TSTP Solution File: SET065+1 by Princess---230619
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%------------------------------------------------------------------------------
% File : Princess---230619
% Problem : SET065+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 : n014.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 15:23:32 EDT 2023
% Result : Theorem 17.74s 3.17s
% Output : Proof 72.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : SET065+1 : TPTP v8.1.2. Bugfixed v5.4.0.
% 0.00/0.14 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s
% 0.13/0.35 % Computer : n014.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Sat Aug 26 14:26:17 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.21/0.63 ________ _____
% 0.21/0.63 ___ __ \_________(_)________________________________
% 0.21/0.63 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/
% 0.21/0.63 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ )
% 0.21/0.63 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/
% 0.21/0.63
% 0.21/0.63 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic
% 0.21/0.63 (2023-06-19)
% 0.21/0.63
% 0.21/0.63 (c) Philipp Rümmer, 2009-2023
% 0.21/0.63 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen,
% 0.21/0.63 Amanda Stjerna.
% 0.21/0.63 Free software under BSD-3-Clause.
% 0.21/0.63
% 0.21/0.63 For more information, visit http://www.philipp.ruemmer.org/princess.shtml
% 0.21/0.63
% 0.21/0.63 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 0.21/0.65 Running up to 7 provers in parallel.
% 0.21/0.67 Prover 0: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1042961893
% 0.21/0.67 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1571432423
% 0.21/0.67 Prover 3: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1922548996
% 0.21/0.67 Prover 2: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMinimalAndEmpty -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1065072994
% 0.21/0.67 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=1868514696
% 0.21/0.67 Prover 6: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximalOutermost -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1399714365
% 0.21/0.67 Prover 5: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=complete -randomSeed=1259561288
% 2.93/1.18 Prover 4: Preprocessing ...
% 2.93/1.18 Prover 1: Preprocessing ...
% 3.63/1.23 Prover 0: Preprocessing ...
% 3.63/1.23 Prover 6: Preprocessing ...
% 3.63/1.23 Prover 3: Preprocessing ...
% 3.63/1.23 Prover 2: Preprocessing ...
% 3.63/1.24 Prover 5: Preprocessing ...
% 8.22/1.84 Prover 1: Warning: ignoring some quantifiers
% 8.22/1.87 Prover 5: Proving ...
% 8.22/1.88 Prover 1: Constructing countermodel ...
% 8.22/1.91 Prover 4: Warning: ignoring some quantifiers
% 8.22/1.91 Prover 3: Warning: ignoring some quantifiers
% 8.83/1.93 Prover 3: Constructing countermodel ...
% 8.83/1.94 Prover 6: Proving ...
% 8.83/1.96 Prover 4: Constructing countermodel ...
% 8.83/1.98 Prover 2: Proving ...
% 9.33/2.03 Prover 0: Proving ...
% 17.74/3.16 Prover 0: proved (2504ms)
% 17.74/3.16
% 17.74/3.17 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% 17.74/3.17
% 18.34/3.17 Prover 2: stopped
% 18.34/3.17 Prover 6: stopped
% 18.38/3.19 Prover 5: stopped
% 18.38/3.19 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470
% 18.38/3.19 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089
% 18.58/3.21 Prover 10: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=919308125
% 18.58/3.22 Prover 11: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1509710984
% 18.70/3.23 Prover 3: stopped
% 18.70/3.25 Prover 13: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=1138197443
% 18.70/3.26 Prover 10: Preprocessing ...
% 18.70/3.27 Prover 8: Preprocessing ...
% 18.70/3.28 Prover 7: Preprocessing ...
% 18.70/3.31 Prover 11: Preprocessing ...
% 18.70/3.31 Prover 13: Preprocessing ...
% 20.02/3.43 Prover 10: Warning: ignoring some quantifiers
% 20.02/3.44 Prover 10: Constructing countermodel ...
% 20.02/3.45 Prover 7: Warning: ignoring some quantifiers
% 20.02/3.45 Prover 13: Warning: ignoring some quantifiers
% 20.02/3.46 Prover 7: Constructing countermodel ...
% 20.02/3.47 Prover 13: Constructing countermodel ...
% 20.02/3.49 Prover 8: Warning: ignoring some quantifiers
% 20.02/3.51 Prover 11: Warning: ignoring some quantifiers
% 20.02/3.51 Prover 8: Constructing countermodel ...
% 20.02/3.54 Prover 11: Constructing countermodel ...
% 22.43/3.73 Prover 10: gave up
% 22.43/3.75 Prover 16: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=completeFrugal -randomSeed=-2043353683
% 22.43/3.77 Prover 16: Preprocessing ...
% 23.75/3.90 Prover 16: Warning: ignoring some quantifiers
% 24.08/3.93 Prover 16: Constructing countermodel ...
% 64.25/9.13 Prover 13: stopped
% 64.25/9.14 Prover 19: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=-1780594085
% 64.25/9.17 Prover 19: Preprocessing ...
% 65.45/9.32 Prover 19: Warning: ignoring some quantifiers
% 65.45/9.34 Prover 19: Constructing countermodel ...
% 71.39/10.08 Prover 11: Found proof (size 187)
% 71.39/10.08 Prover 11: proved (6855ms)
% 71.39/10.08 Prover 19: stopped
% 71.39/10.08 Prover 1: stopped
% 71.39/10.08 Prover 8: stopped
% 71.39/10.08 Prover 7: stopped
% 71.39/10.08 Prover 4: stopped
% 71.39/10.08 Prover 16: stopped
% 71.39/10.08
% 71.39/10.08 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% 71.39/10.08
% 71.68/10.10 % SZS output start Proof for theBenchmark
% 71.68/10.11 Assumptions after simplification:
% 71.68/10.11 ---------------------------------
% 71.68/10.11
% 71.68/10.11 (choice)
% 71.68/10.13 $i(null_class) & $i(universal_class) & ? [v0: $i] : (function(v0) = 0 &
% 71.68/10.13 $i(v0) & ! [v1: $i] : ! [v2: $i] : (v1 = null_class | ~ (apply(v0, v1) =
% 71.68/10.13 v2) | ~ $i(v1) | ? [v3: int] : ? [v4: int] : ((v4 = 0 & member(v2,
% 71.68/10.13 v1) = 0) | ( ~ (v3 = 0) & member(v1, universal_class) = v3))) & !
% 71.68/10.13 [v1: $i] : (v1 = null_class | ~ (member(v1, universal_class) = 0) | ~
% 71.68/10.13 $i(v1) | ? [v2: $i] : (apply(v0, v1) = v2 & member(v2, v1) = 0 &
% 71.68/10.14 $i(v2))))
% 71.68/10.14
% 71.68/10.14 (complement)
% 71.90/10.14 $i(universal_class) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: int] :
% 71.90/10.14 (v3 = 0 | ~ (complement(v0) = v2) | ~ (member(v1, v2) = v3) | ~ $i(v1) | ~
% 71.90/10.14 $i(v0) | ? [v4: int] : ? [v5: int] : ((v5 = 0 & member(v1, v0) = 0) | ( ~
% 71.90/10.14 (v4 = 0) & member(v1, universal_class) = v4))) & ! [v0: $i] : ! [v1:
% 71.90/10.14 $i] : ! [v2: int] : (v2 = 0 | ~ (member(v1, v0) = v2) | ~ $i(v1) | ~
% 71.90/10.14 $i(v0) | ? [v3: int] : ? [v4: $i] : ? [v5: int] : ((v5 = 0 &
% 71.90/10.14 complement(v0) = v4 & member(v1, v4) = 0 & $i(v4)) | ( ~ (v3 = 0) &
% 71.90/10.14 member(v1, universal_class) = v3))) & ! [v0: $i] : ! [v1: $i] : !
% 71.90/10.14 [v2: $i] : ( ~ (complement(v0) = v2) | ~ (member(v1, v2) = 0) | ~ $i(v1) |
% 71.90/10.14 ~ $i(v0) | ? [v3: int] : ( ~ (v3 = 0) & member(v1, v0) = v3 & member(v1,
% 71.90/10.14 universal_class) = 0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: any] : ( ~
% 71.90/10.14 (member(v1, v0) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: int] :
% 71.90/10.14 ? [v5: int] : ((v5 = 0 & ~ (v2 = 0) & member(v1, universal_class) = 0) | (
% 71.90/10.14 ~ (v4 = 0) & complement(v0) = v3 & member(v1, v3) = v4 & $i(v3))))
% 71.90/10.14
% 71.90/10.14 (compose_defn1)
% 71.90/10.14 $i(universal_class) & ? [v0: $i] : (cross_product(universal_class,
% 71.90/10.14 universal_class) = v0 & $i(v0) & ! [v1: $i] : ! [v2: $i] : ! [v3: $i] :
% 71.90/10.14 ( ~ (compose(v2, v1) = v3) | ~ $i(v2) | ~ $i(v1) | subclass(v3, v0) = 0))
% 71.90/10.14
% 71.90/10.14 (cross_product_defn)
% 71.93/10.14 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5:
% 71.93/10.14 $i] : ! [v6: int] : (v6 = 0 | ~ (cross_product(v2, v3) = v5) | ~
% 71.93/10.14 (ordered_pair(v0, v1) = v4) | ~ (member(v4, v5) = v6) | ~ $i(v3) | ~
% 71.93/10.14 $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v7: int] : ? [v8: int] : (( ~ (v8 = 0)
% 71.93/10.14 & member(v1, v3) = v8) | ( ~ (v7 = 0) & member(v0, v2) = v7))) & ! [v0:
% 71.93/10.14 $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] :
% 71.93/10.14 ( ~ (cross_product(v2, v3) = v5) | ~ (ordered_pair(v0, v1) = v4) | ~
% 71.93/10.14 (member(v4, v5) = 0) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) |
% 71.93/10.14 (member(v1, v3) = 0 & member(v0, v2) = 0))
% 71.93/10.14
% 71.93/10.14 (element_relation)
% 71.93/10.14 $i(element_relation) & $i(universal_class) & ? [v0: $i] :
% 71.93/10.14 (cross_product(universal_class, universal_class) = v0 &
% 71.93/10.14 subclass(element_relation, v0) = 0 & $i(v0))
% 71.93/10.14
% 71.93/10.14 (element_relation_defn)
% 71.93/10.15 $i(element_relation) & $i(universal_class) & ! [v0: $i] : ! [v1: $i] : !
% 71.93/10.15 [v2: $i] : ( ~ (ordered_pair(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3:
% 71.93/10.15 int] : ? [v4: int] : ? [v5: int] : ((v5 = 0 & v4 = 0 & member(v1,
% 71.93/10.15 universal_class) = 0 & member(v0, v1) = 0) | ( ~ (v3 = 0) & member(v2,
% 71.93/10.15 element_relation) = v3))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] :
% 71.93/10.15 ( ~ (ordered_pair(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: int] : ?
% 71.93/10.15 [v4: int] : ? [v5: int] : ((v5 = 0 & member(v2, element_relation) = 0) | (
% 71.93/10.15 ~ (v4 = 0) & member(v0, v1) = v4) | ( ~ (v3 = 0) & member(v1,
% 71.93/10.15 universal_class) = v3))) & ! [v0: $i] : ! [v1: $i] : ! [v2: any] :
% 71.93/10.15 ( ~ (member(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: int]
% 71.93/10.15 : ? [v5: int] : ((v5 = 0 & v2 = 0 & member(v1, universal_class) = 0) | ( ~
% 71.93/10.15 (v4 = 0) & ordered_pair(v0, v1) = v3 & member(v3, element_relation) = v4
% 71.93/10.15 & $i(v3)))) & ! [v0: $i] : ! [v1: $i] : ( ~ (member(v0, v1) = 0) | ~
% 71.93/10.15 $i(v1) | ~ $i(v0) | ? [v2: int] : ? [v3: $i] : ? [v4: int] : ((v4 = 0 &
% 71.93/10.15 ordered_pair(v0, v1) = v3 & member(v3, element_relation) = 0 & $i(v3)) |
% 71.93/10.15 ( ~ (v2 = 0) & member(v1, universal_class) = v2)))
% 71.93/10.15
% 71.93/10.15 (flip)
% 71.93/10.15 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0,
% 71.93/10.15 universal_class) = v1 & cross_product(universal_class, universal_class) =
% 71.93/10.15 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (flip(v2) = v3) | ~
% 71.93/10.15 $i(v2) | subclass(v3, v1) = 0))
% 71.93/10.15
% 71.93/10.15 (flip_defn)
% 71.93/10.15 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0,
% 71.93/10.15 universal_class) = v1 & cross_product(universal_class, universal_class) =
% 71.93/10.15 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i]
% 71.93/10.15 : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ! [v9: int] : (v9 = 0 | ~
% 71.93/10.15 (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v2,
% 71.93/10.15 v3) = v6) | ~ (member(v7, v8) = v9) | ~ $i(v5) | ~ $i(v4) | ~
% 71.93/10.15 $i(v3) | ~ $i(v2) | ? [v10: int] : ? [v11: $i] : ? [v12: $i] : ?
