TSTP Solution File: SET083+1 by Princess---230619
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- Process Solution
%------------------------------------------------------------------------------
% File : Princess---230619
% Problem : SET083+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 : n016.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 65.59s 9.42s
% Output : Proof 70.44s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SET083+1 : TPTP v8.1.2. Bugfixed v5.4.0.
% 0.00/0.12 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s
% 0.13/0.33 % Computer : n016.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Sat Aug 26 09:30:41 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.60 ________ _____
% 0.19/0.60 ___ __ \_________(_)________________________________
% 0.19/0.60 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/
% 0.19/0.60 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ )
% 0.19/0.60 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/
% 0.19/0.60
% 0.19/0.60 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic
% 0.19/0.60 (2023-06-19)
% 0.19/0.60
% 0.19/0.60 (c) Philipp Rümmer, 2009-2023
% 0.19/0.60 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen,
% 0.19/0.60 Amanda Stjerna.
% 0.19/0.60 Free software under BSD-3-Clause.
% 0.19/0.60
% 0.19/0.60 For more information, visit http://www.philipp.ruemmer.org/princess.shtml
% 0.19/0.60
% 0.19/0.60 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 0.19/0.61 Running up to 7 provers in parallel.
% 0.19/0.63 Prover 0: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1042961893
% 0.19/0.63 Prover 2: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMinimalAndEmpty -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1065072994
% 0.19/0.63 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1571432423
% 0.19/0.63 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=1868514696
% 0.19/0.63 Prover 3: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1922548996
% 0.19/0.63 Prover 5: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=complete -randomSeed=1259561288
% 0.19/0.63 Prover 6: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximalOutermost -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1399714365
% 3.18/1.14 Prover 1: Preprocessing ...
% 3.18/1.14 Prover 4: Preprocessing ...
% 3.67/1.18 Prover 6: Preprocessing ...
% 3.67/1.18 Prover 5: Preprocessing ...
% 3.67/1.18 Prover 0: Preprocessing ...
% 3.67/1.18 Prover 2: Preprocessing ...
% 3.67/1.18 Prover 3: Preprocessing ...
% 8.77/1.89 Prover 1: Warning: ignoring some quantifiers
% 9.35/1.97 Prover 3: Warning: ignoring some quantifiers
% 9.35/2.02 Prover 3: Constructing countermodel ...
% 9.35/2.02 Prover 4: Warning: ignoring some quantifiers
% 9.35/2.02 Prover 1: Constructing countermodel ...
% 9.35/2.03 Prover 6: Proving ...
% 10.03/2.04 Prover 5: Proving ...
% 10.03/2.09 Prover 4: Constructing countermodel ...
% 10.93/2.18 Prover 0: Proving ...
% 10.93/2.18 Prover 2: Proving ...
% 65.59/9.42 Prover 5: proved (8793ms)
% 65.59/9.42
% 65.59/9.42 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% 65.59/9.42
% 65.59/9.42 Prover 6: stopped
% 65.59/9.43 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470
% 65.59/9.43 Prover 2: stopped
% 65.59/9.43 Prover 3: stopped
% 65.59/9.45 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089
% 65.59/9.45 Prover 10: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=919308125
% 65.59/9.45 Prover 11: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1509710984
% 65.59/9.48 Prover 0: stopped
% 65.59/9.48 Prover 13: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=1138197443
% 65.59/9.52 Prover 7: Preprocessing ...
% 65.59/9.56 Prover 13: Preprocessing ...
% 65.59/9.56 Prover 11: Preprocessing ...
% 65.59/9.57 Prover 8: Preprocessing ...
% 66.85/9.58 Prover 10: Preprocessing ...
% 66.85/9.63 Prover 7: Warning: ignoring some quantifiers
% 67.72/9.64 Prover 7: Constructing countermodel ...
% 68.42/9.73 Prover 8: Warning: ignoring some quantifiers
% 68.42/9.73 Prover 10: Warning: ignoring some quantifiers
% 68.42/9.74 Prover 8: Constructing countermodel ...
% 68.42/9.74 Prover 10: Constructing countermodel ...
% 68.42/9.76 Prover 13: Warning: ignoring some quantifiers
% 68.92/9.81 Prover 13: Constructing countermodel ...
% 68.92/9.86 Prover 11: Warning: ignoring some quantifiers
% 69.49/9.87 Prover 11: Constructing countermodel ...
% 69.84/9.91 Prover 7: Found proof (size 74)
% 69.84/9.91 Prover 7: proved (487ms)
% 69.84/9.91 Prover 11: stopped
% 69.84/9.91 Prover 10: stopped
% 69.84/9.91 Prover 13: stopped
% 69.84/9.92 Prover 4: stopped
% 69.84/9.92 Prover 8: stopped
% 69.84/9.92 Prover 1: stopped
% 69.84/9.92
% 69.84/9.92 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% 69.84/9.92
% 70.00/9.94 % SZS output start Proof for theBenchmark
% 70.00/9.94 Assumptions after simplification:
% 70.00/9.94 ---------------------------------
% 70.00/9.94
% 70.00/9.94 (compose_defn1)
% 70.00/9.96 $i(universal_class) & ? [v0: $i] : (cross_product(universal_class,
% 70.00/9.97 universal_class) = v0 & $i(v0) & ! [v1: $i] : ! [v2: $i] : ! [v3: $i] :
% 70.00/9.97 ( ~ (compose(v2, v1) = v3) | ~ $i(v2) | ~ $i(v1) | subclass(v3, v0)))
% 70.00/9.97
% 70.00/9.97 (element_relation)
% 70.00/9.97 $i(element_relation) & $i(universal_class) & ? [v0: $i] :
% 70.00/9.97 (cross_product(universal_class, universal_class) = v0 & $i(v0) &
% 70.00/9.97 subclass(element_relation, v0))
% 70.00/9.97
% 70.00/9.97 (flip)
% 70.00/9.97 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0,
% 70.00/9.97 universal_class) = v1 & cross_product(universal_class, universal_class) =
% 70.00/9.97 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (flip(v2) = v3) | ~
% 70.00/9.97 $i(v2) | subclass(v3, v1)))
% 70.00/9.97
% 70.00/9.97 (flip_defn)
% 70.00/9.98 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0,
% 70.00/9.98 universal_class) = v1 & cross_product(universal_class, universal_class) =
% 70.00/9.98 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i]
% 70.00/9.98 : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~
% 70.00/9.98 (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v3, v2) = v6) | ~ $i(v5) |
% 70.00/9.98 ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v5) | ? [v9: $i] : ?
