 | % 19.38/3.30 | Case 1: % 19.38/3.30 | | % 19.38/3.30 | | (65) ~ (cross_product(all_53_1, universal_class) = all_64_0) % 19.38/3.30 | | % 19.38/3.30 | | REDUCE: (54), (65) imply: % 19.38/3.30 | | (66) ~ (cross_product(all_37_0, universal_class) = all_64_0) % 19.38/3.30 | | % 19.38/3.30 | | PRED_UNIFY: (64), (66) imply: % 19.38/3.30 | | (67) $false % 19.38/3.30 | | % 19.38/3.30 | | CLOSE: (67) is inconsistent. % 19.38/3.30 | | % 19.38/3.30 | Case 2: % 19.38/3.30 | | % 19.38/3.30 | | % 19.38/3.30 | | GROUND_INST: instantiating (2) with all_59_4, all_59_2, all_59_1, % 19.38/3.30 | | simplifying with (29), (31), (33) gives: % 19.38/3.30 | | (68) member(all_59_1, universal_class) % 19.38/3.30 | | % 19.38/3.30 | | GROUND_INST: instantiating (4) with all_59_4, all_59_3, all_59_2, all_59_1, % 19.38/3.30 | | simplifying with (29), (30), (33), (34) gives: % 19.38/3.30 | | (69) ? [v0: $i] : ? [v1: $i] : (ordered_pair(all_59_4, all_59_3) = v0 & % 19.38/3.30 | | singleton(all_59_4) = v1 & unordered_pair(v1, all_59_1) = v0 & % 19.38/3.30 | | $i(v1) & $i(v0)) % 19.38/3.30 | | % 19.38/3.30 | | GROUND_INST: instantiating (3) with all_59_4, all_59_3, all_59_0, % 19.38/3.30 | | simplifying with (29), (30), (35) gives: % 19.38/3.30 | | (70) ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : (singleton(all_59_3) = v1 % 19.38/3.30 | | & singleton(all_59_4) = v0 & unordered_pair(v0, v2) = all_59_0 & % 19.38/3.30 | | unordered_pair(all_59_4, v1) = v2 & $i(v2) & $i(v1) & $i(v0) & % 19.38/3.30 | | $i(all_59_0)) % 19.38/3.30 | | % 19.38/3.30 | | DELTA: instantiating (69) with fresh symbols all_98_0, all_98_1 gives: % 19.38/3.30 | | (71) ordered_pair(all_59_4, all_59_3) = all_98_1 & singleton(all_59_4) = % 19.38/3.30 | | all_98_0 & unordered_pair(all_98_0, all_59_1) = all_98_1 & % 19.38/3.30 | | $i(all_98_0) & $i(all_98_1) % 19.38/3.30 | | % 19.38/3.30 | | ALPHA: (71) implies: % 19.38/3.30 | | (72) unordered_pair(all_98_0, all_59_1) = all_98_1 % 19.38/3.30 | | (73) singleton(all_59_4) = all_98_0 % 19.38/3.30 | | (74) ordered_pair(all_59_4, all_59_3) = all_98_1 % 19.38/3.30 | | % 19.38/3.30 | | DELTA: instantiating (70) with fresh symbols all_100_0, all_100_1, all_100_2 % 19.38/3.30 | | gives: % 19.38/3.30 | | (75) singleton(all_59_3) = all_100_1 & singleton(all_59_4) = all_100_2 & % 19.38/3.30 | | unordered_pair(all_100_2, all_100_0) = all_59_0 & % 19.38/3.30 | | unordered_pair(all_59_4, all_100_1) = all_100_0 & $i(all_100_0) & % 19.38/3.30 | | $i(all_100_1) & $i(all_100_2) & $i(all_59_0) % 19.38/3.30 | | % 19.38/3.30 | | ALPHA: (75) implies: % 19.38/3.30 | | (76) $i(all_100_2) % 19.38/3.30 | | (77) singleton(all_59_4) = all_100_2 % 19.38/3.30 | | % 19.38/3.30 | | GROUND_INST: instantiating (13) with all_98_0, all_100_2, all_59_4, % 19.38/3.30 | | simplifying with (73), (77) gives: % 19.38/3.30 | | (78) all_100_2 = all_98_0 % 19.38/3.30 | | % 19.38/3.30 | | GROUND_INST: instantiating (14) with all_59_0, all_98_1, all_59_3, all_59_4, % 19.38/3.30 | | simplifying with (35), (74) gives: % 19.38/3.30 | | (79) all_98_1 = all_59_0 % 19.38/3.30 | | % 19.38/3.30 | | REDUCE: (72), (79) imply: % 19.38/3.30 | | (80) unordered_pair(all_98_0, all_59_1) = all_59_0 % 19.38/3.30 | | % 19.38/3.30 | | REDUCE: (76), (78) imply: % 19.38/3.30 | | (81) $i(all_98_0) % 19.38/3.30 | | % 19.38/3.30 | | GROUND_INST: instantiating (1) with all_59_1, all_98_0, all_59_0, % 19.38/3.30 | | simplifying with (28), (32), (68), (80), (81) gives: % 19.38/3.30 | | (82) $false % 19.38/3.30 | | % 19.38/3.30 | | CLOSE: (82) is inconsistent. % 19.38/3.30 | | % 19.38/3.30 | End of split % 19.38/3.30 | % 19.38/3.30 End of proof % 19.38/3.30 % SZS output end Proof for theBenchmark % 19.38/3.30 % 19.38/3.30 2680ms %------------------------------------------------------------------------------