% 71.93/10.15 [v13: int] : (( ~ (v13 = 0) & ordered_pair(v11, v4) = v12 &
% 71.93/10.15 ordered_pair(v3, v2) = v11 & member(v12, v5) = v13 & $i(v12) &
% 71.93/10.15 $i(v11)) | ( ~ (v10 = 0) & member(v7, v1) = v10))) & ! [v2: $i] : !
% 71.93/10.15 [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8:
% 71.93/10.15 $i] : ( ~ (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~
% 71.93/10.15 (ordered_pair(v2, v3) = v6) | ~ (member(v7, v8) = 0) | ~ $i(v5) | ~
% 71.93/10.15 $i(v4) | ~ $i(v3) | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] :
% 71.93/10.15 (ordered_pair(v9, v4) = v10 & ordered_pair(v3, v2) = v9 & member(v10, v5)
% 71.93/10.15 = 0 & member(v7, v1) = 0 & $i(v10) & $i(v9))) & ! [v2: $i] : ! [v3:
% 71.93/10.15 $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: any]
% 71.93/10.15 : ( ~ (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v3, v2) = v6) | ~
% 71.93/10.15 (member(v7, v5) = v8) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ?
% 71.93/10.15 [v9: $i] : ? [v10: $i] : ? [v11: $i] : ? [v12: int] : ? [v13: int] :
% 71.93/10.15 (ordered_pair(v9, v4) = v10 & ordered_pair(v2, v3) = v9 & $i(v10) & $i(v9)
% 71.93/10.15 & ((v13 = 0 & v8 = 0 & member(v10, v1) = 0) | ( ~ (v12 = 0) & flip(v5) =
% 71.93/10.15 v11 & member(v10, v11) = v12 & $i(v11))))) & ! [v2: $i] : ! [v3:
% 71.93/10.15 $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ( ~
% 71.93/10.15 (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v3, v2) = v6) | ~
% 71.93/10.15 (member(v7, v5) = 0) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ?
% 71.93/10.15 [v8: $i] : ? [v9: $i] : ? [v10: int] : ? [v11: $i] : ? [v12: int] :
% 71.93/10.15 (ordered_pair(v8, v4) = v9 & ordered_pair(v2, v3) = v8 & $i(v9) & $i(v8) &
% 71.93/10.15 ((v12 = 0 & flip(v5) = v11 & member(v9, v11) = 0 & $i(v11)) | ( ~ (v10 =
% 71.93/10.15 0) & member(v9, v1) = v10)))))
% 71.93/10.15
% 71.93/10.15 (function_defn)
% 71.93/10.16 $i(identity_relation) & $i(universal_class) & ? [v0: $i] :
% 71.93/10.16 (cross_product(universal_class, universal_class) = v0 & $i(v0) & ! [v1: $i] :
% 71.93/10.16 ! [v2: int] : (v2 = 0 | ~ (function(v1) = v2) | ~ $i(v1) | ? [v3: int] :
% 71.93/10.16 ? [v4: $i] : ? [v5: $i] : ? [v6: int] : (( ~ (v6 = 0) & compose(v1, v4)
% 71.93/10.16 = v5 & inverse(v1) = v4 & subclass(v5, identity_relation) = v6 &
% 71.93/10.16 $i(v5) & $i(v4)) | ( ~ (v3 = 0) & subclass(v1, v0) = v3))) & ! [v1:
% 71.93/10.16 $i] : ! [v2: $i] : ( ~ (inverse(v1) = v2) | ~ $i(v1) | ? [v3: int] : ?
% 71.93/10.16 [v4: int] : ? [v5: $i] : ? [v6: int] : ((v6 = 0 & v4 = 0 & compose(v1,
% 71.93/10.16 v2) = v5 & subclass(v5, identity_relation) = 0 & subclass(v1, v0) =
% 71.93/10.16 0 & $i(v5)) | ( ~ (v3 = 0) & function(v1) = v3))) & ! [v1: $i] : !
% 71.93/10.16 [v2: $i] : ( ~ (inverse(v1) = v2) | ~ $i(v1) | ? [v3: int] : ? [v4: $i] :
% 71.93/10.16 ? [v5: int] : ? [v6: int] : ((v6 = 0 & function(v1) = 0) | ( ~ (v5 = 0)
% 71.93/10.16 & compose(v1, v2) = v4 & subclass(v4, identity_relation) = v5 &
% 71.93/10.16 $i(v4)) | ( ~ (v3 = 0) & subclass(v1, v0) = v3))) & ! [v1: $i] : !
% 71.93/10.16 [v2: any] : ( ~ (subclass(v1, v0) = v2) | ~ $i(v1) | ? [v3: int] : ? [v4:
% 71.93/10.16 $i] : ? [v5: $i] : ? [v6: int] : ((v6 = 0 & v2 = 0 & compose(v1, v4) =
% 71.93/10.16 v5 & inverse(v1) = v4 & subclass(v5, identity_relation) = 0 & $i(v5) &
% 71.93/10.16 $i(v4)) | ( ~ (v3 = 0) & function(v1) = v3))) & ! [v1: $i] : ( ~
% 71.93/10.16 (function(v1) = 0) | ~ $i(v1) | ? [v2: $i] : ? [v3: $i] : (compose(v1,
% 71.93/10.16 v2) = v3 & inverse(v1) = v2 & subclass(v3, identity_relation) = 0 &
% 71.93/10.16 subclass(v1, v0) = 0 & $i(v3) & $i(v2))) & ! [v1: $i] : ( ~
% 71.93/10.16 (subclass(v1, v0) = 0) | ~ $i(v1) | ? [v2: $i] : ? [v3: $i] : ? [v4:
% 71.93/10.16 int] : ? [v5: int] : ((v5 = 0 & function(v1) = 0) | ( ~ (v4 = 0) &
% 71.93/10.16 compose(v1, v2) = v3 & inverse(v1) = v2 & subclass(v3,
% 71.93/10.16 identity_relation) = v4 & $i(v3) & $i(v2)))))
% 71.93/10.16
% 71.93/10.16 (inductive_defn)
% 71.93/10.16 $i(successor_relation) & $i(null_class) & ! [v0: $i] : ! [v1: int] : (v1 = 0
% 71.93/10.16 | ~ (inductive(v0) = v1) | ~ $i(v0) | ? [v2: int] : ? [v3: $i] : ? [v4:
% 71.93/10.16 int] : (( ~ (v4 = 0) & image(successor_relation, v0) = v3 & subclass(v3,
% 71.93/10.16 v0) = v4 & $i(v3)) | ( ~ (v2 = 0) & member(null_class, v0) = v2))) &
% 71.93/10.16 ! [v0: $i] : ! [v1: $i] : ( ~ (image(successor_relation, v0) = v1) | ~
% 71.93/10.16 $i(v0) | ? [v2: int] : ? [v3: int] : ? [v4: int] : ((v4 = 0 & v3 = 0 &
% 71.93/10.16 subclass(v1, v0) = 0 & member(null_class, v0) = 0) | ( ~ (v2 = 0) &
% 71.93/10.16 inductive(v0) = v2))) & ! [v0: $i] : ! [v1: $i] : ( ~
% 71.93/10.16 (image(successor_relation, v0) = v1) | ~ $i(v0) | ? [v2: int] : ? [v3:
% 71.93/10.16 int] : ? [v4: int] : ((v4 = 0 & inductive(v0) = 0) | ( ~ (v3 = 0) &
% 71.93/10.16 subclass(v1, v0) = v3) | ( ~ (v2 = 0) & member(null_class, v0) = v2))) &
% 71.93/10.16 ! [v0: $i] : ! [v1: any] : ( ~ (member(null_class, v0) = v1) | ~ $i(v0) |
% 71.93/10.16 ? [v2: int] : ? [v3: $i] : ? [v4: int] : ((v4 = 0 & v1 = 0 &
% 71.93/10.16 image(successor_relation, v0) = v3 & subclass(v3, v0) = 0 & $i(v3)) | (
% 71.93/10.16 ~ (v2 = 0) & inductive(v0) = v2))) & ! [v0: $i] : ( ~ (inductive(v0) =
% 71.93/10.16 0) | ~ $i(v0) | ? [v1: $i] : (image(successor_relation, v0) = v1 &
% 71.93/10.16 subclass(v1, v0) = 0 & member(null_class, v0) = 0 & $i(v1))) & ! [v0: $i]
% 71.93/10.16 : ( ~ (member(null_class, v0) = 0) | ~ $i(v0) | ? [v1: $i] : ? [v2: int] :
% 71.93/10.16 ? [v3: int] : ((v3 = 0 & inductive(v0) = 0) | ( ~ (v2 = 0) &
% 71.93/10.16 image(successor_relation, v0) = v1 & subclass(v1, v0) = v2 & $i(v1))))
% 71.93/10.16
% 71.93/10.16 (infinity)
% 71.93/10.16 $i(universal_class) & ? [v0: $i] : (inductive(v0) = 0 & member(v0,
% 71.93/10.16 universal_class) = 0 & $i(v0) & ! [v1: $i] : ! [v2: int] : (v2 = 0 | ~
% 71.93/10.16 (subclass(v0, v1) = v2) | ~ $i(v1) | ? [v3: int] : ( ~ (v3 = 0) &
% 71.93/10.16 inductive(v1) = v3)) & ! [v1: $i] : ( ~ (inductive(v1) = 0) | ~ $i(v1)
% 71.93/10.16 | subclass(v0, v1) = 0))
% 71.93/10.16
% 71.93/10.16 (null_class_is_a_set)
% 71.93/10.16 $i(null_class) & $i(universal_class) & ? [v0: int] : ( ~ (v0 = 0) &
% 71.93/10.16 member(null_class, universal_class) = v0)
% 71.93/10.16
% 71.93/10.16 (power_class_defn)
% 71.93/10.16 $i(universal_class) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: int] :
% 71.93/10.16 (v3 = 0 | ~ (power_class(v1) = v2) | ~ (member(v0, v2) = v3) | ~ $i(v1) |
% 71.93/10.16 ~ $i(v0) | ? [v4: int] : ? [v5: int] : (( ~ (v5 = 0) & subclass(v0, v1) =
% 71.93/10.16 v5) | ( ~ (v4 = 0) & member(v0, universal_class) = v4))) & ! [v0: $i] :
% 71.93/10.16 ! [v1: $i] : ! [v2: $i] : ( ~ (power_class(v1) = v2) | ~ (member(v0, v2) =
% 71.93/10.16 0) | ~ $i(v1) | ~ $i(v0) | (subclass(v0, v1) = 0 & member(v0,
% 71.93/10.16 universal_class) = 0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: any] : ( ~
% 71.93/10.16 (subclass(v0, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: int]
% 71.93/10.16 : ? [v5: int] : ((v5 = 0 & v2 = 0 & member(v0, universal_class) = 0) | ( ~
% 71.93/10.16 (v4 = 0) & power_class(v1) = v3 & member(v0, v3) = v4 & $i(v3)))) & !
% 71.93/10.16 [v0: $i] : ! [v1: $i] : ( ~ (subclass(v0, v1) = 0) | ~ $i(v1) | ~ $i(v0) |
% 71.93/10.16 ? [v2: int] : ? [v3: $i] : ? [v4: int] : ((v4 = 0 & power_class(v1) = v3 &
% 71.93/10.16 member(v0, v3) = 0 & $i(v3)) | ( ~ (v2 = 0) & member(v0,
% 71.93/10.16 universal_class) = v2)))
% 71.93/10.16
% 71.93/10.16 (rotate)
% 71.93/10.17 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0,
% 71.93/10.17 universal_class) = v1 & cross_product(universal_class, universal_class) =
% 71.93/10.17 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (rotate(v2) = v3) |
% 71.93/10.17 ~ $i(v2) | subclass(v3, v1) = 0))
% 71.93/10.17
% 71.93/10.17 (rotate_defn)
% 71.93/10.17 $i(universal_class) & ? [v0: $i] : ? [v1: $i] : (cross_product(v0,
% 71.93/10.17 universal_class) = v1 & cross_product(universal_class, universal_class) =
% 71.93/10.17 v0 & $i(v1) & $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i]
% 71.93/10.17 : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ! [v9: int] : (v9 = 0 | ~
% 71.93/10.17 (rotate(v2) = v8) | ~ (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3,
% 71.93/10.17 v4) = v6) | ~ (member(v7, v8) = v9) | ~ $i(v5) | ~ $i(v4) | ~
% 71.93/10.17 $i(v3) | ~ $i(v2) | ? [v10: int] : ? [v11: $i] : ? [v12: $i] : ?
% 71.93/10.17 [v13: int] : (( ~ (v13 = 0) & ordered_pair(v11, v3) = v12 &
% 71.93/10.17 ordered_pair(v4, v5) = v11 & member(v12, v2) = v13 & $i(v12) &
% 71.93/10.17 $i(v11)) | ( ~ (v10 = 0) & member(v7, v1) = v10))) & ! [v2: $i] : !