% 70.00/9.98 [v10: $i] : (ordered_pair(v9, v4) = v10 & ordered_pair(v2, v3) = v9 &
% 70.00/9.98 $i(v10) & $i(v9) & ( ~ member(v10, v1) | member(v10, v8)))) & ! [v2:
% 70.00/9.98 $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i]
% 70.00/9.98 : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~
% 70.00/9.98 (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~
% 70.00/9.98 $i(v2) | ~ member(v7, v8) | member(v7, v1)) & ! [v2: $i] : ! [v3: $i] :
% 70.00/9.98 ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~
% 70.00/9.98 (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v2,
% 70.00/9.98 v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~
% 70.00/9.98 member(v7, v8) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v4) = v10
% 70.00/9.98 & ordered_pair(v3, v2) = v9 & $i(v10) & $i(v9) & member(v10, v5))) & !
% 70.00/9.98 [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7:
% 70.00/9.98 $i] : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) |
% 70.00/9.98 ~ (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~
% 70.00/9.98 $i(v2) | ~ member(v7, v1) | member(v7, v8) | ? [v9: $i] : ? [v10: $i] :
% 70.00/9.98 (ordered_pair(v9, v4) = v10 & ordered_pair(v3, v2) = v9 & $i(v10) & $i(v9)
% 70.00/9.98 & ~ member(v10, v5))) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : !
% 70.00/9.98 [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (flip(v5) = v6) |
% 70.00/9.98 ~ (ordered_pair(v7, v4) = v8) | ~ (ordered_pair(v3, v2) = v7) | ~ $i(v5)
% 70.00/9.98 | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] :
% 70.00/9.98 (ordered_pair(v9, v4) = v10 & ordered_pair(v2, v3) = v9 & $i(v10) & $i(v9)
% 70.00/9.98 & ( ~ member(v10, v6) | (member(v10, v1) & member(v8, v5))))))
% 70.00/9.98
% 70.00/9.98 (function_defn)
% 70.00/9.98 $i(identity_relation) & $i(universal_class) & ? [v0: $i] :
% 70.00/9.98 (cross_product(universal_class, universal_class) = v0 & $i(v0) & ! [v1: $i] :
% 70.00/9.98 ! [v2: $i] : ( ~ (inverse(v1) = v2) | ~ $i(v1) | ~ function(v1) |
% 70.00/9.98 subclass(v1, v0)) & ! [v1: $i] : ! [v2: $i] : ( ~ (inverse(v1) = v2) |
% 70.00/9.98 ~ $i(v1) | ~ function(v1) | ? [v3: $i] : (compose(v1, v2) = v3 & $i(v3)
% 70.00/9.98 & subclass(v3, identity_relation))) & ! [v1: $i] : ! [v2: $i] : ( ~
% 70.00/9.98 (inverse(v1) = v2) | ~ $i(v1) | ~ subclass(v1, v0) | function(v1) | ?
% 70.00/9.98 [v3: $i] : (compose(v1, v2) = v3 & $i(v3) & ~ subclass(v3,
% 70.00/9.98 identity_relation))))
% 70.00/9.98
% 70.00/9.98 (rotate)
% 70.00/9.98 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0,
% 70.00/9.98 universal_class) = v1 & cross_product(universal_class, universal_class) =
% 70.00/9.98 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (rotate(v2) = v3) |
% 70.00/9.98 ~ $i(v2) | subclass(v3, v1)))
% 70.00/9.98
% 70.00/9.98 (rotate_defn)
% 70.00/9.99 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0,
% 70.00/9.99 universal_class) = v1 & cross_product(universal_class, universal_class) =
% 70.00/9.99 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i]
% 70.00/9.99 : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~
% 70.00/9.99 (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = v6) | ~ $i(v5) |
% 70.00/9.99 ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v8) | member(v7, v1)) &
% 70.00/9.99 ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7:
% 70.00/9.99 $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~ (ordered_pair(v6, v5) = v7)
% 70.00/9.99 | ~ (ordered_pair(v3, v4) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~
% 70.00/9.99 $i(v2) | ~ member(v7, v8) | ? [v9: $i] : ? [v10: $i] :
% 70.00/9.99 (ordered_pair(v9, v3) = v10 & ordered_pair(v4, v5) = v9 & $i(v10) & $i(v9)
% 70.00/9.99 & member(v10, v2))) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5:
% 70.00/9.99 $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~
% 70.00/9.99 (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = v6) | ~ $i(v5) |
% 70.00/9.99 ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v1) | member(v7, v8) |
% 70.00/9.99 ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v3) = v10 & ordered_pair(v4,
% 70.00/9.99 v5) = v9 & $i(v10) & $i(v9) & ~ member(v10, v2))) & ! [v2: $i] : !
% 70.00/9.99 [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8:
% 70.00/9.99 $i] : ( ~ (rotate(v2) = v8) | ~ (ordered_pair(v6, v3) = v7) | ~
% 70.00/9.99 (ordered_pair(v4, v5) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~
% 70.00/9.99 $i(v2) | ~ member(v7, v2) | ? [v9: $i] : ? [v10: $i] :
% 70.00/9.99 (ordered_pair(v9, v5) = v10 & ordered_pair(v3, v4) = v9 & $i(v10) & $i(v9)
% 70.00/9.99 & ( ~ member(v10, v1) | member(v10, v8)))) & ! [v2: $i] : ! [v3: $i] :
% 70.00/9.99 ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~
% 70.00/9.99 (rotate(v2) = v6) | ~ (ordered_pair(v7, v3) = v8) | ~ (ordered_pair(v4,
% 70.00/9.99 v5) = v7) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ? [v9:
% 70.00/9.99 $i] : ? [v10: $i] : (ordered_pair(v9, v5) = v10 & ordered_pair(v3, v4)
% 70.00/9.99 = v9 & $i(v10) & $i(v9) & ( ~ member(v10, v6) | (member(v10, v1) &
% 70.00/9.99 member(v8, v2))))))
% 70.00/9.99
% 70.00/9.99 (singleton_identified_by_element1)
% 70.00/9.99 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ( ~ (v1 = v0)
% 70.00/9.99 & singleton(v1) = v2 & singleton(v0) = v2 & $i(v2) & $i(v1) & $i(v0) &
% 70.00/9.99 member(v0, universal_class))
% 70.00/9.99
% 70.00/9.99 (singleton_set_defn)
% 70.00/9.99 ! [v0: $i] : ! [v1: $i] : ( ~ (singleton(v0) = v1) | ~ $i(v0) |
% 70.00/9.99 (unordered_pair(v0, v0) = v1 & $i(v1))) & ! [v0: $i] : ! [v1: $i] : ( ~
% 70.00/9.99 (unordered_pair(v0, v0) = v1) | ~ $i(v0) | (singleton(v0) = v1 & $i(v1)))
% 70.00/9.99
% 70.00/9.99 (successor_relation_defn1)
% 70.00/9.99 $i(successor_relation) & $i(universal_class) & ? [v0: $i] :
% 70.00/9.99 (cross_product(universal_class, universal_class) = v0 & $i(v0) &
% 70.00/9.99 subclass(successor_relation, v0))
% 70.00/9.99
% 70.00/9.99 (unordered_pair_defn)
% 70.00/9.99 $i(universal_class) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] :
% 70.00/9.99 (v2 = v0 | v1 = v0 | ~ (unordered_pair(v1, v2) = v3) | ~ $i(v2) | ~ $i(v1)
% 70.00/9.99 | ~ $i(v0) | ~ member(v0, v3)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] :
% 70.00/9.99 ! [v3: $i] : ( ~ (unordered_pair(v1, v2) = v3) | ~ $i(v2) | ~ $i(v1) | ~
% 70.00/9.99 $i(v0) | ~ member(v0, v3) | member(v0, universal_class)) & ! [v0: $i] : !
