TSTP Solution File: SET102+1 by Princess---230619
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- Process Solution
%------------------------------------------------------------------------------
% 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
%------------------------------------------------------------------------------