% 71.93/10.17 [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8:
% 71.93/10.17 $i] : ( ~ (rotate(v2) = v8) | ~ (ordered_pair(v6, v5) = v7) | ~
% 71.93/10.17 (ordered_pair(v3, v4) = v6) | ~ (member(v7, v8) = 0) | ~ $i(v5) | ~
% 71.93/10.17 $i(v4) | ~ $i(v3) | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] :
% 71.93/10.17 (ordered_pair(v9, v3) = v10 & ordered_pair(v4, v5) = v9 & member(v10, v2)
% 71.93/10.17 = 0 & member(v7, v1) = 0 & $i(v10) & $i(v9))) & ! [v2: $i] : ! [v3:
% 71.93/10.17 $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: any]
% 71.93/10.17 : ( ~ (ordered_pair(v6, v3) = v7) | ~ (ordered_pair(v4, v5) = v6) | ~
% 71.93/10.17 (member(v7, v2) = v8) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ?
% 71.93/10.17 [v9: $i] : ? [v10: $i] : ? [v11: $i] : ? [v12: int] : ? [v13: int] :
% 71.93/10.17 (ordered_pair(v9, v5) = v10 & ordered_pair(v3, v4) = v9 & $i(v10) & $i(v9)
% 71.93/10.17 & ((v13 = 0 & v8 = 0 & member(v10, v1) = 0) | ( ~ (v12 = 0) & rotate(v2)
% 71.93/10.17 = v11 & member(v10, v11) = v12 & $i(v11))))) & ! [v2: $i] : ! [v3:
% 71.93/10.17 $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ( ~
% 71.93/10.17 (ordered_pair(v6, v3) = v7) | ~ (ordered_pair(v4, v5) = v6) | ~
% 71.93/10.17 (member(v7, v2) = 0) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ?
% 71.93/10.17 [v8: $i] : ? [v9: $i] : ? [v10: int] : ? [v11: $i] : ? [v12: int] :
% 71.93/10.17 (ordered_pair(v8, v5) = v9 & ordered_pair(v3, v4) = v8 & $i(v9) & $i(v8) &
% 71.93/10.17 ((v12 = 0 & rotate(v2) = v11 & member(v9, v11) = 0 & $i(v11)) | ( ~ (v10
% 71.93/10.17 = 0) & member(v9, v1) = v10)))))
% 71.93/10.17
% 71.93/10.17 (subclass_defn)
% 71.93/10.17 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: int] : (v3 = 0 | ~
% 71.93/10.17 (subclass(v0, v1) = 0) | ~ (member(v2, v1) = v3) | ~ $i(v2) | ~ $i(v1) |
% 71.93/10.17 ~ $i(v0) | ? [v4: int] : ( ~ (v4 = 0) & member(v2, v0) = v4)) & ! [v0: $i]
% 71.93/10.17 : ! [v1: $i] : ! [v2: int] : (v2 = 0 | ~ (subclass(v0, v1) = v2) | ~
% 71.93/10.17 $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: int] : ( ~ (v4 = 0) & member(v3,
% 71.93/10.17 v1) = v4 & member(v3, v0) = 0 & $i(v3))) & ! [v0: $i] : ! [v1: $i] :
% 71.93/10.17 ! [v2: $i] : ( ~ (subclass(v0, v1) = 0) | ~ (member(v2, v0) = 0) | ~ $i(v2)
% 71.93/10.17 | ~ $i(v1) | ~ $i(v0) | member(v2, v1) = 0)
% 71.93/10.17
% 71.93/10.17 (successor_relation_defn1)
% 71.93/10.17 $i(successor_relation) & $i(universal_class) & ? [v0: $i] :
% 71.93/10.17 (cross_product(universal_class, universal_class) = v0 &
% 71.93/10.17 subclass(successor_relation, v0) = 0 & $i(v0))
% 71.93/10.17
% 71.93/10.17 (function-axioms)
% 71.93/10.18 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : (v1 = v0
% 71.93/10.18 | ~ (restrict(v4, v3, v2) = v1) | ~ (restrict(v4, v3, v2) = v0)) & ! [v0:
% 71.93/10.18 $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (apply(v3, v2)
% 71.93/10.18 = v1) | ~ (apply(v3, v2) = v0)) & ! [v0: MultipleValueBool] : ! [v1:
% 71.93/10.18 MultipleValueBool] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (disjoint(v3,
% 71.93/10.18 v2) = v1) | ~ (disjoint(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : !
% 71.93/10.18 [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (compose(v3, v2) = v1) | ~
% 71.93/10.18 (compose(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3:
% 71.93/10.18 $i] : (v1 = v0 | ~ (image(v3, v2) = v1) | ~ (image(v3, v2) = v0)) & !
% 71.93/10.18 [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (union(v3,
% 71.93/10.18 v2) = v1) | ~ (union(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : !
% 71.93/10.18 [v2: $i] : ! [v3: $i] : (v1 = v0 | ~ (intersection(v3, v2) = v1) | ~
% 71.93/10.18 (intersection(v3, v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : !
% 71.93/10.18 [v3: $i] : (v1 = v0 | ~ (cross_product(v3, v2) = v1) | ~ (cross_product(v3,
% 71.93/10.18 v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1
% 71.93/10.18 = v0 | ~ (ordered_pair(v3, v2) = v1) | ~ (ordered_pair(v3, v2) = v0)) & !
% 71.93/10.18 [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~
% 71.93/10.18 (unordered_pair(v3, v2) = v1) | ~ (unordered_pair(v3, v2) = v0)) & ! [v0:
% 71.93/10.18 MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: $i] : ! [v3: $i]
% 71.93/10.18 : (v1 = v0 | ~ (subclass(v3, v2) = v1) | ~ (subclass(v3, v2) = v0)) & !
% 71.93/10.18 [v0: MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: $i] : ! [v3:
% 71.93/10.18 $i] : (v1 = v0 | ~ (member(v3, v2) = v1) | ~ (member(v3, v2) = v0)) & !
% 71.93/10.18 [v0: MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: $i] : (v1 = v0
% 71.93/10.18 | ~ (function(v2) = v1) | ~ (function(v2) = v0)) & ! [v0: $i] : ! [v1:
% 71.93/10.18 $i] : ! [v2: $i] : (v1 = v0 | ~ (power_class(v2) = v1) | ~
% 71.93/10.18 (power_class(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0
% 71.93/10.18 | ~ (sum_class(v2) = v1) | ~ (sum_class(v2) = v0)) & ! [v0:
% 71.93/10.18 MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: $i] : (v1 = v0 |
% 71.93/10.18 ~ (inductive(v2) = v1) | ~ (inductive(v2) = v0)) & ! [v0: $i] : ! [v1:
% 71.93/10.18 $i] : ! [v2: $i] : (v1 = v0 | ~ (range_of(v2) = v1) | ~ (range_of(v2) =
% 71.93/10.18 v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~
% 71.93/10.18 (inverse(v2) = v1) | ~ (inverse(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : !
% 71.93/10.18 [v2: $i] : (v1 = v0 | ~ (successor(v2) = v1) | ~ (successor(v2) = v0)) & !
% 71.93/10.18 [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (flip(v2) = v1) | ~
% 71.93/10.18 (flip(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~
% 71.93/10.18 (rotate(v2) = v1) | ~ (rotate(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : !
% 71.93/10.18 [v2: $i] : (v1 = v0 | ~ (domain_of(v2) = v1) | ~ (domain_of(v2) = v0)) & !
% 71.93/10.18 [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (complement(v2) = v1) |
% 71.93/10.18 ~ (complement(v2) = v0)) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 =
% 71.93/10.18 v0 | ~ (first(v2) = v1) | ~ (first(v2) = v0)) & ! [v0: $i] : ! [v1: $i]
% 71.93/10.18 : ! [v2: $i] : (v1 = v0 | ~ (second(v2) = v1) | ~ (second(v2) = v0)) & !
% 71.93/10.18 [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~ (singleton(v2) = v1) | ~
% 71.93/10.18 (singleton(v2) = v0))
% 71.93/10.18
% 71.93/10.18 Further assumptions not needed in the proof:
% 71.93/10.18 --------------------------------------------
% 71.93/10.18 apply_defn, class_elements_are_sets, compose_defn2, cross_product,
% 71.93/10.18 disjoint_defn, domain_of, extensionality, first_second, identity_relation,
% 71.93/10.18 image_defn, intersection, inverse_defn, null_class_defn, ordered_pair_defn,
% 71.93/10.18 power_class, range_of_defn, regularity, replacement, restrict_defn,
% 71.93/10.18 singleton_set_defn, successor_defn, successor_relation_defn2, sum_class,
% 71.93/10.18 sum_class_defn, union_defn, unordered_pair, unordered_pair_defn
% 71.93/10.18
% 71.93/10.18 Those formulas are unsatisfiable:
% 71.93/10.18 ---------------------------------
% 71.93/10.18
% 71.93/10.18 Begin of proof
% 71.93/10.18 |
% 71.93/10.18 | ALPHA: (subclass_defn) implies:
% 71.93/10.18 | (1) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (subclass(v0, v1) = 0) |
% 71.93/10.18 | ~ (member(v2, v0) = 0) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) |
% 71.93/10.18 | member(v2, v1) = 0)
% 71.93/10.18 |
% 71.93/10.18 | ALPHA: (cross_product_defn) implies:
% 71.93/10.18 | (2) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i] :
% 71.93/10.18 | ! [v5: $i] : ( ~ (cross_product(v2, v3) = v5) | ~ (ordered_pair(v0,
% 71.93/10.18 | v1) = v4) | ~ (member(v4, v5) = 0) | ~ $i(v3) | ~ $i(v2) | ~
% 71.93/10.18 | $i(v1) | ~ $i(v0) | (member(v1, v3) = 0 & member(v0, v2) = 0))
% 71.93/10.18 |
% 71.93/10.18 | ALPHA: (element_relation_defn) implies:
% 71.93/10.18 | (3) ! [v0: $i] : ! [v1: $i] : ( ~ (member(v0, v1) = 0) | ~ $i(v1) | ~
% 71.93/10.18 | $i(v0) | ? [v2: int] : ? [v3: $i] : ? [v4: int] : ((v4 = 0 &
% 71.93/10.18 | ordered_pair(v0, v1) = v3 & member(v3, element_relation) = 0 &
% 71.93/10.18 | $i(v3)) | ( ~ (v2 = 0) & member(v1, universal_class) = v2)))
% 71.93/10.18 |
% 71.93/10.18 | ALPHA: (element_relation) implies:
% 71.93/10.19 | (4) $i(element_relation)
% 71.93/10.19 | (5) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 &
% 71.93/10.19 | subclass(element_relation, v0) = 0 & $i(v0))
% 71.93/10.19 |
% 71.93/10.19 | ALPHA: (complement) implies:
% 71.93/10.19 | (6) ! [v0: $i] : ! [v1: $i] : ! [v2: any] : ( ~ (member(v1, v0) = v2) |
% 71.93/10.19 | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: int] : ? [v5: int] :
% 71.93/10.19 | ((v5 = 0 & ~ (v2 = 0) & member(v1, universal_class) = 0) | ( ~ (v4 =
% 71.93/10.19 | 0) & complement(v0) = v3 & member(v1, v3) = v4 & $i(v3))))
% 71.93/10.19 | (7) ! [v0: $i] : ! [v1: $i] : ! [v2: int] : (v2 = 0 | ~ (member(v1, v0)
% 71.93/10.19 | = v2) | ~ $i(v1) | ~ $i(v0) | ? [v3: int] : ? [v4: $i] : ?
% 71.93/10.19 | [v5: int] : ((v5 = 0 & complement(v0) = v4 & member(v1, v4) = 0 &
% 71.93/10.19 | $i(v4)) | ( ~ (v3 = 0) & member(v1, universal_class) = v3)))
% 71.93/10.19 |
% 71.93/10.19 | ALPHA: (rotate_defn) implies:
% 71.93/10.19 | (8) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 &
% 71.93/10.19 | cross_product(universal_class, universal_class) = v0 & $i(v1) &
% 71.93/10.19 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : !
% 71.93/10.19 | [v6: $i] : ! [v7: $i] : ! [v8: $i] : ! [v9: int] : (v9 = 0 | ~
% 71.93/10.19 | (rotate(v2) = v8) | ~ (ordered_pair(v6, v5) = v7) | ~
% 71.93/10.19 | (ordered_pair(v3, v4) = v6) | ~ (member(v7, v8) = v9) | ~ $i(v5)
% 71.93/10.19 | | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ? [v10: int] : ? [v11: $i]
% 71.93/10.19 | : ? [v12: $i] : ? [v13: int] : (( ~ (v13 = 0) & ordered_pair(v11,
% 71.93/10.19 | v3) = v12 & ordered_pair(v4, v5) = v11 & member(v12, v2) =
% 71.93/10.19 | v13 & $i(v12) & $i(v11)) | ( ~ (v10 = 0) & member(v7, v1) =
% 71.93/10.19 | v10))) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i]
% 71.93/10.19 | : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~ (rotate(v2) = v8) |
% 71.93/10.19 | ~ (ordered_pair(v6, v5) = v7) | ~ (ordered_pair(v3, v4) = v6) | ~
% 71.93/10.19 | (member(v7, v8) = 0) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~
% 71.93/10.19 | $i(v2) | ? [v9: $i] : ? [v10: $i] : (ordered_pair(v9, v3) = v10 &
% 71.93/10.19 | ordered_pair(v4, v5) = v9 & member(v10, v2) = 0 & member(v7, v1)
% 71.93/10.19 | = 0 & $i(v10) & $i(v9))) & ! [v2: $i] : ! [v3: $i] : ! [v4:
% 71.93/10.19 | $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: any] : ( ~
% 71.93/10.19 | (ordered_pair(v6, v3) = v7) | ~ (ordered_pair(v4, v5) = v6) | ~
% 71.93/10.19 | (member(v7, v2) = v8) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~
% 71.93/10.19 | $i(v2) | ? [v9: $i] : ? [v10: $i] : ? [v11: $i] : ? [v12: int]
% 71.93/10.19 | : ? [v13: int] : (ordered_pair(v9, v5) = v10 & ordered_pair(v3,
% 71.93/10.19 | v4) = v9 & $i(v10) & $i(v9) & ((v13 = 0 & v8 = 0 & member(v10,
% 71.93/10.19 | v1) = 0) | ( ~ (v12 = 0) & rotate(v2) = v11 & member(v10,
% 71.93/10.19 | v11) = v12 & $i(v11))))) & ! [v2: $i] : ! [v3: $i] : !
% 71.93/10.19 | [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ( ~
% 71.93/10.19 | (ordered_pair(v6, v3) = v7) | ~ (ordered_pair(v4, v5) = v6) | ~
% 71.93/10.19 | (member(v7, v2) = 0) | ~ $i(v5) | ~ $i(v4) | ~ $i(v3) | ~
% 71.93/10.19 | $i(v2) | ? [v8: $i] : ? [v9: $i] : ? [v10: int] : ? [v11: $i] :
% 71.93/10.19 | ? [v12: int] : (ordered_pair(v8, v5) = v9 & ordered_pair(v3, v4) =
% 71.93/10.19 | v8 & $i(v9) & $i(v8) & ((v12 = 0 & rotate(v2) = v11 & member(v9,
% 71.93/10.19 | v11) = 0 & $i(v11)) | ( ~ (v10 = 0) & member(v9, v1) =
% 71.93/10.19 | v10)))))
% 71.93/10.19 |
% 71.93/10.19 | ALPHA: (rotate) implies:
% 71.93/10.19 | (9) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 &
% 71.93/10.19 | cross_product(universal_class, universal_class) = v0 & $i(v1) &
% 71.93/10.19 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (rotate(v2) = v3) | ~
% 71.93/10.19 | $i(v2) | subclass(v3, v1) = 0))
% 71.93/10.19 |
% 71.93/10.19 | ALPHA: (flip_defn) implies:
% 71.93/10.19 | (10) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 &
% 71.93/10.19 | cross_product(universal_class, universal_class) = v0 & $i(v1) &
% 71.93/10.19 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : !
% 71.93/10.19 | [v6: $i] : ! [v7: $i] : ! [v8: $i] : ! [v9: int] : (v9 = 0 | ~
% 71.93/10.19 | (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~
% 71.93/10.19 | (ordered_pair(v2, v3) = v6) | ~ (member(v7, v8) = v9) | ~ $i(v5)
% 71.93/10.19 | | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ? [v10: int] : ? [v11: $i]
% 71.93/10.19 | : ? [v12: $i] : ? [v13: int] : (( ~ (v13 = 0) &
% 71.93/10.19 | ordered_pair(v11, v4) = v12 & ordered_pair(v3, v2) = v11 &
% 71.93/10.19 | member(v12, v5) = v13 & $i(v12) & $i(v11)) | ( ~ (v10 = 0) &
% 71.93/10.19 | member(v7, v1) = v10))) & ! [v2: $i] : ! [v3: $i] : ! [v4:
% 71.93/10.19 | $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: $i] : ! [v8: $i] : ( ~
% 71.93/10.19 | (flip(v5) = v8) | ~ (ordered_pair(v6, v4) = v7) | ~
% 71.93/10.19 | (ordered_pair(v2, v3) = v6) | ~ (member(v7, v8) = 0) | ~ $i(v5)
% 71.93/10.20 | | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] :
% 71.93/10.20 | (ordered_pair(v9, v4) = v10 & ordered_pair(v3, v2) = v9 &
% 71.93/10.20 | member(v10, v5) = 0 & member(v7, v1) = 0 & $i(v10) & $i(v9))) &
% 71.93/10.20 | ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] :
% 71.93/10.20 | ! [v7: $i] : ! [v8: any] : ( ~ (ordered_pair(v6, v4) = v7) | ~
% 71.93/10.20 | (ordered_pair(v3, v2) = v6) | ~ (member(v7, v5) = v8) | ~ $i(v5)
% 71.93/10.20 | | ~ $i(v4) | ~ $i(v3) | ~ $i(v2) | ? [v9: $i] : ? [v10: $i] :
% 71.93/10.20 | ? [v11: $i] : ? [v12: int] : ? [v13: int] : (ordered_pair(v9,
% 71.93/10.20 | v4) = v10 & ordered_pair(v2, v3) = v9 & $i(v10) & $i(v9) &
% 71.93/10.20 | ((v13 = 0 & v8 = 0 & member(v10, v1) = 0) | ( ~ (v12 = 0) &
% 71.93/10.20 | flip(v5) = v11 & member(v10, v11) = v12 & $i(v11))))) & !
% 71.93/10.20 | [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] :
% 71.93/10.20 | ! [v7: $i] : ( ~ (ordered_pair(v6, v4) = v7) | ~ (ordered_pair(v3,
% 71.93/10.20 | v2) = v6) | ~ (member(v7, v5) = 0) | ~ $i(v5) | ~ $i(v4) |
% 71.93/10.20 | ~ $i(v3) | ~ $i(v2) | ? [v8: $i] : ? [v9: $i] : ? [v10: int] :
% 71.93/10.20 | ? [v11: $i] : ? [v12: int] : (ordered_pair(v8, v4) = v9 &
% 71.93/10.20 | ordered_pair(v2, v3) = v8 & $i(v9) & $i(v8) & ((v12 = 0 &
% 71.93/10.20 | flip(v5) = v11 & member(v9, v11) = 0 & $i(v11)) | ( ~ (v10 =
% 71.93/10.20 | 0) & member(v9, v1) = v10)))))
% 71.93/10.20 |
% 71.93/10.20 | ALPHA: (flip) implies:
% 71.93/10.20 | (11) ? [v0: $i] : ? [v1: $i] : (cross_product(v0, universal_class) = v1 &
% 71.93/10.20 | cross_product(universal_class, universal_class) = v0 & $i(v1) &
% 71.93/10.20 | $i(v0) & ! [v2: $i] : ! [v3: $i] : ( ~ (flip(v2) = v3) | ~ $i(v2)
% 71.93/10.20 | | subclass(v3, v1) = 0))
% 71.93/10.20 |
% 71.93/10.20 | ALPHA: (successor_relation_defn1) implies:
% 71.93/10.20 | (12) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 &
% 71.93/10.20 | subclass(successor_relation, v0) = 0 & $i(v0))
% 71.93/10.20 |
% 71.93/10.20 | ALPHA: (inductive_defn) implies:
% 71.93/10.20 | (13) ! [v0: $i] : ( ~ (inductive(v0) = 0) | ~ $i(v0) | ? [v1: $i] :
% 71.93/10.20 | (image(successor_relation, v0) = v1 & subclass(v1, v0) = 0 &
% 71.93/10.20 | member(null_class, v0) = 0 & $i(v1)))
% 71.93/10.20 |
% 71.93/10.20 | ALPHA: (infinity) implies:
% 71.93/10.20 | (14) ? [v0: $i] : (inductive(v0) = 0 & member(v0, universal_class) = 0 &
% 71.93/10.20 | $i(v0) & ! [v1: $i] : ! [v2: int] : (v2 = 0 | ~ (subclass(v0, v1)
% 71.93/10.20 | = v2) | ~ $i(v1) | ? [v3: int] : ( ~ (v3 = 0) & inductive(v1)
% 71.93/10.20 | = v3)) & ! [v1: $i] : ( ~ (inductive(v1) = 0) | ~ $i(v1) |
% 71.93/10.20 | subclass(v0, v1) = 0))
% 71.93/10.20 |
% 71.93/10.20 | ALPHA: (power_class_defn) implies:
% 71.93/10.20 | (15) ! [v0: $i] : ! [v1: $i] : ( ~ (subclass(v0, v1) = 0) | ~ $i(v1) |
% 71.93/10.20 | ~ $i(v0) | ? [v2: int] : ? [v3: $i] : ? [v4: int] : ((v4 = 0 &
% 71.93/10.20 | power_class(v1) = v3 & member(v0, v3) = 0 & $i(v3)) | ( ~ (v2 =
% 71.93/10.20 | 0) & member(v0, universal_class) = v2)))
% 71.93/10.20 | (16) ! [v0: $i] : ! [v1: $i] : ! [v2: any] : ( ~ (subclass(v0, v1) = v2)
% 71.93/10.20 | | ~ $i(v1) | ~ $i(v0) | ? [v3: $i] : ? [v4: int] : ? [v5: int]
% 71.93/10.20 | : ((v5 = 0 & v2 = 0 & member(v0, universal_class) = 0) | ( ~ (v4 =
% 71.93/10.20 | 0) & power_class(v1) = v3 & member(v0, v3) = v4 & $i(v3))))
% 71.93/10.20 |
% 71.93/10.20 | ALPHA: (compose_defn1) implies:
% 71.93/10.20 | (17) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 &
% 71.93/10.20 | $i(v0) & ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~ (compose(v2,
% 71.93/10.20 | v1) = v3) | ~ $i(v2) | ~ $i(v1) | subclass(v3, v0) = 0))
% 71.93/10.20 |
% 71.93/10.20 | ALPHA: (function_defn) implies:
% 71.93/10.20 | (18) $i(identity_relation)
% 71.93/10.20 | (19) ? [v0: $i] : (cross_product(universal_class, universal_class) = v0 &
% 71.93/10.20 | $i(v0) & ! [v1: $i] : ! [v2: int] : (v2 = 0 | ~ (function(v1) =
% 71.93/10.20 | v2) | ~ $i(v1) | ? [v3: int] : ? [v4: $i] : ? [v5: $i] : ?
% 71.93/10.20 | [v6: int] : (( ~ (v6 = 0) & compose(v1, v4) = v5 & inverse(v1) =
% 71.93/10.20 | v4 & subclass(v5, identity_relation) = v6 & $i(v5) & $i(v4)) |
% 71.93/10.20 | ( ~ (v3 = 0) & subclass(v1, v0) = v3))) & ! [v1: $i] : ! [v2:
% 71.93/10.20 | $i] : ( ~ (inverse(v1) = v2) | ~ $i(v1) | ? [v3: int] : ? [v4:
% 72.22/10.20 | int] : ? [v5: $i] : ? [v6: int] : ((v6 = 0 & v4 = 0 &
% 72.22/10.20 | compose(v1, v2) = v5 & subclass(v5, identity_relation) = 0 &
% 72.22/10.20 | subclass(v1, v0) = 0 & $i(v5)) | ( ~ (v3 = 0) & function(v1) =
% 72.22/10.20 | v3))) & ! [v1: $i] : ! [v2: $i] : ( ~ (inverse(v1) = v2) |
% 72.22/10.20 | ~ $i(v1) | ? [v3: int] : ? [v4: $i] : ? [v5: int] : ? [v6:
% 72.22/10.20 | int] : ((v6 = 0 & function(v1) = 0) | ( ~ (v5 = 0) & compose(v1,
% 72.22/10.20 | v2) = v4 & subclass(v4, identity_relation) = v5 & $i(v4)) |
% 72.22/10.20 | ( ~ (v3 = 0) & subclass(v1, v0) = v3))) & ! [v1: $i] : ! [v2:
% 72.22/10.20 | any] : ( ~ (subclass(v1, v0) = v2) | ~ $i(v1) | ? [v3: int] : ?