% 70.00/9.99 [v1: $i] : ! [v2: $i] : ( ~ (unordered_pair(v1, v0) = v2) | ~ $i(v1) | ~
% 70.00/9.99 $i(v0) | ~ member(v0, universal_class) | member(v0, v2)) & ! [v0: $i] : !
% 70.00/9.99 [v1: $i] : ! [v2: $i] : ( ~ (unordered_pair(v0, v1) = v2) | ~ $i(v1) | ~
% 70.00/9.99 $i(v0) | ~ member(v0, universal_class) | member(v0, v2))
% 70.00/9.99
% 70.00/9.99 (function-axioms)
% 70.00/10.00 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : (v1 = v0
% 70.00/10.00 | ~ (restrict(v4, v3, v2) = v1) | ~ (restrict(v4, v3, v2) = v0)) & ! [v0:
% 70.00/10.00 $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (apply(v3, v2)
% 70.00/10.00 = v1) | ~ (apply(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i]
% 70.00/10.00 : ! [v3: $i] : (v1 = v0 | ~ (compose(v3, v2) = v1) | ~ (compose(v3, v2) =
% 70.00/10.00 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 |
% 70.00/10.00 ~ (image(v3, v2) = v1) | ~ (image(v3, v2) = v0)) & ! [v0: $i] : ! [v1:
% 70.00/10.00 $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (union(v3, v2) = v1) | ~
% 70.00/10.00 (union(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3:
% 70.00/10.00 $i] : (v1 = v0 | ~ (intersection(v3, v2) = v1) | ~ (intersection(v3, v2) =
% 70.00/10.00 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 |
% 70.00/10.00 ~ (cross_product(v3, v2) = v1) | ~ (cross_product(v3, v2) = v0)) & ! [v0:
% 70.00/10.00 $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~
% 70.00/10.00 (ordered_pair(v3, v2) = v1) | ~ (ordered_pair(v3, v2) = v0)) & ! [v0: $i]
% 70.00/10.00 : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (unordered_pair(v3,
% 70.00/10.00 v2) = v1) | ~ (unordered_pair(v3, v2) = v0)) & ! [v0: $i] : ! [v1:
% 70.00/10.00 $i] : ! [v2: $i] : (v1 = v0 | ~ (power_class(v2) = v1) | ~
% 70.00/10.00 (power_class(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0
% 70.00/10.00 | ~ (sum_class(v2) = v1) | ~ (sum_class(v2) = v0)) & ! [v0: $i] : ! [v1:
% 70.00/10.00 $i] : ! [v2: $i] : (v1 = v0 | ~ (range_of(v2) = v1) | ~ (range_of(v2) =
% 70.00/10.00 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~
% 70.00/10.00 (inverse(v2) = v1) | ~ (inverse(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : !
% 70.00/10.00 [v2: $i] : (v1 = v0 | ~ (successor(v2) = v1) | ~ (successor(v2) = v0)) & !
% 70.00/10.00 [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (flip(v2) = v1) | ~
% 70.00/10.00 (flip(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~
% 70.00/10.00 (rotate(v2) = v1) | ~ (rotate(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : !
% 70.00/10.00 [v2: $i] : (v1 = v0 | ~ (domain_of(v2) = v1) | ~ (domain_of(v2) = v0)) & !
% 70.00/10.00 [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (complement(v2) = v1) |
% 70.00/10.00 ~ (complement(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 =
% 70.00/10.00 v0 | ~ (first(v2) = v1) | ~ (first(v2) = v0)) & ! [v0: $i] : ! [v1: $i]
% 70.00/10.00 : ! [v2: $i] : (v1 = v0 | ~ (second(v2) = v1) | ~ (second(v2) = v0)) & !
