%------------------------------------------------------------------------------ % File : Princess---230619 % Problem : SET084+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 : n018.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 15:23:40 EDT 2023 % Result : Theorem 63.93s 9.21s % Output : Proof 69.13s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.13 % Problem : SET084+1 : TPTP v8.1.2. Bugfixed v5.4.0. % 0.00/0.13 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s % 0.14/0.34 % Computer : n018.cluster.edu % 0.14/0.34 % Model : x86_64 x86_64 % 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.34 % Memory : 8042.1875MB % 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.34 % CPULimit : 300 % 0.14/0.34 % WCLimit : 300 % 0.14/0.34 % DateTime : Sat Aug 26 10:31:00 EDT 2023 % 0.14/0.34 % CPUTime : % 0.19/0.61 ________ _____ % 0.19/0.61 ___ __ \_________(_)________________________________ % 0.19/0.61 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/ % 0.19/0.61 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ ) % 0.19/0.61 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/ % 0.19/0.61 % 0.19/0.61 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic % 0.19/0.61 (2023-06-19) % 0.19/0.61 % 0.19/0.61 (c) Philipp Rümmer, 2009-2023 % 0.19/0.61 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen, % 0.19/0.61 Amanda Stjerna. % 0.19/0.61 Free software under BSD-3-Clause. % 0.19/0.61 % 0.19/0.61 For more information, visit http://www.philipp.ruemmer.org/princess.shtml % 0.19/0.61 % 0.19/0.61 Loading /export/starexec/sandbox2/benchmark/theBenchmark.p ... % 0.19/0.63 Running up to 7 provers in parallel. % 0.19/0.65 Prover 2: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMinimalAndEmpty -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1065072994 % 0.19/0.65 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1571432423 % 0.19/0.65 Prover 0: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1042961893 % 0.19/0.65 Prover 3: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1922548996 % 0.19/0.65 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=1868514696 % 0.19/0.65 Prover 5: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=complete -randomSeed=1259561288 % 0.19/0.65 Prover 6: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximalOutermost -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1399714365 % 2.88/1.14 Prover 4: Preprocessing ... % 2.88/1.15 Prover 1: Preprocessing ... % 3.51/1.18 Prover 6: Preprocessing ... % 3.51/1.18 Prover 3: Preprocessing ... % 3.51/1.18 Prover 2: Preprocessing ... % 3.51/1.18 Prover 5: Preprocessing ... % 3.51/1.19 Prover 0: Preprocessing ... % 8.54/1.95 Prover 1: Warning: ignoring some quantifiers % 9.32/1.99 Prover 5: Proving ... % 9.44/2.01 Prover 3: Warning: ignoring some quantifiers % 9.44/2.01 Prover 6: Proving ... % 9.44/2.02 Prover 1: Constructing countermodel ... % 9.44/2.04 Prover 3: Constructing countermodel ... % 9.44/2.07 Prover 4: Warning: ignoring some quantifiers % 10.18/2.16 Prover 4: Constructing countermodel ... % 10.18/2.18 Prover 2: Proving ... % 10.18/2.18 Prover 0: Proving ... % 63.93/9.21 Prover 5: proved (8565ms) % 63.93/9.21 % 63.93/9.21 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p % 63.93/9.21 % 63.93/9.21 Prover 6: stopped % 63.93/9.21 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470 % 63.93/9.22 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089 % 63.93/9.23 Prover 3: stopped % 63.93/9.23 Prover 10: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=919308125 % 63.93/9.23 Prover 2: stopped % 63.93/9.24 Prover 11: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1509710984 % 63.93/9.27 Prover 0: stopped % 64.68/9.27 Prover 13: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=1138197443 % 64.83/9.31 Prover 7: Preprocessing ... % 65.02/9.33 Prover 8: Preprocessing ... % 65.02/9.35 Prover 11: Preprocessing ... % 65.02/9.36 Prover 13: Preprocessing ... % 65.02/9.37 Prover 10: Preprocessing ... % 65.63/9.45 Prover 7: Warning: ignoring some quantifiers % 66.24/9.48 Prover 7: Constructing countermodel ... % 66.24/9.49 Prover 10: Warning: ignoring some quantifiers % 66.24/9.54 Prover 10: Constructing countermodel ... % 66.24/9.55 Prover 8: Warning: ignoring some quantifiers % 66.24/9.56 Prover 8: Constructing countermodel ... % 67.00/9.59 Prover 11: Warning: ignoring some quantifiers % 67.00/9.60 Prover 11: Constructing countermodel ... % 67.00/9.61 Prover 13: Warning: ignoring some quantifiers % 67.00/9.62 Prover 13: Constructing countermodel ... % 67.00/9.72 Prover 10: gave up % 67.00/9.72 Prover 16: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=completeFrugal -randomSeed=-2043353683 % 67.30/9.77 Prover 7: Found proof (size 74) % 67.30/9.77 Prover 7: proved (555ms) % 67.30/9.77 Prover 11: stopped % 67.30/9.77 Prover 13: stopped % 67.30/9.77 Prover 8: stopped % 67.30/9.77 Prover 4: stopped % 67.30/9.77 Prover 16: Preprocessing ... % 67.30/9.77 Prover 1: stopped % 68.56/9.81 Prover 16: stopped % 68.63/9.81 % 68.63/9.81 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p % 68.63/9.81 % 68.63/9.83 % SZS output start Proof for theBenchmark % 68.63/9.84 Assumptions after simplification: % 68.63/9.84 --------------------------------- % 68.63/9.84 % 68.63/9.84 (compose_defn1) % 68.63/9.86 $i(universal_class) & ? [v0: $i] : (cross_product(universal_class, % 68.63/9.86 universal_class) = v0 & $i(v0) & ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : % 68.63/9.86 ( ~ (compose(v2, v1) = v3) | ~ $i(v2) | ~ $i(v1) | subclass(v3, v0))) % 68.63/9.86 % 68.63/9.86 (element_relation) % 68.63/9.86 $i(element_relation) & $i(universal_class) & ? [v0: $i] : % 68.63/9.86 (cross_product(universal_class, universal_class) = v0 & $i(v0) & % 68.63/9.86 subclass(element_relation, v0)) % 68.63/9.86 % 68.63/9.86 (flip) % 68.63/9.86 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0, % 68.63/9.86 universal_class) = v1 & cross_product(universal_class, universal_class) = % 68.63/9.86 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (flip(v2) = v3) | ~ % 68.63/9.87 $i(v2) | subclass(v3, v1))) % 68.63/9.87 % 68.63/9.87 (flip_defn) % 68.63/9.87 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0, % 68.63/9.87 universal_class) = v1 & cross_product(universal_class, universal_class) = % 68.63/9.87 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] % 68.63/9.87 : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~ % 68.63/9.87 (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v3, v2) = v6) | ~ $i(v5) | % 68.63/9.87 ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v5) | ? [v9: $i] : ? % 68.63/9.87 [v10: $i] : (ordered_pair(v9, v4) = v10 & ordered_pair(v2, v3) = v9 & % 68.63/9.87 $i(v10) & $i(v9) & ( ~ member(v10, v1) | member(v10, v8)))) & ! [v2: % 68.63/9.87 $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] % 68.63/9.87 : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ % 68.63/9.87 (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ % 68.63/9.87 $i(v2) | ~ member(v7, v8) | member(v7, v1)) & ! [v2: $i] : ! [v3: $i] : % 68.63/9.87 ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ % 68.63/9.87 (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v2, % 68.63/9.87 v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ % 68.63/9.87 member(v7, v8) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v4) = v10 % 68.63/9.87 & ordered_pair(v3, v2) = v9 & $i(v10) & $i(v9) & member(v10, v5))) & ! % 68.63/9.88 [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: % 68.63/9.88 $i] : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | % 68.63/9.88 ~ (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ % 68.63/9.88 $i(v2) | ~ member(v7, v1) | member(v7, v8) | ? [v9: $i] : ? [v10: $i] : % 68.63/9.88 (ordered_pair(v9, v4) = v10 & ordered_pair(v3, v2) = v9 & $i(v10) & $i(v9) % 68.63/9.88 & ~ member(v10, v5))) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! % 68.63/9.88 [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (flip(v5) = v6) | % 68.63/9.88 ~ (ordered_pair(v7, v4) = v8) | ~ (ordered_pair(v3, v2) = v7) | ~ $i(v5) % 68.63/9.88 | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] : % 68.63/9.88 (ordered_pair(v9, v4) = v10 & ordered_pair(v2, v3) = v9 & $i(v10) & $i(v9) % 68.63/9.88 & ( ~ member(v10, v6) | (member(v10, v1) & member(v8, v5)))))) % 68.63/9.88 % 68.63/9.88 (function_defn) % 68.63/9.88 $i(identity_relation) & $i(universal_class) & ? [v0: $i] : % 68.63/9.88 (cross_product(universal_class, universal_class) = v0 & $i(v0) & ! [v1: $i] : % 68.63/9.88 ! [v2: $i] : ( ~ (inverse(v1) = v2) | ~ $i(v1) | ~ function(v1) | % 68.63/9.88 subclass(v1, v0)) & ! [v1: $i] : ! [v2: $i] : ( ~ (inverse(v1) = v2) | % 68.63/9.88 ~ $i(v1) | ~ function(v1) | ? [v3: $i] : (compose(v1, v2) = v3 & $i(v3) % 68.63/9.88 & subclass(v3, identity_relation))) & ! [v1: $i] : ! [v2: $i] : ( ~ % 68.63/9.88 (inverse(v1) = v2) | ~ $i(v1) | ~ subclass(v1, v0) | function(v1) | ? % 68.63/9.88 [v3: $i] : (compose(v1, v2) = v3 & $i(v3) & ~ subclass(v3, % 68.63/9.88 identity_relation)))) % 68.63/9.88 % 68.63/9.88 (rotate) % 68.63/9.88 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0, % 68.63/9.88 universal_class) = v1 & cross_product(universal_class, universal_class) = % 68.63/9.88 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (rotate(v2) = v3) | % 68.63/9.88 ~ $i(v2) | subclass(v3, v1))) % 68.63/9.88 % 68.63/9.88 (rotate_defn) % 68.63/9.88 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0, % 68.63/9.89 universal_class) = v1 & cross_product(universal_class, universal_class) = % 68.63/9.89 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] % 68.63/9.89 : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~ % 68.63/9.89 (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = v6) | ~ $i(v5) | % 68.63/9.89 ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v8) | member(v7, v1)) & % 68.63/9.89 ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: % 68.63/9.89 $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~ (ordered_pair(v6, v5) = v7) % 68.63/9.89 | ~ (ordered_pair(v3, v4) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ % 68.63/9.89 $i(v2) | ~ member(v7, v8) | ? [v9: $i] : ? [v10: $i] : % 68.63/9.89 (ordered_pair(v9, v3) = v10 & ordered_pair(v4, v5) = v9 & $i(v10) & $i(v9) % 68.63/9.89 & member(v10, v2))) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: % 68.63/9.89 $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~ % 68.63/9.89 (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = v6) | ~ $i(v5) | % 68.63/9.89 ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v1) | member(v7, v8) | % 68.63/9.89 ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v3) = v10 & ordered_pair(v4, % 68.63/9.89 v5) = v9 & $i(v10) & $i(v9) & ~ member(v10, v2))) & ! [v2: $i] : ! % 68.63/9.89 [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: % 68.63/9.89 $i] : ( ~ (rotate(v2) = v8) | ~ (ordered_pair(v6, v3) = v7) | ~ % 68.63/9.89 (ordered_pair(v4, v5) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ % 68.63/9.89 $i(v2) | ~ member(v7, v2) | ? [v9: $i] : ? [v10: $i] : % 68.63/9.89 (ordered_pair(v9, v5) = v10 & ordered_pair(v3, v4) = v9 & $i(v10) & $i(v9) % 68.63/9.89 & ( ~ member(v10, v1) | member(v10, v8)))) & ! [v2: $i] : ! [v3: $i] : % 68.63/9.89 ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ % 68.63/9.89 (rotate(v2) = v6) | ~ (ordered_pair(v7, v3) = v8) | ~ (ordered_pair(v4, % 68.63/9.89 v5) = v7) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ? [v9: % 68.63/9.89 $i] : ? [v10: $i] : (ordered_pair(v9, v5) = v10 & ordered_pair(v3, v4) % 68.63/9.89 = v9 & $i(v10) & $i(v9) & ( ~ member(v10, v6) | (member(v10, v1) & % 68.63/9.89 member(v8, v2)))))) % 68.63/9.89 % 68.63/9.89 (singleton_identified_by_element2) % 68.63/9.89 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ( ~ (v1 = v0) % 68.63/9.89 & singleton(v1) = v2 & singleton(v0) = v2 & $i(v2) & $i(v1) & $i(v0) & % 68.63/9.89 member(v1, universal_class)) % 68.63/9.89 % 68.63/9.89 (singleton_set_defn) % 68.63/9.89 ! [v0: $i] : ! [v1: $i] : ( ~ (singleton(v0) = v1) | ~ $i(v0) | % 68.63/9.89 (unordered_pair(v0, v0) = v1 & $i(v1))) & ! [v0: $i] : ! [v1: $i] : ( ~ % 68.63/9.89 (unordered_pair(v0, v0) = v1) | ~ $i(v0) | (singleton(v0) = v1 & $i(v1))) % 68.63/9.89 % 68.63/9.89 (successor_relation_defn1) % 68.63/9.89 $i(successor_relation) & $i(universal_class) & ? [v0: $i] : % 68.63/9.89 (cross_product(universal_class, universal_class) = v0 & $i(v0) & % 68.63/9.89 subclass(successor_relation, v0)) % 68.63/9.89 % 68.63/9.89 (unordered_pair_defn) % 68.63/9.89 $i(universal_class) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : % 68.63/9.89 (v2 = v0 | v1 = v0 | ~ (unordered_pair(v1, v2) = v3) | ~ $i(v2) | ~ $i(v1) % 68.63/9.89 | ~ $i(v0) | ~ member(v0, v3)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : % 68.63/9.89 ! [v3: $i] : ( ~ (unordered_pair(v1, v2) = v3) | ~ $i(v2) | ~ $i(v1) | ~ % 68.63/9.89 $i(v0) | ~ member(v0, v3) | member(v0, universal_class)) & ! [v0: $i] : ! % 68.63/9.89 [v1: $i] : ! [v2: $i] : ( ~ (unordered_pair(v1, v0) = v2) | ~ $i(v1) | ~ % 68.63/9.89 $i(v0) | ~ member(v0, universal_class) | member(v0, v2)) & ! [v0: $i] : ! % 68.63/9.89 [v1: $i] : ! [v2: $i] : ( ~ (unordered_pair(v0, v1) = v2) | ~ $i(v1) | ~ % 68.63/9.89 $i(v0) | ~ member(v0, universal_class) | member(v0, v2)) % 68.63/9.89 % 68.63/9.89 (function-axioms) % 68.63/9.90 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : (v1 = v0 % 68.63/9.90 | ~ (restrict(v4, v3, v2) = v1) | ~ (restrict(v4, v3, v2) = v0)) & ! [v0: % 68.63/9.90 $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (apply(v3, v2) % 68.63/9.90 = v1) | ~ (apply(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] % 68.63/9.90 : ! [v3: $i] : (v1 = v0 | ~ (compose(v3, v2) = v1) | ~ (compose(v3, v2) = % 68.63/9.90 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | % 68.63/9.90 ~ (image(v3, v2) = v1) | ~ (image(v3, v2) = v0)) & ! [v0: $i] : ! [v1: % 68.63/9.90 $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (union(v3, v2) = v1) | ~ % 68.63/9.90 (union(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: % 68.63/9.90 $i] : (v1 = v0 | ~ (intersection(v3, v2) = v1) | ~ (intersection(v3, v2) = % 68.63/9.90 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | % 68.63/9.90 ~ (cross_product(v3, v2) = v1) | ~ (cross_product(v3, v2) = v0)) & ! [v0: % 68.63/9.90 $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ % 68.63/9.90 (ordered_pair(v3, v2) = v1) | ~ (ordered_pair(v3, v2) = v0)) & ! [v0: $i] % 68.63/9.90 : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (unordered_pair(v3, % 68.63/9.90 v2) = v1) | ~ (unordered_pair(v3, v2) = v0)) & ! [v0: $i] : ! [v1: % 68.63/9.90 $i] : ! [v2: $i] : (v1 = v0 | ~ (power_class(v2) = v1) | ~ % 68.63/9.90 (power_class(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 % 68.63/9.90 | ~ (sum_class(v2) = v1) | ~ (sum_class(v2) = v0)) & ! [v0: $i] : ! [v1: % 68.63/9.90 $i] : ! [v2: $i] : (v1 = v0 | ~ (range_of(v2) = v1) | ~ (range_of(v2) = % 68.63/9.90 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ % 68.63/9.90 (inverse(v2) = v1) | ~ (inverse(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! % 68.63/9.90 [v2: $i] : (v1 = v0 | ~ (successor(v2) = v1) | ~ (successor(v2) = v0)) & ! % 68.63/9.90 [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (flip(v2) = v1) | ~ % 68.63/9.90 (flip(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ % 68.63/9.90 (rotate(v2) = v1) | ~ (rotate(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! % 68.63/9.90 [v2: $i] : (v1 = v0 | ~ (domain_of(v2) = v1) | ~ (domain_of(v2) = v0)) & ! % 68.63/9.90 [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (complement(v2) = v1) | % 68.63/9.90 ~ (complement(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = % 68.63/9.90 v0 | ~ (first(v2) = v1) | ~ (first(v2) = v0)) & ! [v0: $i] : ! [v1: $i] % 68.63/9.90 : ! [v2: $i] : (v1 = v0 | ~ (second(v2) = v1) | ~ (second(v2) = v0)) & ! % 68.63/9.90 [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (singleton(v2) = v1) | ~ % 68.63/9.90 (singleton(v2) = v0)) % 68.63/9.90 % 68.63/9.90 Further assumptions not needed in the proof: % 68.63/9.90 -------------------------------------------- % 68.63/9.90 apply_defn, choice, class_elements_are_sets, complement, compose_defn2, % 68.63/9.90 cross_product, cross_product_defn, disjoint_defn, domain_of, % 68.63/9.90 element_relation_defn, extensionality, first_second, identity_relation, % 68.63/9.90 image_defn, inductive_defn, infinity, intersection, inverse_defn, % 68.63/9.90 null_class_defn, ordered_pair_defn, power_class, power_class_defn, % 68.63/9.90 range_of_defn, regularity, replacement, restrict_defn, subclass_defn, % 68.63/9.90 successor_defn, successor_relation_defn2, sum_class, sum_class_defn, union_defn, % 68.63/9.90 unordered_pair % 68.63/9.90 % 68.63/9.90 Those formulas are unsatisfiable: % 68.63/9.90 --------------------------------- % 68.63/9.90 % 68.63/9.90 Begin of proof % 68.63/9.90 | % 68.63/9.90 | ALPHA: (unordered_pair_defn) implies: % 68.63/9.90 | (1) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (unordered_pair(v1, v0) = % 68.63/9.90 | v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v0, universal_class) | % 68.63/9.90 | member(v0, v2)) % 68.63/9.90 | (2) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v2 = v0 | v1 = % 68.63/9.90 | v0 | ~ (unordered_pair(v1, v2) = v3) | ~ $i(v2) | ~ $i(v1) | ~ % 68.63/9.90 | $i(v0) | ~ member(v0, v3)) % 68.63/9.90 | % 68.63/9.90 | ALPHA: (singleton_set_defn) implies: % 68.63/9.90 | (3) ! [v0: $i] : ! [v1: $i] : ( ~ (singleton(v0) = v1) | ~ $i(v0) | % 68.63/9.90 | (unordered_pair(v0, v0) = v1 & $i(v1))) % 68.63/9.90 | % 68.63/9.90 | ALPHA: (element_relation) implies: % 68.63/9.90 | (4) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 & % 68.63/9.90 | $i(v0) & subclass(element_relation, v0)) % 68.63/9.90 | % 68.63/9.90 | ALPHA: (rotate_defn) implies: % 68.63/9.91 | (5) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 & % 68.63/9.91 | cross_product(universal_class, universal_class) = v0 & $i(v1) & % 68.63/9.91 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! % 68.63/9.91 | [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~ % 68.63/9.91 | (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = v6) | ~ % 68.63/9.91 | $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v8) | % 68.63/9.91 | member(v7, v1)) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: % 68.63/9.91 | $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = % 68.63/9.91 | v8) | ~ (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = % 68.63/9.91 | v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ % 68.63/9.91 | member(v7, v8) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v3) % 68.63/9.91 | = v10 & ordered_pair(v4, v5) = v9 & $i(v10) & $i(v9) & % 68.63/9.91 | member(v10, v2))) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! % 68.63/9.91 | [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) % 68.63/9.91 | = v8) | ~ (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) % 68.63/9.91 | = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ % 68.63/9.91 | member(v7, v1) | member(v7, v8) | ? [v9: $i] : ? [v10: $i] : % 68.63/9.91 | (ordered_pair(v9, v3) = v10 & ordered_pair(v4, v5) = v9 & $i(v10) & % 68.63/9.91 | $i(v9) & ~ member(v10, v2))) & ! [v2: $i] : ! [v3: $i] : ! % 68.63/9.91 | [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( % 68.63/9.91 | ~ (rotate(v2) = v8) | ~ (ordered_pair(v6, v3) = v7) | ~ % 68.63/9.91 | (ordered_pair(v4, v5) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 68.63/9.91 | ~ $i(v2) | ~ member(v7, v2) | ? [v9: $i] : ? [v10: $i] : % 68.63/9.91 | (ordered_pair(v9, v5) = v10 & ordered_pair(v3, v4) = v9 & $i(v10) & % 68.63/9.91 | $i(v9) & ( ~ member(v10, v1) | member(v10, v8)))) & ! [v2: $i] : % 68.63/9.91 | ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : % 68.63/9.91 | ! [v8: $i] : ( ~ (rotate(v2) = v6) | ~ (ordered_pair(v7, v3) = v8) % 68.63/9.91 | | ~ (ordered_pair(v4, v5) = v7) | ~ $i(v5) | ~ $i(v4) | ~ % 68.63/9.91 | $i(v3) | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, % 68.63/9.91 | v5) = v10 & ordered_pair(v3, v4) = v9 & $i(v10) & $i(v9) & ( ~ % 68.63/9.91 | member(v10, v6) | (member(v10, v1) & member(v8, v2)))))) % 68.63/9.91 | % 68.63/9.91 | ALPHA: (rotate) implies: % 68.63/9.91 | (6) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 & % 68.63/9.91 | cross_product(universal_class, universal_class) = v0 & $i(v1) & % 68.63/9.91 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (rotate(v2) = v3) | ~ % 68.63/9.91 | $i(v2) | subclass(v3, v1))) % 68.63/9.91 | % 68.63/9.91 | ALPHA: (flip_defn) implies: % 68.63/9.91 | (7) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 & % 68.63/9.91 | cross_product(universal_class, universal_class) = v0 & $i(v1) & % 68.63/9.91 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! % 68.63/9.91 | [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~ % 68.63/9.91 | (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v3, v2) = v6) | ~ % 68.63/9.91 | $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v5) | ? % 68.63/9.91 | [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v4) = v10 & % 68.63/9.91 | ordered_pair(v2, v3) = v9 & $i(v10) & $i(v9) & ( ~ member(v10, % 68.63/9.91 | v1) | member(v10, v8)))) & ! [v2: $i] : ! [v3: $i] : ! % 68.63/9.91 | [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( % 68.63/9.91 | ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ % 68.63/9.91 | (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 68.63/9.91 | ~ $i(v2) | ~ member(v7, v8) | member(v7, v1)) & ! [v2: $i] : ! % 68.63/9.91 | [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! % 68.63/9.91 | [v8: $i] : ( ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ % 68.63/9.91 | (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 68.63/9.91 | ~ $i(v2) | ~ member(v7, v8) | ? [v9: $i] : ? [v10: $i] : % 68.63/9.91 | (ordered_pair(v9, v4) = v10 & ordered_pair(v3, v2) = v9 & $i(v10) & % 69.13/9.91 | $i(v9) & member(v10, v5))) & ! [v2: $i] : ! [v3: $i] : ! [v4: % 69.13/9.91 | $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ % 69.13/9.91 | (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ % 69.13/9.91 | (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 69.13/9.91 | ~ $i(v2) | ~ member(v7, v1) | member(v7, v8) | ? [v9: $i] : ? % 69.13/9.91 | [v10: $i] : (ordered_pair(v9, v4) = v10 & ordered_pair(v3, v2) = v9 % 69.13/9.91 | & $i(v10) & $i(v9) & ~ member(v10, v5))) & ! [v2: $i] : ! [v3: % 69.13/9.91 | $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! % 69.13/9.91 | [v8: $i] : ( ~ (flip(v5) = v6) | ~ (ordered_pair(v7, v4) = v8) | ~ % 69.13/9.91 | (ordered_pair(v3, v2) = v7) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | % 69.13/9.91 | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v4) = v10 % 69.13/9.91 | & ordered_pair(v2, v3) = v9 & $i(v10) & $i(v9) & ( ~ member(v10, % 69.13/9.91 | v6) | (member(v10, v1) & member(v8, v5)))))) % 69.13/9.91 | % 69.13/9.91 | ALPHA: (flip) implies: % 69.13/9.91 | (8) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 & % 69.13/9.91 | cross_product(universal_class, universal_class) = v0 & $i(v1) & % 69.13/9.91 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (flip(v2) = v3) | ~ $i(v2) % 69.13/9.91 | | subclass(v3, v1))) % 69.13/9.91 | % 69.13/9.91 | ALPHA: (successor_relation_defn1) implies: % 69.13/9.91 | (9) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 & % 69.13/9.91 | $i(v0) & subclass(successor_relation, v0)) % 69.13/9.91 | % 69.13/9.91 | ALPHA: (compose_defn1) implies: % 69.13/9.91 | (10) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 & % 69.13/9.91 | $i(v0) & ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~ (compose(v2, % 69.13/9.91 | v1) = v3) | ~ $i(v2) | ~ $i(v1) | subclass(v3, v0))) % 69.13/9.91 | % 69.13/9.91 | ALPHA: (function_defn) implies: % 69.13/9.91 | (11) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 & % 69.13/9.91 | $i(v0) & ! [v1: $i] : ! [v2: $i] : ( ~ (inverse(v1) = v2) | ~ % 69.13/9.91 | $i(v1) | ~ function(v1) | subclass(v1, v0)) & ! [v1: $i] : ! % 69.13/9.92 | [v2: $i] : ( ~ (inverse(v1) = v2) | ~ $i(v1) | ~ function(v1) | ? % 69.13/9.92 | [v3: $i] : (compose(v1, v2) = v3 & $i(v3) & subclass(v3, % 69.13/9.92 | identity_relation))) & ! [v1: $i] : ! [v2: $i] : ( ~ % 69.13/9.92 | (inverse(v1) = v2) | ~ $i(v1) | ~ subclass(v1, v0) | % 69.13/9.92 | function(v1) | ? [v3: $i] : (compose(v1, v2) = v3 & $i(v3) & ~ % 69.13/9.92 | subclass(v3, identity_relation)))) % 69.13/9.92 | % 69.13/9.92 | ALPHA: (singleton_identified_by_element2) implies: % 69.13/9.92 | (12) ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ( ~ (v1 = v0) & % 69.13/9.92 | singleton(v1) = v2 & singleton(v0) = v2 & $i(v2) & $i(v1) & $i(v0) & % 69.13/9.92 | member(v1, universal_class)) % 69.13/9.92 | % 69.13/9.92 | ALPHA: (function-axioms) implies: % 69.13/9.92 | (13) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ % 69.13/9.92 | (cross_product(v3, v2) = v1) | ~ (cross_product(v3, v2) = v0)) % 69.13/9.92 | % 69.13/9.92 | DELTA: instantiating (4) with fresh symbol all_37_0 gives: % 69.13/9.92 | (14) cross_product(universal_class, universal_class) = all_37_0 & % 69.13/9.92 | $i(all_37_0) & subclass(element_relation, all_37_0) % 69.13/9.92 | % 69.13/9.92 | ALPHA: (14) implies: % 69.13/9.92 | (15) cross_product(universal_class, universal_class) = all_37_0 % 69.13/9.92 | % 69.13/9.92 | DELTA: instantiating (9) with fresh symbol