% 72.22/10.20 | [v4: $i] : ? [v5: $i] : ? [v6: int] : ((v6 = 0 & v2 = 0 &
% 72.22/10.20 | compose(v1, v4) = v5 & inverse(v1) = v4 & subclass(v5,
% 72.22/10.20 | identity_relation) = 0 & $i(v5) & $i(v4)) | ( ~ (v3 = 0) &
% 72.22/10.20 | function(v1) = v3))) & ! [v1: $i] : ( ~ (function(v1) = 0) |
% 72.22/10.20 | ~ $i(v1) | ? [v2: $i] : ? [v3: $i] : (compose(v1, v2) = v3 &
% 72.22/10.20 | inverse(v1) = v2 & subclass(v3, identity_relation) = 0 &
% 72.22/10.20 | subclass(v1, v0) = 0 & $i(v3) & $i(v2))) & ! [v1: $i] : ( ~
% 72.22/10.20 | (subclass(v1, v0) = 0) | ~ $i(v1) | ? [v2: $i] : ? [v3: $i] :
% 72.22/10.20 | ? [v4: int] : ? [v5: int] : ((v5 = 0 & function(v1) = 0) | ( ~
% 72.22/10.20 | (v4 = 0) & compose(v1, v2) = v3 & inverse(v1) = v2 &
% 72.22/10.20 | subclass(v3, identity_relation) = v4 & $i(v3) & $i(v2)))))
% 72.22/10.20 |
% 72.22/10.20 | ALPHA: (choice) implies:
% 72.22/10.20 | (20) ? [v0: $i] : (function(v0) = 0 & $i(v0) & ! [v1: $i] : ! [v2: $i] :
% 72.22/10.20 | (v1 = null_class | ~ (apply(v0, v1) = v2) | ~ $i(v1) | ? [v3:
% 72.22/10.20 | int] : ? [v4: int] : ((v4 = 0 & member(v2, v1) = 0) | ( ~ (v3 =
% 72.22/10.20 | 0) & member(v1, universal_class) = v3))) & ! [v1: $i] : (v1
% 72.22/10.20 | = null_class | ~ (member(v1, universal_class) = 0) | ~ $i(v1) |
% 72.22/10.20 | ? [v2: $i] : (apply(v0, v1) = v2 & member(v2, v1) = 0 & $i(v2))))
% 72.22/10.20 |
% 72.22/10.20 | ALPHA: (null_class_is_a_set) implies:
% 72.22/10.20 | (21) $i(universal_class)
% 72.22/10.20 | (22) $i(null_class)
% 72.22/10.20 | (23) ? [v0: int] : ( ~ (v0 = 0) & member(null_class, universal_class) =
% 72.22/10.20 | v0)
% 72.22/10.20 |
% 72.22/10.20 | ALPHA: (function-axioms) implies:
% 72.22/10.20 | (24) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~
% 72.22/10.20 | (complement(v2) = v1) | ~ (complement(v2) = v0))
% 72.22/10.20 | (25) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : (v1 = v0 | ~
% 72.22/10.20 | (power_class(v2) = v1) | ~ (power_class(v2) = v0))
% 72.22/10.20 | (26) ! [v0: MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: $i]
% 72.22/10.20 | : ! [v3: $i] : (v1 = v0 | ~ (member(v3, v2) = v1) | ~ (member(v3,
% 72.22/10.20 | v2) = v0))
% 72.22/10.21 | (27) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : (v1 = v0 | ~
% 72.22/10.21 | (cross_product(v3, v2) = v1) | ~ (cross_product(v3, v2) = v0))
% 72.22/10.21 |
% 72.22/10.21 | DELTA: instantiating (23) with fresh symbol all_36_0 gives:
% 72.22/10.21 | (28) ~ (all_36_0 = 0) & member(null_class, universal_class) = all_36_0
% 72.22/10.21 |
% 72.22/10.21 | ALPHA: (28) implies:
% 72.22/10.21 | (29) ~ (all_36_0 = 0)
% 72.22/10.21 | (30) member(null_class, universal_class) = all_36_0
% 72.22/10.21 |
% 72.22/10.21 | DELTA: instantiating (12) with fresh symbol all_38_0 gives:
% 72.22/10.21 | (31) cross_product(universal_class, universal_class) = all_38_0 &
% 72.22/10.21 | subclass(successor_relation, all_38_0) = 0 & $i(all_38_0)
% 72.22/10.21 |
% 72.22/10.21 | ALPHA: (31) implies:
% 72.22/10.21 | (32) cross_product(universal_class, universal_class) = all_38_0
% 72.22/10.21 |
% 72.22/10.21 | DELTA: instantiating (5) with fresh symbol all_40_0 gives:
% 72.22/10.21 | (33) cross_product(universal_class, universal_class) = all_40_0 &
% 72.22/10.21 | subclass(element_relation, all_40_0) = 0 & $i(all_40_0)
% 72.22/10.21 |
% 72.22/10.21 | ALPHA: (33) implies:
% 72.22/10.21 | (34) $i(all_40_0)
% 72.22/10.21 | (35) subclass(element_relation, all_40_0) = 0
% 72.22/10.21 | (36) cross_product(universal_class, universal_class) = all_40_0
% 72.22/10.21 |
% 72.22/10.21 | DELTA: instantiating (17) with fresh symbol all_42_0 gives:
% 72.22/10.21 | (37) cross_product(universal_class, universal_class) = all_42_0 &
% 72.22/10.21 | $i(all_42_0) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 72.22/10.21 | (compose(v1, v0) = v2) | ~ $i(v1) | ~ $i(v0) | subclass(v2,
% 72.22/10.21 | all_42_0) = 0)
% 72.22/10.21 |
% 72.22/10.21 | ALPHA: (37) implies:
% 72.22/10.21 | (38) cross_product(universal_class, universal_class) = all_42_0
% 72.22/10.21 | (39) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ (compose(v1, v0) = v2) |
% 72.22/10.21 | ~ $i(v1) | ~ $i(v0) | subclass(v2, all_42_0) = 0)
% 72.22/10.21 |
% 72.22/10.21 | DELTA: instantiating (9) with fresh symbols all_46_0, all_46_1 gives:
% 72.22/10.21 | (40) cross_product(all_46_1, universal_class) = all_46_0 &
% 72.22/10.21 | cross_product(universal_class, universal_class) = all_46_1 &
% 72.22/10.21 | $i(all_46_0) & $i(all_46_1) & ! [v0: $i] : ! [v1: $i] : ( ~
% 72.22/10.21 | (rotate(v0) = v1) | ~ $i(v0) | subclass(v1, all_46_0) = 0)
% 72.22/10.21 |
% 72.22/10.21 | ALPHA: (40) implies:
% 72.22/10.21 | (41) cross_product(universal_class, universal_class) = all_46_1
% 72.22/10.21 |
% 72.22/10.21 | DELTA: instantiating (11) with fresh symbols all_49_0, all_49_1 gives:
% 72.22/10.21 | (42) cross_product(all_49_1, universal_class) = all_49_0 &
% 72.22/10.21 | cross_product(universal_class, universal_class) = all_49_1 &
% 72.22/10.21 | $i(all_49_0) & $i(all_49_1) & ! [v0: $i] : ! [v1: $i] : ( ~
% 72.22/10.21 | (flip(v0) = v1) | ~ $i(v0) | subclass(v1, all_49_0) = 0)
% 72.22/10.21 |
% 72.22/10.21 | ALPHA: (42) implies:
% 72.22/10.21 | (43) cross_product(universal_class, universal_class) = all_49_1
% 72.22/10.21 |
% 72.22/10.21 | DELTA: instantiating (14) with fresh symbol all_52_0 gives:
% 72.22/10.21 | (44) inductive(all_52_0) = 0 & member(all_52_0, universal_class) = 0 &
% 72.22/10.21 | $i(all_52_0) & ! [v0: $i] : ! [v1: int] : (v1 = 0 | ~
% 72.22/10.21 | (subclass(all_52_0, v0) = v1) | ~ $i(v0) | ? [v2: int] : ( ~ (v2 =
% 72.22/10.21 | 0) & inductive(v0) = v2)) & ! [v0: $i] : ( ~ (inductive(v0) =
% 72.22/10.21 | 0) | ~ $i(v0) | subclass(all_52_0, v0) = 0)
% 72.22/10.21 |
% 72.22/10.21 | ALPHA: (44) implies:
% 72.22/10.21 | (45) $i(all_52_0)
% 72.22/10.21 | (46) member(all_52_0, universal_class) = 0
% 72.22/10.21 | (47) inductive(all_52_0) = 0
% 72.22/10.21 |
% 72.22/10.21 | DELTA: instantiating (20) with fresh symbol all_55_0 gives:
% 72.22/10.21 | (48) function(all_55_0) = 0 & $i(all_55_0) & ! [v0: $i] : ! [v1: $i] :
% 72.22/10.21 | (v0 = null_class | ~ (apply(all_55_0, v0) = v1) | ~ $i(v0) | ? [v2:
% 72.22/10.21 | int] : ? [v3: int] : ((v3 = 0 & member(v1, v0) = 0) | ( ~ (v2 =
% 72.22/10.21 | 0) & member(v0, universal_class) = v2))) & ! [v0: $i] : (v0 =
% 72.22/10.21 | null_class | ~ (member(v0, universal_class) = 0) | ~ $i(v0) | ?
% 72.22/10.21 | [v1: $i] : (apply(all_55_0, v0) = v1 & member(v1, v0) = 0 & $i(v1)))
% 72.22/10.21 |
% 72.22/10.21 | ALPHA: (48) implies:
% 72.22/10.21 | (49) $i(all_55_0)
% 72.22/10.21 | (50) function(all_55_0) = 0
% 72.22/10.21 |
% 72.22/10.21 | DELTA: instantiating (19) with fresh symbol all_58_0 gives:
% 72.22/10.21 | (51) cross_product(universal_class, universal_class) = all_58_0 &
% 72.22/10.21 | $i(all_58_0) & ! [v0: $i] : ! [v1: int] : (v1 = 0 | ~ (function(v0)
% 72.22/10.21 | = v1) | ~ $i(v0) | ? [v2: int] : ? [v3: $i] : ? [v4: $i] : ?
% 72.22/10.21 | [v5: int] : (( ~ (v5 = 0) & compose(v0, v3) = v4 & inverse(v0) = v3
% 72.22/10.21 | & subclass(v4, identity_relation) = v5 & $i(v4) & $i(v3)) | ( ~
% 72.22/10.21 | (v2 = 0) & subclass(v0, all_58_0) = v2))) & ! [v0: $i] : !
% 72.22/10.21 | [v1: $i] : ( ~ (inverse(v0) = v1) | ~ $i(v0) | ? [v2: int] : ? [v3:
% 72.22/10.21 | int] : ? [v4: $i] : ? [v5: int] : ((v5 = 0 & v3 = 0 &
% 72.22/10.21 | compose(v0, v1) = v4 & subclass(v4, identity_relation) = 0 &
% 72.22/10.21 | subclass(v0, all_58_0) = 0 & $i(v4)) | ( ~ (v2 = 0) &
% 72.22/10.21 | function(v0) = v2))) & ! [v0: $i] : ! [v1: $i] : ( ~
% 72.22/10.21 | (inverse(v0) = v1) | ~ $i(v0) | ? [v2: int] : ? [v3: $i] : ?
% 72.22/10.21 | [v4: int] : ? [v5: int] : ((v5 = 0 & function(v0) = 0) | ( ~ (v4 =
% 72.22/10.21 | 0) & compose(v0, v1) = v3 & subclass(v3, identity_relation) =
% 72.22/10.21 | v4 & $i(v3)) | ( ~ (v2 = 0) & subclass(v0, all_58_0) = v2))) &
% 72.22/10.21 | ! [v0: $i] : ! [v1: any] : ( ~ (subclass(v0, all_58_0) = v1) | ~
% 72.22/10.21 | $i(v0) | ? [v2: int] : ? [v3: $i] : ? [v4: $i] : ? [v5: int] :
% 72.22/10.21 | ((v5 = 0 & v1 = 0 & compose(v0, v3) = v4 & inverse(v0) = v3 &
% 72.22/10.21 | subclass(v4, identity_relation) = 0 & $i(v4) & $i(v3)) | ( ~ (v2
% 72.22/10.21 | = 0) & function(v0) = v2))) & ! [v0: $i] : ( ~ (function(v0)
% 72.22/10.21 | = 0) | ~ $i(v0) | ? [v1: $i] : ? [v2: $i] : (compose(v0, v1) =
% 72.22/10.21 | v2 & inverse(v0) = v1 & subclass(v2, identity_relation) = 0 &
% 72.22/10.21 | subclass(v0, all_58_0) = 0 & $i(v2) & $i(v1))) & ! [v0: $i] : ( ~
% 72.22/10.21 | (subclass(v0, all_58_0) = 0) | ~ $i(v0) | ? [v1: $i] : ? [v2: $i]
% 72.22/10.21 | : ? [v3: int] : ? [v4: int] : ((v4 = 0 & function(v0) = 0) | ( ~
% 72.22/10.21 | (v3 = 0) & compose(v0, v1) = v2 & inverse(v0) = v1 &
% 72.22/10.21 | subclass(v2, identity_relation) = v3 & $i(v2) & $i(v1))))
% 72.22/10.21 |
% 72.22/10.21 | ALPHA: (51) implies:
% 72.22/10.21 | (52) cross_product(universal_class, universal_class) = all_58_0
% 72.22/10.21 | (53) ! [v0: $i] : ( ~ (function(v0) = 0) | ~ $i(v0) | ? [v1: $i] : ?