% 70.00/10.00 [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (singleton(v2) = v1) | ~
% 70.00/10.00 (singleton(v2) = v0))
% 70.00/10.00
% 70.00/10.00 Further assumptions not needed in the proof:
% 70.00/10.00 --------------------------------------------
% 70.00/10.00 apply_defn, choice, class_elements_are_sets, complement, compose_defn2,
% 70.00/10.00 cross_product, cross_product_defn, disjoint_defn, domain_of,
% 70.00/10.00 element_relation_defn, extensionality, first_second, identity_relation,
% 70.00/10.00 image_defn, inductive_defn, infinity, intersection, inverse_defn,
% 70.00/10.00 null_class_defn, ordered_pair_defn, power_class, power_class_defn,
% 70.00/10.00 range_of_defn, regularity, replacement, restrict_defn, subclass_defn,
% 70.00/10.00 successor_defn, successor_relation_defn2, sum_class, sum_class_defn, union_defn,
% 70.00/10.00 unordered_pair
% 70.00/10.00
% 70.00/10.00 Those formulas are unsatisfiable:
% 70.00/10.00 ---------------------------------
% 70.00/10.00
% 70.00/10.00 Begin of proof
% 70.00/10.00 |
% 70.00/10.00 | ALPHA: (unordered_pair_defn) implies:
% 70.00/10.01 | (1) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (unordered_pair(v1, v0) =
% 70.00/10.01 | v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v0, universal_class) |
% 70.00/10.01 | member(v0, v2))
% 70.00/10.01 | (2) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v2 = v0 | v1 =
% 70.00/10.01 | v0 | ~ (unordered_pair(v1, v2) = v3) | ~ $i(v2) | ~ $i(v1) | ~
% 70.00/10.01 | $i(v0) | ~ member(v0, v3))
% 70.00/10.01 |
% 70.00/10.01 | ALPHA: (singleton_set_defn) implies:
% 70.00/10.01 | (3) ! [v0: $i] : ! [v1: $i] : ( ~ (singleton(v0) = v1) | ~ $i(v0) |
% 70.00/10.01 | (unordered_pair(v0, v0) = v1 & $i(v1)))
% 70.00/10.01 |
% 70.00/10.01 | ALPHA: (element_relation) implies:
% 70.00/10.01 | (4) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 &
% 70.00/10.01 | $i(v0) & subclass(element_relation, v0))
% 70.00/10.01 |
% 70.00/10.01 | ALPHA: (rotate_defn) implies:
% 70.00/10.01 | (5) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 &
% 70.00/10.01 | cross_product(universal_class, universal_class) = v0 & $i(v1) &
% 70.00/10.01 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : !
% 70.00/10.01 | [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) | ~
% 70.00/10.01 | (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = v6) | ~
% 70.00/10.01 | $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v8) |
% 70.00/10.01 | member(v7, v1)) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5:
% 70.00/10.01 | $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) =
% 70.00/10.01 | v8) | ~ (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) =
% 70.00/10.01 | v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~
% 70.00/10.01 | member(v7, v8) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v3)
% 70.00/10.01 | = v10 & ordered_pair(v4, v5) = v9 & $i(v10) & $i(v9) &
% 70.00/10.01 | member(v10, v2))) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : !
% 70.00/10.01 | [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2)
% 70.00/10.01 | = v8) | ~ (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4)
% 70.00/10.01 | = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~
% 70.00/10.01 | member(v7, v1) | member(v7, v8) | ? [v9: $i] : ? [v10: $i] :
% 70.00/10.01 | (ordered_pair(v9, v3) = v10 & ordered_pair(v4, v5) = v9 & $i(v10) &
% 70.00/10.01 | $i(v9) & ~ member(v10, v2))) & ! [v2: $i] : ! [v3: $i] : !
% 70.00/10.01 | [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : (
% 70.00/10.01 | ~ (rotate(v2) = v8) | ~ (ordered_pair(v6, v3) = v7) | ~
% 70.00/10.01 | (ordered_pair(v4, v5) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) |
% 70.00/10.01 | ~ $i(v2) | ~ member(v7, v2) | ? [v9: $i] : ? [v10: $i] :
% 70.00/10.01 | (ordered_pair(v9, v5) = v10 & ordered_pair(v3, v4) = v9 & $i(v10) &
% 70.00/10.01 | $i(v9) & ( ~ member(v10, v1) | member(v10, v8)))) & ! [v2: $i] :
% 70.00/10.01 | ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] :
% 70.00/10.01 | ! [v8: $i] : ( ~ (rotate(v2) = v6) | ~ (ordered_pair(v7, v3) = v8)
% 70.00/10.01 | | ~ (ordered_pair(v4, v5) = v7) | ~ $i(v5) | ~ $i(v4) | ~
% 70.00/10.01 | $i(v3) | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9,
% 70.00/10.01 | v5) = v10 & ordered_pair(v3, v4) = v9 & $i(v10) & $i(v9) & ( ~
% 70.00/10.01 | member(v10, v6) | (member(v10, v1) & member(v8, v2))))))
% 70.00/10.01 |
% 70.00/10.01 | ALPHA: (rotate) implies:
% 70.00/10.01 | (6) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 &
% 70.00/10.01 | cross_product(universal_class, universal_class) = v0 & $i(v1) &
% 70.00/10.01 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (rotate(v2) = v3) | ~
% 70.00/10.01 | $i(v2) | subclass(v3, v1)))
% 70.00/10.01 |
% 70.00/10.01 | ALPHA: (flip_defn) implies:
% 70.00/10.02 | (7) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 &
% 70.00/10.02 | cross_product(universal_class, universal_class) = v0 & $i(v1) &
% 70.00/10.02 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : !
% 70.00/10.02 | [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (flip(v5) = v8) | ~
% 70.00/10.02 | (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v3, v2) = v6) | ~
% 70.00/10.02 | $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ~ member(v7, v5) | ?
% 70.00/10.02 | [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v4) = v10 &
% 70.00/10.02 | ordered_pair(v2, v3) = v9 & $i(v10) & $i(v9) & ( ~ member(v10,
% 70.00/10.02 | v1) | member(v10, v8)))) & ! [v2: $i] : ! [v3: $i] : !
% 70.00/10.02 | [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : (
% 70.00/10.02 | ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~
% 70.00/10.02 | (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) |
% 70.00/10.02 | ~ $i(v2) | ~ member(v7, v8) | member(v7, v1)) & ! [v2: $i] : !
% 70.00/10.02 | [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : !
% 70.00/10.02 | [v8: $i] : ( ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~
% 70.00/10.02 | (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) |
% 70.00/10.02 | ~ $i(v2) | ~ member(v7, v8) | ? [v9: $i] : ? [v10: $i] :
% 70.00/10.02 | (ordered_pair(v9, v4) = v10 & ordered_pair(v3, v2) = v9 & $i(v10) &
% 70.00/10.02 | $i(v9) & member(v10, v5))) & ! [v2: $i] : ! [v3: $i] : ! [v4:
% 70.00/10.02 | $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~
% 70.00/10.02 | (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~
% 70.00/10.02 | (ordered_pair(v2, v3) = v6) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) |
% 70.00/10.02 | ~ $i(v2) | ~ member(v7, v1) | member(v7, v8) | ? [v9: $i] : ?
% 70.00/10.02 | [v10: $i] : (ordered_pair(v9, v4) = v10 & ordered_pair(v3, v2) = v9
% 70.00/10.02 | & $i(v10) & $i(v9) & ~ member(v10, v5))) & ! [v2: $i] : ! [v3:
% 70.00/10.02 | $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : !
% 70.00/10.02 | [v8: $i] : ( ~ (flip(v5) = v6) | ~ (ordered_pair(v7, v4) = v8) | ~
% 70.00/10.02 | (ordered_pair(v3, v2) = v7) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) |
% 70.00/10.02 | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v4) = v10
% 70.00/10.02 | & ordered_pair(v2, v3) = v9 & $i(v10) & $i(v9) & ( ~ member(v10,
% 70.00/10.02 | v6) | (member(v10, v1) & member(v8, v5))))))
% 70.00/10.02 |
% 70.00/10.02 | ALPHA: (flip) implies:
% 70.00/10.02 | (8) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 &
% 70.00/10.02 | cross_product(universal_class, universal_class) = v0 & $i(v1) &
% 70.00/10.02 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (flip(v2) = v3) | ~ $i(v2)
% 70.00/10.02 | | subclass(v3, v1)))
% 70.00/10.02 |
% 70.00/10.02 | ALPHA: (successor_relation_defn1) implies:
% 70.00/10.02 | (9) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 &
% 70.00/10.02 | $i(v0) & subclass(successor_relation, v0))
% 70.00/10.02 |
% 70.00/10.02 | ALPHA: (compose_defn1) implies:
% 70.00/10.02 | (10) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 &
% 70.00/10.02 | $i(v0) & ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~ (compose(v2,
% 70.44/10.02 | v1) = v3) | ~ $i(v2) | ~ $i(v1) | subclass(v3, v0)))
% 70.44/10.02 |
% 70.44/10.02 | ALPHA: (function_defn) implies:
% 70.44/10.02 | (11) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 &
% 70.44/10.02 | $i(v0) & ! [v1: $i] : ! [v2: $i] : ( ~ (inverse(v1) = v2) | ~
% 70.44/10.02 | $i(v1) | ~ function(v1) | subclass(v1, v0)) & ! [v1: $i] : !
% 70.44/10.02 | [v2: $i] : ( ~ (inverse(v1) = v2) | ~ $i(v1) | ~ function(v1) | ?
% 70.44/10.02 | [v3: $i] : (compose(v1, v2) = v3 & $i(v3) & subclass(v3,
% 70.44/10.02 | identity_relation))) & ! [v1: $i] : ! [v2: $i] : ( ~
% 70.44/10.02 | (inverse(v1) = v2) | ~ $i(v1) | ~ subclass(v1, v0) |
% 70.44/10.02 | function(v1) | ? [v3: $i] : (compose(v1, v2) = v3 & $i(v3) & ~
% 70.44/10.02 | subclass(v3, identity_relation))))
% 70.44/10.02 |
% 70.44/10.02 | ALPHA: (singleton_identified_by_element1) implies:
% 70.44/10.03 | (12) ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : ( ~ (v1 = v0) &
% 70.44/10.03 | singleton(v1) = v2 & singleton(v0) = v2 & $i(v2) & $i(v1) & $i(v0) &
% 70.44/10.03 | member(v0, universal_class))
% 70.44/10.03 |
% 70.44/10.03 | ALPHA: (function-axioms) implies:
% 70.44/10.03 | (13) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~
% 70.44/10.03 | (cross_product(v3, v2) = v1) | ~ (cross_product(v3, v2) = v0))
% 70.44/10.03 |
% 70.44/10.03 | DELTA: instantiating (4) with fresh symbol all_37_0 gives:
% 70.44/10.03 | (14) cross_product(universal_class, universal_class) = all_37_0 &
% 70.44/10.03 | $i(all_37_0) & subclass(element_relation, all_37_0)
% 70.44/10.03 |
% 70.44/10.03 | ALPHA: (14) implies:
% 70.44/10.03 | (15) cross_product(universal_class, universal_class) = all_37_0
% 70.44/10.03 |
% 70.44/10.03 | DELTA: instantiating (9) with fresh symbol all_39_0 gives:
% 70.44/10.03 | (16) cross_product(universal_class, universal_class) = all_39_0 &
% 70.44/10.03 | $i(all_39_0) & subclass(successor_relation, all_39_0)
% 70.44/10.03 |
% 70.44/10.03 | ALPHA: (16) implies:
% 70.44/10.03 | (17) cross_product(universal_class, universal_class) = all_39_0
% 70.44/10.03 |
% 70.44/10.03 | DELTA: instantiating (10) with fresh symbol all_41_0 gives:
% 70.44/10.03 | (18) cross_product(universal_class, universal_class) = all_41_0 &
% 70.44/10.03 | $i(all_41_0) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 70.44/10.03 | (compose(v1, v0) = v2) | ~ $i(v1) | ~ $i(v0) | subclass(v2,
% 70.44/10.03 | all_41_0))
% 70.44/10.03 |
% 70.44/10.03 | ALPHA: (18) implies:
% 70.44/10.03 | (19) cross_product(universal_class, universal_class) = all_41_0
% 70.44/10.03 |
% 70.44/10.03 | DELTA: instantiating (12) with fresh symbols all_53_0, all_53_1, all_53_2
% 70.44/10.03 | gives:
% 70.44/10.03 | (20) ~ (all_53_1 = all_53_2) & singleton(all_53_1) = all_53_0 &
% 70.44/10.03 | singleton(all_53_2) = all_53_0 & $i(all_53_0) & $i(all_53_1) &
% 70.44/10.03 | $i(all_53_2) & member(all_53_2, universal_class)
% 70.44/10.03 |
% 70.44/10.03 | ALPHA: (20) implies:
% 70.44/10.03 | (21) ~ (all_53_1 = all_53_2)
% 70.44/10.03 | (22) member(all_53_2, universal_class)
% 70.44/10.03 | (23) $i(all_53_2)
% 70.44/10.03 | (24) $i(all_53_1)
% 70.44/10.03 | (25) singleton(all_53_2) = all_53_0
% 70.44/10.03 | (26) singleton(all_53_1) = all_53_0
% 70.44/10.03 |
% 70.44/10.03 | DELTA: instantiating (6) with fresh symbols all_55_0, all_55_1 gives:
% 70.44/10.03 | (27) cross_product(all_55_1, universal_class) = all_55_0 &
% 70.44/10.03 | cross_product(universal_class, universal_class) = all_55_1 &
% 70.44/10.03 | $i(all_55_0) & $i(all_55_1) & ! [v0: $i] : ! [v1: $i] : ( ~
% 70.44/10.03 | (rotate(v0) = v1) | ~ $i(v0) | subclass(v1, all_55_0))
% 70.44/10.03 |
% 70.44/10.03 | ALPHA: (27) implies:
% 70.44/10.03 | (28) cross_product(universal_class, universal_class) = all_55_1
% 70.44/10.03 | (29) cross_product(all_55_1, universal_class) = all_55_0
% 70.44/10.03 |
% 70.44/10.03 | DELTA: instantiating (8) with fresh symbols all_58_0, all_58_1 gives:
% 70.44/10.03 | (30) cross_product(all_58_1, universal_class) = all_58_0 &
% 70.44/10.03 | cross_product(universal_class, universal_class) = all_58_1 &
% 70.44/10.03 | $i(all_58_0) & $i(all_58_1) & ! [v0: $i] : ! [v1: $i] : ( ~
% 70.44/10.03 | (flip(v0) = v1) | ~ $i(v0) | subclass(v1, all_58_0))
% 70.44/10.03 |
% 70.44/10.03 | ALPHA: (30) implies:
% 70.44/10.03 | (31) cross_product(universal_class, universal_class) = all_58_1
% 70.44/10.03 |
% 70.44/10.03 | DELTA: instantiating (11) with fresh symbol all_61_0 gives:
% 70.44/10.03 | (32) cross_product(universal_class, universal_class) = all_61_0 &
% 70.44/10.03 | $i(all_61_0) & ! [v0: $i] : ! [v1: $i] : ( ~ (inverse(v0) = v1) | ~
% 70.44/10.03 | $i(v0) | ~ function(v0) | subclass(v0, all_61_0)) & ! [v0: $i] :
% 70.44/10.03 | ! [v1: $i] : ( ~ (inverse(v0) = v1) | ~ $i(v0) | ~ function(v0) | ?