all_39_0 gives: % 69.13/9.92 | (16) cross_product(universal_class, universal_class) = all_39_0 & % 69.13/9.92 | $i(all_39_0) & subclass(successor_relation, all_39_0) % 69.13/9.92 | % 69.13/9.92 | ALPHA: (16) implies: % 69.13/9.92 | (17) cross_product(universal_class, universal_class) = all_39_0 % 69.13/9.92 | % 69.13/9.92 | DELTA: instantiating (10) with fresh symbol all_41_0 gives: % 69.13/9.92 | (18) cross_product(universal_class, universal_class) = all_41_0 & % 69.13/9.92 | $i(all_41_0) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ % 69.13/9.92 | (compose(v1, v0) = v2) | ~ $i(v1) | ~ $i(v0) | subclass(v2, % 69.13/9.92 | all_41_0)) % 69.13/9.92 | % 69.13/9.92 | ALPHA: (18) implies: % 69.13/9.92 | (19) cross_product(universal_class, universal_class) = all_41_0 % 69.13/9.92 | % 69.13/9.92 | DELTA: instantiating (12) with fresh symbols all_53_0, all_53_1, all_53_2 % 69.13/9.92 | gives: % 69.13/9.92 | (20) ~ (all_53_1 = all_53_2) & singleton(all_53_1) = all_53_0 & % 69.13/9.92 | singleton(all_53_2) = all_53_0 & $i(all_53_0) & $i(all_53_1) & % 69.13/9.92 | $i(all_53_2) & member(all_53_1, universal_class) % 69.13/9.92 | % 69.13/9.92 | ALPHA: (20) implies: % 69.13/9.92 | (21) ~ (all_53_1 = all_53_2) % 69.13/9.92 | (22) member(all_53_1, universal_class) % 69.13/9.92 | (23) $i(all_53_2) % 69.13/9.92 | (24) $i(all_53_1) % 69.13/9.92 | (25) singleton(all_53_2) = all_53_0 % 69.13/9.92 | (26) singleton(all_53_1) = all_53_0 % 69.13/9.92 | % 69.13/9.92 | DELTA: instantiating (6) with fresh symbols all_55_0, all_55_1 gives: % 69.13/9.92 | (27) cross_product(all_55_1, universal_class) = all_55_0 & % 69.13/9.92 | cross_product(universal_class, universal_class) = all_55_1 & % 69.13/9.92 | $i(all_55_0) & $i(all_55_1) & ! [v0: $i] : ! [v1: $i] : ( ~ % 69.13/9.92 | (rotate(v0) = v1) | ~ $i(v0) | subclass(v1, all_55_0)) % 69.13/9.92 | % 69.13/9.92 | ALPHA: (27) implies: % 69.13/9.92 | (28) cross_product(universal_class, universal_class) = all_55_1 % 69.13/9.92 | (29) cross_product(all_55_1, universal_class) = all_55_0 % 69.13/9.92 | % 69.13/9.92 | DELTA: instantiating (8) with fresh symbols all_58_0, all_58_1 gives: % 69.13/9.92 | (30) cross_product(all_58_1, universal_class) = all_58_0 & % 69.13/9.92 | cross_product(universal_class, universal_class) = all_58_1 & % 69.13/9.92 | $i(all_58_0) & $i(all_58_1) & ! [v0: $i] : ! [v1: $i] : ( ~ % 69.13/9.92 | (flip(v0) = v1) | ~ $i(v0) | subclass(v1, all_58_0)) % 69.13/9.92 | % 69.13/9.92 | ALPHA: (30) implies: % 69.13/9.92 | (31) cross_product(universal_class, universal_class) = all_58_1 % 69.13/9.92 | % 69.13/9.92 | DELTA: instantiating (11) with fresh symbol all_61_0 gives: % 69.13/9.92 | (32) cross_product(universal_class, universal_class) = all_61_0 & % 69.13/9.92 | $i(all_61_0) & ! [v0: $i] : ! [v1: $i] : ( ~ (inverse(v0) = v1) | ~ % 69.13/9.92 | $i(v0) | ~ function(v0) | subclass(v0, all_61_0)) & ! [v0: $i] : % 69.13/9.92 | ! [v1: $i] : ( ~ (inverse(v0) = v1) | ~ $i(v0) | ~ function(v0) | ? % 69.13/9.92 | [v2: $i] : (compose(v0, v1) = v2 & $i(v2) & subclass(v2, % 69.13/9.92 | identity_relation))) & ! [v0: $i] : ! [v1: $i] : ( ~ % 69.13/9.92 | (inverse(v0) = v1) | ~ $i(v0) | ~ subclass(v0, all_61_0) | % 69.13/9.92 | function(v0) | ? [v2: $i] : (compose(v0, v1) = v2 & $i(v2) & ~ % 69.13/9.92 | subclass(v2, identity_relation))) % 69.13/9.92 | % 69.13/9.92 | ALPHA: (32) implies: % 69.13/9.92 | (33) cross_product(universal_class, universal_class) = all_61_0 % 69.13/9.92 | % 69.13/9.92 | DELTA: instantiating (7) with fresh symbols all_64_0, all_64_1 gives: % 69.13/9.93 | (34) cross_product(all_64_1, universal_class) = all_64_0 & % 69.13/9.93 | cross_product(universal_class, universal_class) = all_64_1 & % 69.13/9.93 | $i(all_64_0) & $i(all_64_1) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] % 69.13/9.93 | : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ % 69.13/9.93 | (flip(v3) = v6) | ~ (ordered_pair(v4, v2) = v5) | ~ % 69.13/9.93 | (ordered_pair(v1, v0) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ % 69.13/9.93 | $i(v0) | ~ member(v5, v3) | ? [v7: $i] : ? [v8: $i] : % 69.13/9.93 | (ordered_pair(v7, v2) = v8 & ordered_pair(v0, v1) = v7 & $i(v8) & % 69.13/9.93 | $i(v7) & ( ~ member(v8, all_64_0) | member(v8, v6)))) & ! [v0: % 69.13/9.93 | $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! % 69.13/9.93 | [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v6) | ~ (ordered_pair(v4, % 69.13/9.93 | v2) = v5) | ~ (ordered_pair(v0, v1) = v4) | ~ $i(v3) | ~ % 69.13/9.93 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v5, v6) | member(v5, % 69.13/9.93 | all_64_0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] % 69.13/9.93 | : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v6) | ~ % 69.13/9.93 | (ordered_pair(v4, v2) = v5) | ~ (ordered_pair(v0, v1) = v4) | ~ % 69.13/9.93 | $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v5, v6) | ? % 69.13/9.93 | [v7: $i] : ? [v8: $i] : (ordered_pair(v7, v2) = v8 & % 69.13/9.93 | ordered_pair(v1, v0) = v7 & $i(v8) & $i(v7) & member(v8, v3))) & % 69.13/9.93 | ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : % 69.13/9.93 | ! [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v6) | ~ (ordered_pair(v4, % 69.13/9.93 | v2) = v5) | ~ (ordered_pair(v0, v1) = v4) | ~ $i(v3) | ~ % 69.13/9.93 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v5, all_64_0) | % 69.13/9.93 | member(v5, v6) | ? [v7: $i] : ? [v8: $i] : (ordered_pair(v7, v2) = % 69.13/9.93 | v8 & ordered_pair(v1, v0) = v7 & $i(v8) & $i(v7) & ~ member(v8, % 69.13/9.93 | v3))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : % 69.13/9.93 | ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v4) | ~ % 69.13/9.93 | (ordered_pair(v5, v2) = v6) | ~ (ordered_pair(v1, v0) = v5) | ~ % 69.13/9.93 | $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v7: $i] : ? [v8: % 69.13/9.93 | $i] : (ordered_pair(v7, v2) = v8 & ordered_pair(v0, v1) = v7 & % 69.13/9.93 | $i(v8) & $i(v7) & ( ~ member(v8, v4) | (member(v8, all_64_0) & % 69.13/9.93 | member(v6, v3))))) % 69.13/9.93 | % 69.13/9.93 | ALPHA: (34) implies: % 69.13/9.93 | (35) cross_product(universal_class, universal_class) = all_64_1 % 69.13/9.93 | (36) cross_product(all_64_1, universal_class) = all_64_0 % 69.13/9.93 | % 69.13/9.93 | DELTA: instantiating (5) with fresh symbols all_67_0, all_67_1 gives: % 69.13/9.93 | (37) cross_product(all_67_1, universal_class) = all_67_0 & % 69.13/9.93 | cross_product(universal_class, universal_class) = all_67_1 & % 69.13/9.93 | $i(all_67_0) & $i(all_67_1) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] % 69.13/9.93 | : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ % 69.13/9.93 | (rotate(v0) = v6) | ~ (ordered_pair(v4, v3) = v5) | ~ % 69.13/9.93 | (ordered_pair(v1, v2) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ % 69.13/9.93 | $i(v0) | ~ member(v5, v6) | member(v5, all_67_0)) & ! [v0: $i] : % 69.13/9.93 | ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : % 69.13/9.93 | ! [v6: $i] : ( ~ (rotate(v0) = v6) | ~ (ordered_pair(v4, v3) = v5) | % 69.13/9.93 | ~ (ordered_pair(v1, v2) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | % 69.13/9.93 | ~ $i(v0) | ~ member(v5, v6) | ? [v7: $i] : ? [v8: $i] : % 69.13/9.93 | (ordered_pair(v7, v1) = v8 & ordered_pair(v2, v3) = v7 & $i(v8) & % 69.13/9.93 | $i(v7) & member(v8, v0))) & ! [v0: $i] : ! [v1: $i] : ! [v2: % 69.13/9.93 | $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ % 69.13/9.93 | (rotate(v0) = v6) | ~ (ordered_pair(v4, v3) = v5) | ~ % 69.13/9.93 | (ordered_pair(v1, v2) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ % 69.13/9.93 | $i(v0) | ~ member(v5, all_67_0) | member(v5, v6) | ? [v7: $i] : ? % 69.13/9.93 | [v8: $i] : (ordered_pair(v7, v1) = v8 & ordered_pair(v2, v3) = v7 & % 69.13/9.93 | $i(v8) & $i(v7) & ~ member(v8, v0))) & ! [v0: $i] : ! [v1: $i] % 69.13/9.93 | : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] % 69.13/9.93 | : ( ~ (rotate(v0) = v6) | ~ (ordered_pair(v4, v1) = v5) | ~ % 69.13/9.93 | (ordered_pair(v2, v3) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ % 69.13/9.93 | $i(v0) | ~ member(v5, v0) | ? [v7: $i] : ? [v8: $i] : % 69.13/9.93 | (ordered_pair(v7, v3) = v8 & ordered_pair(v1, v2) = v7 & $i(v8) & % 69.13/9.93 | $i(v7) & ( ~ member(v8, all_67_0) | member(v8, v6)))) & ! [v0: % 69.13/9.93 | $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! % 69.13/9.93 | [v5: $i] : ! [v6: $i] : ( ~ (rotate(v0) = v4) | ~ (ordered_pair(v5, % 69.13/9.93 | v1) = v6) | ~ (ordered_pair(v2, v3) = v5) | ~ $i(v3) | ~ % 69.13/9.93 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v7: $i] : ? [v8: $i] : % 69.13/9.93 | (ordered_pair(v7, v3) = v8 & ordered_pair(v1, v2) = v7 & $i(v8) & % 69.13/9.93 | $i(v7) & ( ~ member(v8, v4) | (member(v8, all_67_0) & member(v6, % 69.13/9.93 | v0))))) % 69.13/9.93 | % 69.13/9.93 | ALPHA: (37) implies: % 69.13/9.93 | (38) cross_product(universal_class, universal_class) = all_67_1 % 69.13/9.93 | (39) cross_product(all_67_1, universal_class) = all_67_0 % 69.13/9.93 | % 69.13/9.93 | GROUND_INST: instantiating (13) with all_58_1, all_61_0, universal_class, % 69.13/9.93 | universal_class, simplifying with (31), (33) gives: % 69.13/9.93 | (40) all_61_0 = all_58_1 % 69.13/9.93 | % 69.13/9.93 | GROUND_INST: instantiating (13) with all_39_0, all_61_0, universal_class, % 69.13/9.93 | universal_class, simplifying with (17), (33) gives: % 69.13/9.94 | (41) all_61_0 = all_39_0 % 69.13/9.94 | % 69.13/9.94 | GROUND_INST: instantiating (13) with all_58_1, all_64_1, universal_class, % 69.13/9.94 | universal_class, simplifying with (31), (35) gives: % 69.13/9.94 | (42) all_64_1 = all_58_1 % 69.13/9.94 | % 69.13/9.94 | GROUND_INST: instantiating (13) with all_55_1, all_64_1, universal_class, % 69.13/9.94 | universal_class, simplifying with (28), (35) gives: % 69.13/9.94 | (43) all_64_1 = all_55_1 % 69.13/9.94 | % 69.13/9.94 | GROUND_INST: instantiating (13) with all_41_0, all_64_1, universal_class, % 69.13/9.94 | universal_class, simplifying with (19), (35) gives: % 69.13/9.94 | (44) all_64_1 = all_41_0 % 69.13/9.94 | % 69.13/9.94 | GROUND_INST: instantiating (13) with all_64_1, all_67_1, universal_class, % 69.13/9.94 | universal_class, simplifying with (35), (38) gives: % 