% 72.22/10.21 | [v2: $i] : (compose(v0, v1) = v2 & inverse(v0) = v1 & subclass(v2,
% 72.22/10.21 | identity_relation) = 0 & subclass(v0, all_58_0) = 0 & $i(v2) &
% 72.22/10.21 | $i(v1)))
% 72.22/10.21 |
% 72.22/10.21 | DELTA: instantiating (8) with fresh symbols all_61_0, all_61_1 gives:
% 72.22/10.22 | (54) cross_product(all_61_1, universal_class) = all_61_0 &
% 72.22/10.22 | cross_product(universal_class, universal_class) = all_61_1 &
% 72.22/10.22 | $i(all_61_0) & $i(all_61_1) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i]
% 72.22/10.22 | : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: int]
% 72.22/10.22 | : (v7 = 0 | ~ (rotate(v0) = v6) | ~ (ordered_pair(v4, v3) = v5) | ~
% 72.22/10.22 | (ordered_pair(v1, v2) = v4) | ~ (member(v5, v6) = v7) | ~ $i(v3) |
% 72.22/10.22 | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v8: int] : ? [v9: $i] : ?
% 72.22/10.22 | [v10: $i] : ? [v11: int] : (( ~ (v11 = 0) & ordered_pair(v9, v1) =
% 72.22/10.22 | v10 & ordered_pair(v2, v3) = v9 & member(v10, v0) = v11 &
% 72.22/10.22 | $i(v10) & $i(v9)) | ( ~ (v8 = 0) & member(v5, all_61_0) = v8)))
% 72.22/10.22 | & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i]
% 72.22/10.22 | : ! [v5: $i] : ! [v6: $i] : ( ~ (rotate(v0) = v6) | ~
% 72.22/10.22 | (ordered_pair(v4, v3) = v5) | ~ (ordered_pair(v1, v2) = v4) | ~
% 72.22/10.22 | (member(v5, v6) = 0) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0)
% 72.22/10.22 | | ? [v7: $i] : ? [v8: $i] : (ordered_pair(v7, v1) = v8 &
% 72.22/10.22 | ordered_pair(v2, v3) = v7 & member(v8, v0) = 0 & member(v5,
% 72.22/10.22 | all_61_0) = 0 & $i(v8) & $i(v7))) & ! [v0: $i] : ! [v1: $i] :
% 72.22/10.22 | ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: any] :
% 72.22/10.22 | ( ~ (ordered_pair(v4, v1) = v5) | ~ (ordered_pair(v2, v3) = v4) | ~
% 72.22/10.22 | (member(v5, v0) = v6) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~
% 72.22/10.22 | $i(v0) | ? [v7: $i] : ? [v8: $i] : ? [v9: $i] : ? [v10: int] :
% 72.22/10.22 | ? [v11: int] : (ordered_pair(v7, v3) = v8 & ordered_pair(v1, v2) =
% 72.22/10.22 | v7 & $i(v8) & $i(v7) & ((v11 = 0 & v6 = 0 & member(v8, all_61_0) =
% 72.22/10.22 | 0) | ( ~ (v10 = 0) & rotate(v0) = v9 & member(v8, v9) = v10 &
% 72.22/10.22 | $i(v9))))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3:
% 72.22/10.22 | $i] : ! [v4: $i] : ! [v5: $i] : ( ~ (ordered_pair(v4, v1) = v5) |
% 72.22/10.22 | ~ (ordered_pair(v2, v3) = v4) | ~ (member(v5, v0) = 0) | ~ $i(v3)
% 72.22/10.22 | | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v6: $i] : ? [v7: $i] : ?
% 72.22/10.22 | [v8: int] : ? [v9: $i] : ? [v10: int] : (ordered_pair(v6, v3) = v7
% 72.22/10.22 | & ordered_pair(v1, v2) = v6 & $i(v7) & $i(v6) & ((v10 = 0 &
% 72.22/10.22 | rotate(v0) = v9 & member(v7, v9) = 0 & $i(v9)) | ( ~ (v8 = 0)
% 72.22/10.22 | & member(v7, all_61_0) = v8))))
% 72.22/10.22 |
% 72.22/10.22 | ALPHA: (54) implies:
% 72.22/10.22 | (55) cross_product(universal_class, universal_class) = all_61_1
% 72.22/10.22 |
% 72.22/10.22 | DELTA: instantiating (10) with fresh symbols all_64_0, all_64_1 gives:
% 72.22/10.22 | (56) cross_product(all_64_1, universal_class) = all_64_0 &
% 72.22/10.22 | cross_product(universal_class, universal_class) = all_64_1 &
% 72.22/10.22 | $i(all_64_0) & $i(all_64_1) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i]
% 72.22/10.22 | : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: $i] : ! [v7: int]
% 72.22/10.22 | : (v7 = 0 | ~ (flip(v3) = v6) | ~ (ordered_pair(v4, v2) = v5) | ~
% 72.22/10.22 | (ordered_pair(v0, v1) = v4) | ~ (member(v5, v6) = v7) | ~ $i(v3) |
% 72.22/10.22 | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v8: int] : ? [v9: $i] : ?
% 72.22/10.22 | [v10: $i] : ? [v11: int] : (( ~ (v11 = 0) & ordered_pair(v9, v2) =
% 72.22/10.22 | v10 & ordered_pair(v1, v0) = v9 & member(v10, v3) = v11 &
% 72.22/10.22 | $i(v10) & $i(v9)) | ( ~ (v8 = 0) & member(v5, all_64_0) = v8)))
% 72.22/10.22 | & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ! [v4: $i]
% 72.22/10.22 | : ! [v5: $i] : ! [v6: $i] : ( ~ (flip(v3) = v6) | ~
% 72.22/10.22 | (ordered_pair(v4, v2) = v5) | ~ (ordered_pair(v0, v1) = v4) | ~
% 72.22/10.22 | (member(v5, v6) = 0) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~ $i(v0)
% 72.22/10.22 | | ? [v7: $i] : ? [v8: $i] : (ordered_pair(v7, v2) = v8 &
% 72.22/10.22 | ordered_pair(v1, v0) = v7 & member(v8, v3) = 0 & member(v5,
% 72.22/10.22 | all_64_0) = 0 & $i(v8) & $i(v7))) & ! [v0: $i] : ! [v1: $i] :
% 72.22/10.22 | ! [v2: $i] : ! [v3: $i] : ! [v4: $i] : ! [v5: $i] : ! [v6: any] :
% 72.22/10.22 | ( ~ (ordered_pair(v4, v2) = v5) | ~ (ordered_pair(v1, v0) = v4) | ~
% 72.22/10.22 | (member(v5, v3) = v6) | ~ $i(v3) | ~ $i(v2) | ~ $i(v1) | ~
% 72.22/10.22 | $i(v0) | ? [v7: $i] : ? [v8: $i] : ? [v9: $i] : ? [v10: int] :
% 72.22/10.22 | ? [v11: int] : (ordered_pair(v7, v2) = v8 & ordered_pair(v0, v1) =
% 72.22/10.22 | v7 & $i(v8) & $i(v7) & ((v11 = 0 & v6 = 0 & member(v8, all_64_0) =
% 72.22/10.22 | 0) | ( ~ (v10 = 0) & flip(v3) = v9 & member(v8, v9) = v10 &
% 72.22/10.22 | $i(v9))))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3:
% 72.22/10.22 | $i] : ! [v4: $i] : ! [v5: $i] : ( ~ (ordered_pair(v4, v2) = v5) |
% 72.22/10.22 | ~ (ordered_pair(v1, v0) = v4) | ~ (member(v5, v3) = 0) | ~ $i(v3)
% 72.22/10.22 | | ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | ? [v6: $i] : ? [v7: $i] : ?
% 72.22/10.22 | [v8: int] : ? [v9: $i] : ? [v10: int] : (ordered_pair(v6, v2) = v7
% 72.22/10.22 | & ordered_pair(v0, v1) = v6 & $i(v7) & $i(v6) & ((v10 = 0 &
% 72.22/10.22 | flip(v3) = v9 & member(v7, v9) = 0 & $i(v9)) | ( ~ (v8 = 0) &
% 72.22/10.22 | member(v7, all_64_0) = v8))))
% 72.22/10.22 |
% 72.22/10.22 | ALPHA: (56) implies:
% 72.22/10.22 | (57) cross_product(universal_class, universal_class) = all_64_1
% 72.22/10.22 |
% 72.22/10.22 | GROUND_INST: instantiating (27) with all_40_0, all_46_1, universal_class,
% 72.22/10.22 | universal_class, simplifying with (36), (41) gives:
% 72.22/10.22 | (58) all_46_1 = all_40_0
% 72.22/10.22 |
% 72.22/10.22 | GROUND_INST: instantiating (27) with all_46_1, all_49_1, universal_class,
% 72.22/10.22 | universal_class, simplifying with (41), (43) gives:
% 72.22/10.22 | (59) all_49_1 = all_46_1
% 72.22/10.22 |
% 72.22/10.22 | GROUND_INST: instantiating (27) with all_46_1, all_58_0, universal_class,
% 72.22/10.22 | universal_class, simplifying with (41), (52) gives:
% 72.22/10.22 | (60) all_58_0 = all_46_1
% 72.22/10.22 |
% 72.22/10.22 | GROUND_INST: instantiating (27) with all_42_0, all_58_0, universal_class,
% 72.22/10.22 | universal_class, simplifying with (38), (52) gives:
% 72.22/10.22 | (61) all_58_0 = all_42_0
% 72.22/10.22 |
% 72.22/10.22 | GROUND_INST: instantiating (27) with all_49_1, all_61_1, universal_class,
% 72.22/10.22 | universal_class, simplifying with (43), (55) gives:
% 72.22/10.22 | (62) all_61_1 = all_49_1
% 72.22/10.22 |
% 72.22/10.22 | GROUND_INST: instantiating (27) with all_61_1, all_64_1, universal_class,
% 72.22/10.22 | universal_class, simplifying with (55), (57) gives:
% 72.22/10.22 | (63) all_64_1 = all_61_1
% 72.22/10.22 |
% 72.22/10.22 | GROUND_INST: instantiating (27) with all_38_0, all_64_1, universal_class,
% 72.22/10.22 | universal_class, simplifying with (32), (57) gives:
% 72.22/10.22 | (64) all_64_1 = all_38_0
% 72.22/10.22 |
% 72.22/10.22 | COMBINE_EQS: (63), (64) imply:
% 72.22/10.22 | (65) all_61_1 = all_38_0
% 72.22/10.22 |
% 72.22/10.22 | SIMP: (65) implies:
% 72.22/10.22 | (66) all_61_1 = all_38_0
% 72.22/10.22 |
% 72.22/10.22 | COMBINE_EQS: (62), (66) imply:
% 72.22/10.22 | (67) all_49_1 = all_38_0
% 72.22/10.22 |
% 72.22/10.22 | SIMP: (67) implies:
% 72.22/10.22 | (68) all_49_1 = all_38_0
% 72.22/10.22 |
% 72.22/10.22 | COMBINE_EQS: (60), (61) imply:
% 72.22/10.22 | (69) all_46_1 = all_42_0
% 72.22/10.22 |
% 72.22/10.22 | SIMP: (69) implies:
% 72.22/10.22 | (70) all_46_1 = all_42_0
% 72.22/10.22 |
% 72.22/10.22 | COMBINE_EQS: (59), (68) imply:
% 72.22/10.22 | (71) all_46_1 = all_38_0