% 70.44/10.03 | [v2: $i] : (compose(v0, v1) = v2 & $i(v2) & subclass(v2,
% 70.44/10.03 | identity_relation))) & ! [v0: $i] : ! [v1: $i] : ( ~
% 70.44/10.03 | (inverse(v0) = v1) | ~ $i(v0) | ~ subclass(v0, all_61_0) |
% 70.44/10.03 | function(v0) | ? [v2: $i] : (compose(v0, v1) = v2 & $i(v2) & ~
% 70.44/10.03 | subclass(v2, identity_relation)))
% 70.44/10.03 |
% 70.44/10.03 | ALPHA: (32) implies:
% 70.44/10.03 | (33) cross_product(universal_class, universal_class) = all_61_0
% 70.44/10.03 |
% 70.44/10.03 | DELTA: instantiating (7) with fresh symbols all_64_0, all_64_1 gives:
% 70.44/10.04 | (34) cross_product(all_64_1, universal_class) = all_64_0 &
% 70.44/10.04 | cross_product(universal_class, universal_class) = all_64_1 &
% 70.44/10.04 | $i(all_64_0) & $i(all_64_1) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i]
% 70.44/10.04 | : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~
% 70.44/10.04 | (flip(v3) = v6) | ~ (ordered_pair(v4, v2) = v5) | ~
% 70.44/10.04 | (ordered_pair(v1, v0) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~
% 70.44/10.04 | $i(v0) | ~ member(v5, v3) | ? [v7: $i] : ? [v8: $i] :
% 70.44/10.04 | (ordered_pair(v7, v2) = v8 & ordered_pair(v0, v1) = v7 & $i(v8) &
% 70.44/10.04 | $i(v7) & ( ~ member(v8, all_64_0) | member(v8, v6)))) & ! [v0:
% 70.44/10.04 | $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : !
% 70.44/10.04 | [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v6) | ~ (ordered_pair(v4,
% 70.44/10.04 | v2) = v5) | ~ (ordered_pair(v0, v1) = v4) | ~ $i(v3) | ~
% 70.44/10.04 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v5, v6) | member(v5,
% 70.44/10.04 | all_64_0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i]
% 70.44/10.04 | : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v6) | ~
% 70.44/10.04 | (ordered_pair(v4, v2) = v5) | ~ (ordered_pair(v0, v1) = v4) | ~
% 70.44/10.04 | $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v5, v6) | ?
% 70.44/10.04 | [v7: $i] : ? [v8: $i] : (ordered_pair(v7, v2) = v8 &
% 70.44/10.04 | ordered_pair(v1, v0) = v7 & $i(v8) & $i(v7) & member(v8, v3))) &
% 70.44/10.04 | ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] :
% 70.44/10.04 | ! [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v6) | ~ (ordered_pair(v4,
% 70.44/10.04 | v2) = v5) | ~ (ordered_pair(v0, v1) = v4) | ~ $i(v3) | ~
% 70.44/10.04 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ~ member(v5, all_64_0) |
% 70.44/10.04 | member(v5, v6) | ? [v7: $i] : ? [v8: $i] : (ordered_pair(v7, v2) =
% 70.44/10.04 | v8 & ordered_pair(v1, v0) = v7 & $i(v8) & $i(v7) & ~ member(v8,
% 70.44/10.04 | v3))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] :
% 70.44/10.04 | ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v4) | ~
% 70.44/10.04 | (ordered_pair(v5, v2) = v6) | ~ (ordered_pair(v1, v0) = v5) | ~
% 70.44/10.04 | $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v7: $i] : ? [v8:
% 70.44/10.04 | $i] : (ordered_pair(v7, v2) = v8 & ordered_pair(v0, v1) = v7 &
% 70.44/10.04 | $i(v8) & $i(v7) & ( ~ member(v8, v4) | (member(v8, all_64_0) &
% 70.44/10.04 | member(v6, v3)))))
% 70.44/10.04 |
% 70.44/10.04 | ALPHA: (34) implies:
% 70.44/10.04 | (35) cross_product(universal_class, universal_class) = all_64_1
% 70.44/10.04 | (36) cross_product(all_64_1, universal_class) = all_64_0
% 70.44/10.04 |
% 70.44/10.04 | DELTA: instantiating (5) with fresh symbols all_67_0, all_67_1 gives:
% 70.44/10.04 | (37) cross_product(all_67_1, universal_class) = all_67_0 &
% 70.44/10.04 | cross_product(universal_class, universal_class) = all_67_1 &
% 70.44/10.04 | $i(all_67_0) & $i(all_67_1) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i]
% 70.44/10.04 | : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~
% 70.44/10.04 | (rotate(v0) = v6) | ~ (ordered_pair(v4, v3) = v5) | ~
% 70.44/10.04 | (ordered_pair(v1, v2) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~
% 70.44/10.04 | $i(v0) | ~ member(v5, v6) | member(v5, all_67_0)) & ! [v0: $i] :
% 70.44/10.04 | ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] :
% 70.44/10.04 | ! [v6: $i] : ( ~ (rotate(v0) = v6) | ~ (ordered_pair(v4, v3) = v5) |
% 70.44/10.04 | ~ (ordered_pair(v1, v2) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) |
% 70.44/10.04 | ~ $i(v0) | ~ member(v5, v6) | ? [v7: $i] : ? [v8: $i] :
% 70.44/10.04 | (ordered_pair(v7, v1) = v8 & ordered_pair(v2, v3) = v7 & $i(v8) &
% 70.44/10.04 | $i(v7) & member(v8, v0))) & ! [v0: $i] : ! [v1: $i] : ! [v2:
% 70.44/10.04 | $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ( ~
% 70.44/10.04 | (rotate(v0) = v6) | ~ (ordered_pair(v4, v3) = v5) | ~
% 70.44/10.04 | (ordered_pair(v1, v2) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~
% 70.44/10.04 | $i(v0) | ~ member(v5, all_67_0) | member(v5, v6) | ? [v7: $i] : ?