69.13/9.94 | (45) all_67_1 = all_64_1 % 69.13/9.94 | % 69.13/9.94 | GROUND_INST: instantiating (13) with all_37_0, all_67_1, universal_class, % 69.13/9.94 | universal_class, simplifying with (15), (38) gives: % 69.13/9.94 | (46) all_67_1 = all_37_0 % 69.13/9.94 | % 69.13/9.94 | GROUND_INST: instantiating (13) with all_64_0, all_67_0, universal_class, % 69.13/9.94 | all_64_1, simplifying with (36) gives: % 69.13/9.94 | (47) all_67_0 = all_64_0 | ~ (cross_product(all_64_1, universal_class) = % 69.13/9.94 | all_67_0) % 69.13/9.94 | % 69.13/9.94 | GROUND_INST: instantiating (13) with all_55_0, all_67_0, universal_class, % 69.13/9.94 | all_55_1, simplifying with (29) gives: % 69.13/9.94 | (48) all_67_0 = all_55_0 | ~ (cross_product(all_55_1, universal_class) = % 69.13/9.94 | all_67_0) % 69.13/9.94 | % 69.13/9.94 | COMBINE_EQS: (45), (46) imply: % 69.13/9.94 | (49) all_64_1 = all_37_0 % 69.13/9.94 | % 69.13/9.94 | SIMP: (49) implies: % 69.13/9.94 | (50) all_64_1 = all_37_0 % 69.13/9.94 | % 69.13/9.94 | COMBINE_EQS: (43), (44) imply: % 69.13/9.94 | (51) all_55_1 = all_41_0 % 69.13/9.94 | % 69.13/9.94 | COMBINE_EQS: (43), (50) imply: % 69.13/9.94 | (52) all_55_1 = all_37_0 % 69.13/9.94 | % 69.13/9.94 | COMBINE_EQS: (42), (43) imply: % 69.13/9.94 | (53) all_58_1 = all_55_1 % 69.13/9.94 | % 69.13/9.94 | SIMP: (53) implies: % 69.13/9.94 | (54) all_58_1 = all_55_1 % 69.13/9.94 | % 69.13/9.94 | COMBINE_EQS: (40), (41) imply: % 69.13/9.94 | (55) all_58_1 = all_39_0 % 69.13/9.94 | % 69.13/9.94 | SIMP: (55) implies: % 69.13/9.94 | (56) all_58_1 = all_39_0 % 69.13/9.94 | % 69.13/9.94 | COMBINE_EQS: (54), (56) imply: % 69.13/9.94 | (57) all_55_1 = all_39_0 % 69.13/9.94 | % 69.13/9.94 | SIMP: (57) implies: % 69.13/9.94 | (58) all_55_1 = all_39_0 % 69.13/9.94 | % 69.13/9.94 | COMBINE_EQS: (51), (52) imply: % 69.13/9.94 | (59) all_41_0 = all_37_0 % 69.13/9.94 | % 69.13/9.94 | COMBINE_EQS: (51), (58) imply: % 69.13/9.94 | (60) all_41_0 = all_39_0 % 69.13/9.94 | % 69.13/9.94 | COMBINE_EQS: (59), (60) imply: % 69.13/9.94 | (61) all_39_0 = all_37_0 % 69.13/9.94 | % 69.13/9.94 | REDUCE: (39), (46) imply: % 69.13/9.94 | (62) cross_product(all_37_0, universal_class) = all_67_0 % 69.13/9.94 | % 69.13/9.94 | BETA: splitting (47) gives: % 69.13/9.94 | % 69.13/9.94 | Case 1: % 69.13/9.94 | | % 69.13/9.94 | | (63) ~ (cross_product(all_64_1, universal_class) = all_67_0) % 69.13/9.94 | | % 69.13/9.94 | | REDUCE: (50), (63) imply: % 69.13/9.94 | | (64) ~ (cross_product(all_37_0, universal_class) = all_67_0) % 69.13/9.94 | | % 69.13/9.94 | | PRED_UNIFY: (62), (64) imply: % 69.13/9.94 | | (65) $false % 69.13/9.94 | | % 69.13/9.94 | | CLOSE: (65) is inconsistent. % 69.13/9.94 | | % 69.13/9.94 | Case 2: % 69.13/9.94 | | % 69.13/9.94 | | (66) all_67_0 = all_64_0 % 69.13/9.94 | | % 69.13/9.94 | | REDUCE: (62), (66) imply: % 69.13/9.94 | | (67) cross_product(all_37_0, universal_class) = all_64_0 % 69.13/9.94 | | % 69.13/9.94 | | BETA: splitting (48) gives: % 69.13/9.94 | | % 69.13/9.94 | | Case 1: % 69.13/9.94 | | | % 69.13/9.94 | | | (68) ~ (cross_product(all_55_1, universal_class) = all_67_0) % 69.13/9.94 | | | % 69.13/9.94 | | | REDUCE: (52), (66), (68) imply: % 69.13/9.94 | | | (69) ~ (cross_product(all_37_0, universal_class) = all_64_0) % 69.13/9.94 | | | % 69.13/9.94 | | | PRED_UNIFY: (67), (69) imply: % 69.13/9.94 | | | (70) $false % 69.13/9.94 | | | % 69.13/9.94 | | | CLOSE: (70) is inconsistent. % 69.13/9.94 | | | % 69.13/9.94 | | Case 2: % 69.13/9.94 | | | % 69.13/9.94 | | | % 69.13/9.94 | | | GROUND_INST: instantiating (3) with all_53_2, all_53_0, simplifying with % 69.13/9.94 | | | (23), (25) gives: % 69.13/9.94 | | | (71) unordered_pair(all_53_2, all_53_2) = all_53_0 & $i(all_53_0) % 69.13/9.94 | | | % 69.13/9.94 | | | ALPHA: (71) implies: % 69.13/9.94 | | | (72) unordered_pair(all_53_2, all_53_2) = all_53_0 % 69.13/9.94 | | | % 69.13/9.95 | | | GROUND_INST: instantiating (3) with all_53_1, all_53_0, simplifying with % 69.13/9.95 | | | (24), (26) gives: % 69.13/9.95 | | | (73) unordered_pair(all_53_1, all_53_1) = all_53_0 & $i(all_53_0) % 69.13/9.95 | | | % 69.13/9.95 | | | ALPHA: (73) implies: % 69.13/9.95 | | | (74) unordered_pair(all_53_1, all_53_1) = all_53_0 % 69.13/9.95 | | | % 69.13/9.95 | | | GROUND_INST: instantiating (1) with all_53_1, all_53_1, all_53_0, % 69.13/9.95 | | | simplifying with (22), (24), (74) gives: % 69.13/9.95 | | | (75) member(all_53_1, all_53_0) % 69.13/9.95 | | | % 69.13/9.95 | | | GROUND_INST: instantiating (2) with all_53_1, all_53_2, all_53_2, % 69.13/9.95 | | | all_53_0, simplifying with (23), (24), (72), (75) gives: % 69.13/9.95 | | | (76) all_53_1 = all_53_2 % 69.13/9.95 | | | % 69.13/9.95 | | | REDUCE: (21), (76) imply: % 69.13/9.95 | | | (77) $false % 69.13/9.95 | | | % 69.13/9.95 | | | CLOSE: (77) is inconsistent. % 69.13/9.95 | | | % 69.13/9.95 | | End of split % 69.13/9.95 | | % 69.13/9.95 | End of split % 69.13/9.95 | % 69.13/9.95 End of proof % 69.13/9.95 % SZS output end Proof for theBenchmark % 69.13/9.95 % 69.13/9.95 9332ms %------------------------------------------------------------------------------