% 72.22/10.22 |
% 72.22/10.22 | SIMP: (71) implies:
% 72.22/10.22 | (72) all_46_1 = all_38_0
% 72.22/10.22 |
% 72.22/10.22 | COMBINE_EQS: (70), (72) imply:
% 72.22/10.22 | (73) all_42_0 = all_38_0
% 72.22/10.22 |
% 72.22/10.22 | COMBINE_EQS: (58), (70) imply:
% 72.22/10.22 | (74) all_42_0 = all_40_0
% 72.22/10.22 |
% 72.22/10.22 | COMBINE_EQS: (73), (74) imply:
% 72.22/10.22 | (75) all_40_0 = all_38_0
% 72.22/10.22 |
% 72.22/10.22 | SIMP: (75) implies:
% 72.22/10.22 | (76) all_40_0 = all_38_0
% 72.22/10.22 |
% 72.22/10.22 | REDUCE: (35), (76) imply:
% 72.22/10.22 | (77) subclass(element_relation, all_38_0) = 0
% 72.22/10.22 |
% 72.22/10.22 | REDUCE: (34), (76) imply:
% 72.22/10.22 | (78) $i(all_38_0)
% 72.22/10.22 |
% 72.22/10.22 | GROUND_INST: instantiating (7) with universal_class, null_class, all_36_0,
% 72.22/10.22 | simplifying with (21), (22), (30) gives:
% 72.22/10.22 | (79) all_36_0 = 0 | ? [v0: int] : ? [v1: $i] : ? [v2: int] : ((v2 = 0 &
% 72.22/10.22 | complement(universal_class) = v1 & member(null_class, v1) = 0 &
% 72.22/10.22 | $i(v1)) | ( ~ (v0 = 0) & member(null_class, universal_class) =
% 72.22/10.22 | v0))
% 72.22/10.23 |
% 72.22/10.23 | GROUND_INST: instantiating (6) with universal_class, null_class, all_36_0,
% 72.22/10.23 | simplifying with (21), (22), (30) gives:
% 72.22/10.23 | (80) ? [v0: $i] : ? [v1: int] : ? [v2: int] : ((v2 = 0 & ~ (all_36_0 =
% 72.22/10.23 | 0) & member(null_class, universal_class) = 0) | ( ~ (v1 = 0) &
% 72.22/10.23 | complement(universal_class) = v0 & member(null_class, v0) = v1 &
% 72.22/10.23 | $i(v0)))
% 72.22/10.23 |
% 72.22/10.23 | GROUND_INST: instantiating (6) with universal_class, all_52_0, 0, simplifying
% 72.22/10.23 | with (21), (45), (46) gives:
% 72.22/10.23 | (81) ? [v0: $i] : ? [v1: int] : ( ~ (v1 = 0) &
% 72.22/10.23 | complement(universal_class) = v0 & member(all_52_0, v0) = v1 &
% 72.22/10.23 | $i(v0))
% 72.22/10.23 |
% 72.22/10.23 | GROUND_INST: instantiating (13) with all_52_0, simplifying with (45), (47)
% 72.22/10.23 | gives:
% 72.22/10.23 | (82) ? [v0: $i] : (image(successor_relation, all_52_0) = v0 & subclass(v0,
% 72.22/10.23 | all_52_0) = 0 & member(null_class, all_52_0) = 0 & $i(v0))
% 72.22/10.23 |
% 72.22/10.23 | GROUND_INST: instantiating (53) with all_55_0, simplifying with (49), (50)
% 72.22/10.23 | gives:
% 72.22/10.23 | (83) ? [v0: $i] : ? [v1: $i] : (compose(all_55_0, v0) = v1 &
% 72.22/10.23 | inverse(all_55_0) = v0 & subclass(v1, identity_relation) = 0 &
% 72.22/10.23 | subclass(all_55_0, all_58_0) = 0 & $i(v1) & $i(v0))
% 72.22/10.23 |
% 72.22/10.23 | DELTA: instantiating (81) with fresh symbols all_85_0, all_85_1 gives:
% 72.22/10.23 | (84) ~ (all_85_0 = 0) & complement(universal_class) = all_85_1 &
% 72.22/10.23 | member(all_52_0, all_85_1) = all_85_0 & $i(all_85_1)
% 72.22/10.23 |
% 72.22/10.23 | ALPHA: (84) implies:
% 72.22/10.23 | (85) complement(universal_class) = all_85_1
% 72.22/10.23 |
% 72.22/10.23 | DELTA: instantiating (82) with fresh symbol all_87_0 gives:
% 72.22/10.23 | (86) image(successor_relation, all_52_0) = all_87_0 & subclass(all_87_0,
% 72.22/10.23 | all_52_0) = 0 & member(null_class, all_52_0) = 0 & $i(all_87_0)
% 72.22/10.23 |
% 72.22/10.23 | ALPHA: (86) implies:
% 72.22/10.23 | (87) member(null_class, all_52_0) = 0
% 72.22/10.23 |
% 72.22/10.23 | DELTA: instantiating (83) with fresh symbols all_96_0, all_96_1 gives:
% 72.22/10.23 | (88) compose(all_55_0, all_96_1) = all_96_0 & inverse(all_55_0) = all_96_1
% 72.22/10.23 | & subclass(all_96_0, identity_relation) = 0 & subclass(all_55_0,
% 72.22/10.23 | all_58_0) = 0 & $i(all_96_0) & $i(all_96_1)
% 72.22/10.23 |
% 72.22/10.23 | ALPHA: (88) implies:
% 72.22/10.23 | (89) $i(all_96_1)
% 72.22/10.23 | (90) $i(all_96_0)
% 72.22/10.23 | (91) subclass(all_96_0, identity_relation) = 0
% 72.22/10.23 | (92) compose(all_55_0, all_96_1) = all_96_0
% 72.22/10.23 |
% 72.22/10.23 | DELTA: instantiating (80) with fresh symbols all_101_0, all_101_1, all_101_2
% 72.22/10.23 | gives:
% 72.22/10.23 | (93) (all_101_0 = 0 & ~ (all_36_0 = 0) & member(null_class,
% 72.22/10.23 | universal_class) = 0) | ( ~ (all_101_1 = 0) &
% 72.22/10.23 | complement(universal_class) = all_101_2 & member(null_class,
% 72.22/10.23 | all_101_2) = all_101_1 & $i(all_101_2))
% 72.22/10.23 |
% 72.22/10.23 | BETA: splitting (79) gives:
% 72.22/10.23 |
% 72.22/10.23 | Case 1:
% 72.22/10.23 | |
% 72.22/10.23 | | (94) all_36_0 = 0
% 72.22/10.23 | |
% 72.22/10.23 | | REDUCE: (29), (94) imply:
% 72.22/10.23 | | (95) $false
% 72.22/10.23 | |
% 72.22/10.23 | | CLOSE: (95) is inconsistent.
% 72.22/10.23 | |
% 72.22/10.23 | Case 2:
% 72.22/10.23 | |
% 72.22/10.23 | | (96) ? [v0: int] : ? [v1: $i] : ? [v2: int] : ((v2 = 0 &
% 72.22/10.23 | | complement(universal_class) = v1 & member(null_class, v1) = 0 &
% 72.22/10.23 | | $i(v1)) | ( ~ (v0 = 0) & member(null_class, universal_class) =
% 72.22/10.23 | | v0))
% 72.22/10.23 | |
% 72.22/10.23 | | DELTA: instantiating (96) with fresh symbols all_111_0, all_111_1, all_111_2
% 72.22/10.23 | | gives:
% 72.22/10.23 | | (97) (all_111_0 = 0 & complement(universal_class) = all_111_1 &
% 72.22/10.23 | | member(null_class, all_111_1) = 0 & $i(all_111_1)) | ( ~
% 72.22/10.23 | | (all_111_2 = 0) & member(null_class, universal_class) = all_111_2)
% 72.22/10.23 | |
% 72.22/10.23 | | BETA: splitting (93) gives:
% 72.22/10.23 | |
% 72.22/10.23 | | Case 1:
% 72.22/10.23 | | |
% 72.22/10.23 | | | (98) all_101_0 = 0 & ~ (all_36_0 = 0) & member(null_class,
% 72.22/10.23 | | | universal_class) = 0
% 72.22/10.23 | | |
% 72.22/10.23 | | | ALPHA: (98) implies:
% 72.22/10.23 | | | (99) member(null_class, universal_class) = 0
% 72.22/10.23 | | |
% 72.22/10.23 | | | REF_CLOSE: (26), (29), (30), (99) are inconsistent by sub-proof #1.
% 72.22/10.23 | | |
% 72.22/10.23 | | Case 2:
% 72.22/10.23 | | |
% 72.22/10.23 | | | (100) ~ (all_101_1 = 0) & complement(universal_class) = all_101_2 &
% 72.22/10.23 | | | member(null_class, all_101_2) = all_101_1 & $i(all_101_2)
% 72.22/10.23 | | |
% 72.22/10.23 | | | ALPHA: (100) implies:
% 72.22/10.23 | | | (101) ~ (all_101_1 = 0)
% 72.22/10.23 | | | (102) $i(all_101_2)
% 72.22/10.23 | | | (103) member(null_class, all_101_2) = all_101_1
% 72.22/10.23 | | | (104) complement(universal_class) = all_101_2
% 72.22/10.23 | | |
% 72.22/10.23 | | | GROUND_INST: instantiating (24) with all_85_1, all_101_2, universal_class,
% 72.22/10.23 | | | simplifying with (85), (104) gives:
% 72.22/10.23 | | | (105) all_101_2 = all_85_1
% 72.22/10.23 | | |
% 72.22/10.23 | | | REDUCE: (103), (105) imply:
% 72.22/10.23 | | | (106) member(null_class, all_85_1) = all_101_1
% 72.22/10.23 | | |
% 72.22/10.23 | | | REDUCE: (102), (105) imply:
% 72.22/10.23 | | | (107) $i(all_85_1)
% 72.22/10.23 | | |
% 72.22/10.23 | | | BETA: splitting (97) gives:
% 72.22/10.23 | | |
% 72.22/10.23 | | | Case 1:
% 72.22/10.23 | | | |
% 72.22/10.23 | | | | (108) all_111_0 = 0 & complement(universal_class) = all_111_1 &
% 72.22/10.23 | | | | member(null_class, all_111_1) = 0 & $i(all_111_1)
% 72.22/10.23 | | | |
% 72.22/10.23 | | | | ALPHA: (108) implies:
% 72.22/10.23 | | | | (109) member(null_class, all_111_1) = 0
% 72.22/10.23 | | | | (110) complement(universal_class) = all_111_1
% 72.22/10.23 | | | |
% 72.22/10.23 | | | | GROUND_INST: instantiating (24) with all_85_1, all_111_1,
% 72.22/10.23 | | | | universal_class, simplifying with (85), (110) gives:
% 72.22/10.23 | | | | (111) all_111_1 = all_85_1
% 72.22/10.23 | | | |
% 72.22/10.23 | | | | REDUCE: (109), (111) imply:
% 72.22/10.23 | | | | (112) member(null_class, all_85_1) = 0
% 72.22/10.23 | | | |
% 72.22/10.23 | | | | GROUND_INST: instantiating (26) with all_101_1, 0, all_85_1, null_class,
% 72.22/10.23 | | | | simplifying with (106), (112) gives:
% 72.22/10.23 | | | | (113) all_101_1 = 0
% 72.22/10.23 | | | |
% 72.22/10.23 | | | | REDUCE: (101), (113) imply:
% 72.22/10.23 | | | | (114) $false
% 72.22/10.23 | | | |
% 72.22/10.23 | | | | CLOSE: (114) is inconsistent.