% 70.44/10.04 | [v8: $i] : (ordered_pair(v7, v1) = v8 & ordered_pair(v2, v3) = v7 &
% 70.44/10.04 | $i(v8) & $i(v7) & ~ member(v8, v0))) & ! [v0: $i] : ! [v1: $i]
% 70.44/10.04 | : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i]
% 70.44/10.04 | : ( ~ (rotate(v0) = v6) | ~ (ordered_pair(v4, v1) = v5) | ~
% 70.44/10.04 | (ordered_pair(v2, v3) = v4) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~
% 70.44/10.04 | $i(v0) | ~ member(v5, v0) | ? [v7: $i] : ? [v8: $i] :
% 70.44/10.04 | (ordered_pair(v7, v3) = v8 & ordered_pair(v1, v2) = v7 & $i(v8) &
% 70.44/10.04 | $i(v7) & ( ~ member(v8, all_67_0) | member(v8, v6)))) & ! [v0:
% 70.44/10.04 | $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : !
% 70.44/10.04 | [v5: $i] : ! [v6: $i] : ( ~ (rotate(v0) = v4) | ~ (ordered_pair(v5,
% 70.44/10.04 | v1) = v6) | ~ (ordered_pair(v2, v3) = v5) | ~ $i(v3) | ~
% 70.44/10.04 | $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v7: $i] : ? [v8: $i] :
% 70.44/10.04 | (ordered_pair(v7, v3) = v8 & ordered_pair(v1, v2) = v7 & $i(v8) &
% 70.44/10.04 | $i(v7) & ( ~ member(v8, v4) | (member(v8, all_67_0) & member(v6,
% 70.44/10.04 | v0)))))
% 70.44/10.04 |
% 70.44/10.04 | ALPHA: (37) implies:
% 70.44/10.04 | (38) cross_product(universal_class, universal_class) = all_67_1
% 70.44/10.04 | (39) cross_product(all_67_1, universal_class) = all_67_0
% 70.44/10.04 |
% 70.44/10.04 | GROUND_INST: instantiating (13) with all_58_1, all_61_0, universal_class,
% 70.44/10.04 | universal_class, simplifying with (31), (33) gives:
% 70.44/10.05 | (40) all_61_0 = all_58_1
% 70.44/10.05 |
% 70.44/10.05 | GROUND_INST: instantiating (13) with all_39_0, all_61_0, universal_class,
% 70.44/10.05 | universal_class, simplifying with (17), (33) gives:
% 70.44/10.05 | (41) all_61_0 = all_39_0
% 70.44/10.05 |
% 70.44/10.05 | GROUND_INST: instantiating (13) with all_58_1, all_64_1, universal_class,
% 70.44/10.05 | universal_class, simplifying with (31), (35) gives:
% 70.44/10.05 | (42) all_64_1 = all_58_1
% 70.44/10.05 |
% 70.44/10.05 | GROUND_INST: instantiating (13) with all_55_1, all_64_1, universal_class,
% 70.44/10.05 | universal_class, simplifying with (28), (35) gives:
% 70.44/10.05 | (43) all_64_1 = all_55_1
% 70.44/10.05 |
% 70.44/10.05 | GROUND_INST: instantiating (13) with all_41_0, all_64_1, universal_class,
% 70.44/10.05 | universal_class, simplifying with (19), (35) gives:
% 70.44/10.05 | (44) all_64_1 = all_41_0
% 70.44/10.05 |
% 70.44/10.05 | GROUND_INST: instantiating (13) with all_64_1, all_67_1, universal_class,
% 70.44/10.05 | universal_class, simplifying with (35), (38) gives:
% 70.44/10.05 | (45) all_67_1 = all_64_1
% 70.44/10.05 |
% 70.44/10.05 | GROUND_INST: instantiating (13) with all_37_0, all_67_1, universal_class,
% 70.44/10.05 | universal_class, simplifying with (15), (38) gives:
% 70.44/10.05 | (46) all_67_1 = all_37_0
% 70.44/10.05 |
% 70.44/10.05 | GROUND_INST: instantiating (13) with all_64_0, all_67_0, universal_class,
% 70.44/10.05 | all_64_1, simplifying with (36) gives:
% 70.44/10.05 | (47) all_67_0 = all_64_0 | ~ (cross_product(all_64_1, universal_class) =
% 70.44/10.05 | all_67_0)
% 70.44/10.05 |
% 70.44/10.05 | GROUND_INST: instantiating (13) with all_55_0, all_67_0, universal_class,
% 70.44/10.05 | all_55_1, simplifying with (29) gives:
% 70.44/10.05 | (48) all_67_0 = all_55_0 | ~ (cross_product(all_55_1, universal_class) =
% 70.44/10.05 | all_67_0)
% 70.44/10.05 |
% 70.44/10.05 | COMBINE_EQS: (45), (46) imply:
% 70.44/10.05 | (49) all_64_1 = all_37_0
% 70.44/10.05 |
% 70.44/10.05 | SIMP: (49) implies:
% 70.44/10.05 | (50) all_64_1 = all_37_0
% 70.44/10.05 |
% 70.44/10.05 | COMBINE_EQS: (43), (44) imply:
% 70.44/10.05 | (51) all_55_1 = all_41_0
% 70.44/10.05 |
% 70.44/10.05 | COMBINE_EQS: (43), (50) imply:
% 70.44/10.05 | (52) all_55_1 = all_37_0
% 70.44/10.05 |
% 70.44/10.05 | COMBINE_EQS: (42), (43) imply:
% 70.44/10.05 | (53) all_58_1 = all_55_1
% 70.44/10.05 |
% 70.44/10.05 | SIMP: (53) implies:
% 70.44/10.05 | (54) all_58_1 = all_55_1
% 70.44/10.05 |
% 70.44/10.05 | COMBINE_EQS: (40), (41) imply:
% 70.44/10.05 | (55) all_58_1 = all_39_0
% 70.44/10.05 |
% 70.44/10.05 | SIMP: (55) implies:
% 70.44/10.05 | (56) all_58_1 = all_39_0
% 70.44/10.05 |
% 70.44/10.05 | COMBINE_EQS: (54), (56) imply:
% 70.44/10.05 | (57) all_55_1 = all_39_0
% 70.44/10.05 |
% 70.44/10.05 | SIMP: (57) implies:
% 70.44/10.05 | (58) all_55_1 = all_39_0
% 70.44/10.05 |
% 70.44/10.05 | COMBINE_EQS: (51), (52) imply:
% 70.44/10.05 | (59) all_41_0 = all_37_0
% 70.44/10.05 |
% 70.44/10.05 | COMBINE_EQS: (51), (58) imply:
% 70.44/10.05 | (60) all_41_0 = all_39_0
% 70.44/10.05 |
% 70.44/10.05 | COMBINE_EQS: (59), (60) imply:
% 70.44/10.05 | (61) all_39_0 = all_37_0
% 70.44/10.05 |
% 70.44/10.05 | REDUCE: (39), (46) imply:
% 70.44/10.05 | (62) cross_product(all_37_0, universal_class) = all_67_0
% 70.44/10.05 |
% 70.44/10.05 | BETA: splitting (47) gives:
% 70.44/10.05 |
% 70.44/10.05 | Case 1:
% 70.44/10.05 | |
% 70.44/10.05 | | (63) ~ (cross_product(all_64_1, universal_class) = all_67_0)
% 70.44/10.05 | |
% 70.44/10.05 | | REDUCE: (50), (63) imply:
% 70.44/10.05 | | (64) ~ (cross_product(all_37_0, universal_class) = all_67_0)
% 70.44/10.05 | |
% 70.44/10.05 | | PRED_UNIFY: (62), (64) imply:
% 70.44/10.05 | | (65) $false
% 70.44/10.05 | |
% 70.44/10.05 | | CLOSE: (65) is inconsistent.
% 70.44/10.05 | |
% 70.44/10.05 | Case 2:
% 70.44/10.05 | |
% 70.44/10.05 | | (66) all_67_0 = all_64_0
% 70.44/10.05 | |
% 70.44/10.05 | | REDUCE: (62), (66) imply:
% 70.44/10.05 | | (67) cross_product(all_37_0, universal_class) = all_64_0
% 70.44/10.05 | |
% 70.44/10.05 | | BETA: splitting (48) gives:
% 70.44/10.05 | |
% 70.44/10.05 | | Case 1:
% 70.44/10.05 | | |
% 70.44/10.05 | | | (68) ~ (cross_product(all_55_1, universal_class) = all_67_0)
% 70.44/10.05 | | |
% 70.44/10.05 | | | REDUCE: (52), (66), (68) imply:
% 70.44/10.05 | | | (69) ~ (cross_product(all_37_0, universal_class) = all_64_0)
% 70.44/10.05 | | |
% 70.44/10.05 | | | PRED_UNIFY: (67), (69) imply:
% 70.44/10.05 | | | (70) $false
% 70.44/10.05 | | |
% 70.44/10.05 | | | CLOSE: (70) is inconsistent.
% 70.44/10.05 | | |
% 70.44/10.05 | | Case 2:
% 70.44/10.05 | | |
% 70.44/10.05 | | |
% 70.44/10.05 | | | GROUND_INST: instantiating (3) with all_53_2, all_53_0, simplifying with
% 70.44/10.05 | | | (23), (25) gives:
% 70.44/10.05 | | | (71) unordered_pair(all_53_2, all_53_2) = all_53_0 & $i(all_53_0)
% 70.44/10.05 | | |
% 70.44/10.05 | | | ALPHA: (71) implies:
% 70.44/10.05 | | | (72) unordered_pair(all_53_2, all_53_2) = all_53_0
% 70.44/10.05 | | |
% 70.44/10.05 | | | GROUND_INST: instantiating (3) with all_53_1, all_53_0, simplifying with
% 70.44/10.05 | | | (24), (26) gives:
% 70.44/10.05 | | | (73) unordered_pair(all_53_1, all_53_1) = all_53_0 & $i(all_53_0)
% 70.44/10.05 | | |
% 70.44/10.05 | | | ALPHA: (73) implies:
% 70.44/10.05 | | | (74) unordered_pair(all_53_1, all_53_1) = all_53_0
% 70.44/10.05 | | |
% 70.44/10.05 | | | GROUND_INST: instantiating (1) with all_53_2, all_53_2, all_53_0,
% 70.44/10.05 | | | simplifying with (22), (23), (72) gives:
% 70.44/10.05 | | | (75) member(all_53_2, all_53_0)
% 70.44/10.05 | | |
% 70.44/10.05 | | | GROUND_INST: instantiating (2) with all_53_2, all_53_1, all_53_1,
% 70.44/10.05 | | | all_53_0, simplifying with (23), (24), (74), (75) gives:
% 70.44/10.06 | | | (76) all_53_1 = all_53_2
% 70.44/10.06 | | |
% 70.44/10.06 | | | REDUCE: (21), (76) imply:
% 70.44/10.06 | | | (77) $false
% 70.44/10.06 | | |
% 70.44/10.06 | | | CLOSE: (77) is inconsistent.
% 70.44/10.06 | | |
% 70.44/10.06 | | End of split
% 70.44/10.06 | |
% 70.44/10.06 | End of split
% 70.44/10.06 |
% 70.44/10.06 End of proof
% 70.44/10.06 % SZS output end Proof for theBenchmark
% 70.44/10.06
% 70.44/10.06 9453ms
%------------------------------------------------------------------------------