% 72.22/10.23 | | | |
% 72.22/10.23 | | | Case 2:
% 72.22/10.23 | | | |
% 72.22/10.23 | | | | (115) ~ (all_111_2 = 0) & member(null_class, universal_class) =
% 72.22/10.23 | | | | all_111_2
% 72.22/10.23 | | | |
% 72.22/10.23 | | | | ALPHA: (115) implies:
% 72.22/10.23 | | | | (116) ~ (all_111_2 = 0)
% 72.22/10.23 | | | | (117) member(null_class, universal_class) = all_111_2
% 72.22/10.23 | | | |
% 72.22/10.23 | | | | GROUND_INST: instantiating (26) with all_36_0, all_111_2,
% 72.22/10.23 | | | | universal_class, null_class, simplifying with (30), (117)
% 72.22/10.23 | | | | gives:
% 72.22/10.24 | | | | (118) all_111_2 = all_36_0
% 72.22/10.24 | | | |
% 72.22/10.24 | | | | GROUND_INST: instantiating (3) with null_class, all_52_0, simplifying
% 72.22/10.24 | | | | with (22), (45), (87) gives:
% 72.22/10.24 | | | | (119) ? [v0: int] : ? [v1: $i] : ? [v2: int] : ((v2 = 0 &
% 72.22/10.24 | | | | ordered_pair(null_class, all_52_0) = v1 & member(v1,
% 72.22/10.24 | | | | element_relation) = 0 & $i(v1)) | ( ~ (v0 = 0) &
% 72.22/10.24 | | | | member(all_52_0, universal_class) = v0))
% 72.22/10.24 | | | |
% 72.22/10.24 | | | | GROUND_INST: instantiating (7) with all_85_1, null_class, all_101_1,
% 72.22/10.24 | | | | simplifying with (22), (106), (107) gives:
% 72.22/10.24 | | | | (120) all_101_1 = 0 | ? [v0: int] : ? [v1: $i] : ? [v2: int] :
% 72.22/10.24 | | | | ((v2 = 0 & complement(all_85_1) = v1 & member(null_class, v1) =
% 72.22/10.24 | | | | 0 & $i(v1)) | ( ~ (v0 = 0) & member(null_class,
% 72.22/10.24 | | | | universal_class) = v0))
% 72.22/10.24 | | | |
% 72.22/10.24 | | | | GROUND_INST: instantiating (6) with all_85_1, null_class, all_101_1,
% 72.22/10.24 | | | | simplifying with (22), (106), (107) gives:
% 72.22/10.24 | | | | (121) ? [v0: $i] : ? [v1: int] : ? [v2: int] : ((v2 = 0 & ~
% 72.22/10.24 | | | | (all_101_1 = 0) & member(null_class, universal_class) = 0)
% 72.22/10.24 | | | | | ( ~ (v1 = 0) & complement(all_85_1) = v0 &
% 72.22/10.24 | | | | member(null_class, v0) = v1 & $i(v0)))
% 72.22/10.24 | | | |
% 72.22/10.24 | | | | GROUND_INST: instantiating (15) with all_96_0, identity_relation,
% 72.22/10.24 | | | | simplifying with (18), (90), (91) gives:
% 72.22/10.24 | | | | (122) ? [v0: int] : ? [v1: $i] : ? [v2: int] : ((v2 = 0 &
% 72.22/10.24 | | | | power_class(identity_relation) = v1 & member(all_96_0, v1)
% 72.22/10.24 | | | | = 0 & $i(v1)) | ( ~ (v0 = 0) & member(all_96_0,
% 72.22/10.24 | | | | universal_class) = v0))
% 72.22/10.24 | | | |
% 72.22/10.24 | | | | GROUND_INST: instantiating (16) with all_96_0, identity_relation, 0,
% 72.22/10.24 | | | | simplifying with (18), (90), (91) gives:
% 72.22/10.24 | | | | (123) ? [v0: $i] : ? [v1: int] : ? [v2: int] : ((v2 = 0 &
% 72.22/10.24 | | | | member(all_96_0, universal_class) = 0) | ( ~ (v1 = 0) &
% 72.22/10.24 | | | | power_class(identity_relation) = v0 & member(all_96_0, v0)
% 72.22/10.24 | | | | = v1 & $i(v0)))
% 72.22/10.24 | | | |
% 72.22/10.24 | | | | GROUND_INST: instantiating (39) with all_96_1, all_55_0, all_96_0,
% 72.22/10.24 | | | | simplifying with (49), (89), (92) gives:
% 72.22/10.24 | | | | (124) subclass(all_96_0, all_42_0) = 0
% 72.22/10.24 | | | |
% 72.22/10.24 | | | | DELTA: instantiating (119) with fresh symbols all_161_0, all_161_1,
% 72.22/10.24 | | | | all_161_2 gives:
% 72.22/10.24 | | | | (125) (all_161_0 = 0 & ordered_pair(null_class, all_52_0) = all_161_1
% 72.22/10.24 | | | | & member(all_161_1, element_relation) = 0 & $i(all_161_1)) |
% 72.22/10.24 | | | | ( ~ (all_161_2 = 0) & member(all_52_0, universal_class) =
% 72.22/10.24 | | | | all_161_2)
% 72.22/10.24 | | | |
% 72.22/10.24 | | | | DELTA: instantiating (123) with fresh symbols all_168_0, all_168_1,
% 72.22/10.24 | | | | all_168_2 gives:
% 72.22/10.24 | | | | (126) (all_168_0 = 0 & member(all_96_0, universal_class) = 0) | ( ~
% 72.22/10.24 | | | | (all_168_1 = 0) & power_class(identity_relation) = all_168_2
% 72.22/10.24 | | | | & member(all_96_0, all_168_2) = all_168_1 & $i(all_168_2))
% 72.22/10.24 | | | |
% 72.22/10.24 | | | | DELTA: instantiating (122) with fresh symbols all_176_0, all_176_1,
% 72.22/10.24 | | | | all_176_2 gives:
% 72.22/10.24 | | | | (127) (all_176_0 = 0 & power_class(identity_relation) = all_176_1 &
% 72.22/10.24 | | | | member(all_96_0, all_176_1) = 0 & $i(all_176_1)) | ( ~
% 72.22/10.24 | | | | (all_176_2 = 0) & member(all_96_0, universal_class) =
% 72.22/10.24 | | | | all_176_2)
% 72.22/10.24 | | | |
% 72.22/10.24 | | | | DELTA: instantiating (121) with fresh symbols all_183_0, all_183_1,
% 72.22/10.24 | | | | all_183_2 gives:
% 72.22/10.24 | | | | (128) (all_183_0 = 0 & ~ (all_101_1 = 0) & member(null_class,
% 72.22/10.24 | | | | universal_class) = 0) | ( ~ (all_183_1 = 0) &
% 72.22/10.24 | | | | complement(all_85_1) = all_183_2 & member(null_class,
% 72.22/10.24 | | | | all_183_2) = all_183_1 & $i(all_183_2))
% 72.22/10.24 | | | |
% 72.22/10.24 | | | | REDUCE: (73), (124) imply:
% 72.22/10.24 | | | | (129) subclass(all_96_0, all_38_0) = 0
% 72.22/10.24 | | | |
% 72.22/10.24 | | | | BETA: splitting (125) gives:
% 72.22/10.24 | | | |
% 72.22/10.24 | | | | Case 1:
% 72.22/10.24 | | | | |
% 72.22/10.24 | | | | | (130) all_161_0 = 0 & ordered_pair(null_class, all_52_0) =
% 72.22/10.24 | | | | | all_161_1 & member(all_161_1, element_relation) = 0 &
% 72.22/10.24 | | | | | $i(all_161_1)
% 72.22/10.24 | | | | |
% 72.22/10.24 | | | | | ALPHA: (130) implies:
% 72.22/10.24 | | | | | (131) $i(all_161_1)
% 72.22/10.24 | | | | | (132) member(all_161_1, element_relation) = 0
% 72.22/10.24 | | | | | (133) ordered_pair(null_class, all_52_0) = all_161_1
% 72.22/10.24 | | | | |
% 72.22/10.24 | | | | | BETA: splitting (120) gives:
% 72.22/10.24 | | | | |
% 72.22/10.24 | | | | | Case 1:
% 72.22/10.24 | | | | | |
% 72.22/10.24 | | | | | | (134) all_101_1 = 0
% 72.22/10.24 | | | | | |
% 72.22/10.24 | | | | | | REDUCE: (101), (134) imply:
% 72.22/10.24 | | | | | | (135) $false
% 72.22/10.24 | | | | | |
% 72.22/10.24 | | | | | | CLOSE: (135) is inconsistent.
% 72.22/10.24 | | | | | |
% 72.22/10.24 | | | | | Case 2:
% 72.22/10.24 | | | | | |
% 72.22/10.24 | | | | | | (136) ? [v0: int] : ? [v1: $i] : ? [v2: int] : ((v2 = 0 &
% 72.22/10.24 | | | | | | complement(all_85_1) = v1 & member(null_class, v1) = 0
% 72.22/10.24 | | | | | | & $i(v1)) | ( ~ (v0 = 0) & member(null_class,
% 72.22/10.24 | | | | | | universal_class) = v0))
% 72.22/10.24 | | | | | |
% 72.22/10.24 | | | | | | DELTA: instantiating (136) with fresh symbols all_209_0, all_209_1,
% 72.22/10.24 | | | | | | all_209_2 gives:
% 72.22/10.24 | | | | | | (137) (all_209_0 = 0 & complement(all_85_1) = all_209_1 &
% 72.22/10.24 | | | | | | member(null_class, all_209_1) = 0 & $i(all_209_1)) | ( ~
% 72.22/10.24 | | | | | | (all_209_2 = 0) & member(null_class, universal_class) =
% 72.22/10.24 | | | | | | all_209_2)
% 72.22/10.24 | | | | | |
% 72.22/10.24 | | | | | | BETA: splitting (128) gives:
% 72.22/10.24 | | | | | |
% 72.22/10.24 | | | | | | Case 1:
% 72.22/10.24 | | | | | | |
% 72.22/10.24 | | | | | | | (138) all_183_0 = 0 & ~ (all_101_1 = 0) & member(null_class,
% 72.22/10.24 | | | | | | | universal_class) = 0
% 72.22/10.24 | | | | | | |
% 72.22/10.24 | | | | | | | ALPHA: (138) implies:
% 72.22/10.24 | | | | | | | (139) member(null_class, universal_class) = 0
% 72.22/10.24 | | | | | | |
% 72.22/10.24 | | | | | | | REF_CLOSE: (26), (29), (30), (139) are inconsistent by sub-proof
% 72.22/10.24 | | | | | | | #1.
% 72.22/10.24 | | | | | | |
% 72.22/10.24 | | | | | | Case 2:
% 72.22/10.24 | | | | | | |
% 72.22/10.24 | | | | | | | (140) ~ (all_183_1 = 0) & complement(all_85_1) = all_183_2 &
% 72.22/10.24 | | | | | | | member(null_class, all_183_2) = all_183_1 & $i(all_183_2)
% 72.22/10.24 | | | | | | |
% 72.22/10.24 | | | | | | | ALPHA: (140) implies:
% 72.22/10.24 | | | | | | | (141) ~ (all_183_1 = 0)
% 72.22/10.24 | | | | | | | (142) member(null_class, all_183_2) = all_183_1
% 72.22/10.24 | | | | | | | (143) complement(all_85_1) = all_183_2
% 72.22/10.24 | | | | | | |
% 72.22/10.24 | | | | | | | BETA: splitting (137) gives:
% 72.22/10.24 | | | | | | |
% 72.22/10.24 | | | | | | | Case 1:
% 72.22/10.24 | | | | | | | |
% 72.22/10.24 | | | | | | | | (144) all_209_0 = 0 & complement(all_85_1) = all_209_1 &
% 72.22/10.24 | | | | | | | | member(null_class, all_209_1) = 0 & $i(all_209_1)
% 72.22/10.24 | | | | | | | |
% 72.22/10.24 | | | | | | | | ALPHA: (144) implies:
% 72.22/10.24 | | | | | | | | (145) member(null_class, all_209_1) = 0
% 72.22/10.24 | | | | | | | | (146) complement(all_85_1) = all_209_1
% 72.22/10.24 | | | | | | | |
% 72.22/10.24 | | | | | | | | GROUND_INST: instantiating (24) with all_183_2, all_209_1,
% 72.22/10.24 | | | | | | | | all_85_1, simplifying with (143), (146) gives:
% 72.22/10.24 | | | | | | | | (147) all_209_1 = all_183_2
% 72.22/10.24 | | | | | | | |
% 72.22/10.24 | | | | | | | | REDUCE: (145), (147) imply:
% 72.22/10.24 | | | | | | | | (148) member(null_class, all_183_2) = 0
% 72.22/10.24 | | | | | | | |
% 72.22/10.24 | | | | | | | | GROUND_INST: instantiating (26) with all_183_1, 0, all_183_2,
% 72.22/10.24 | | | | | | | | null_class, simplifying with (142), (148) gives:
% 72.22/10.24 | | | | | | | | (149) all_183_1 = 0
% 72.22/10.24 | | | | | | | |
% 72.22/10.24 | | | | | | | | REDUCE: (141), (149) imply:
% 72.22/10.24 | | | | | | | | (150) $false
% 72.22/10.24 | | | | | | | |
% 72.22/10.24 | | | | | | | | CLOSE: (150) is inconsistent.
% 72.22/10.24 | | | | | | | |
% 72.22/10.24 | | | | | | | Case 2:
% 72.22/10.24 | | | | | | | |
% 72.22/10.24 | | | | | | | | (151) ~ (all_209_2 = 0) & member(null_class,
% 72.22/10.24 | | | | | | | | universal_class) = all_209_2
% 72.22/10.24 | | | | | | | |
% 72.22/10.24 | | | | | | | | ALPHA: (151) implies:
% 72.22/10.24 | | | | | | | | (152) ~ (all_209_2 = 0)
% 72.22/10.24 | | | | | | | | (153) member(null_class, universal_class) = all_209_2
% 72.22/10.24 | | | | | | | |
% 72.22/10.24 | | | | | | | | GROUND_INST: instantiating (26) with all_36_0, all_209_2,
% 72.22/10.24 | | | | | | | | universal_class, null_class, simplifying with (30),
% 72.22/10.24 | | | | | | | | (153) gives:
% 72.22/10.24 | | | | | | | | (154) all_209_2 = all_36_0
% 72.22/10.24 | | | | | | | |
% 72.22/10.24 | | | | | | | | GROUND_INST: instantiating (1) with element_relation, all_38_0,
% 72.22/10.24 | | | | | | | | all_161_1, simplifying with (4), (77), (78), (131),
% 72.22/10.24 | | | | | | | | (132) gives:
% 72.22/10.24 | | | | | | | | (155) member(all_161_1, all_38_0) = 0
% 72.22/10.24 | | | | | | | |
% 72.22/10.24 | | | | | | | | GROUND_INST: instantiating (15) with all_96_0, all_38_0,
% 72.22/10.24 | | | | | | | | simplifying with (78), (90), (129) gives:
% 72.22/10.24 | | | | | | | | (156) ? [v0: int] : ? [v1: $i] : ? [v2: int] : ((v2 = 0 &
% 72.22/10.24 | | | | | | | | power_class(all_38_0) = v1 & member(all_96_0, v1) =
% 72.22/10.24 | | | | | | | | 0 & $i(v1)) | ( ~ (v0 = 0) & member(all_96_0,
% 72.22/10.24 | | | | | | | | universal_class) = v0))
% 72.22/10.24 | | | | | | | |
% 72.22/10.24 | | | | | | | | GROUND_INST: instantiating (16) with all_96_0, all_38_0, 0,
% 72.22/10.24 | | | | | | | | simplifying with (78), (90), (129) gives:
% 72.22/10.25 | | | | | | | | (157) ? [v0: $i] : ? [v1: int] : ? [v2: int] : ((v2 = 0 &
% 72.22/10.25 | | | | | | | | member(all_96_0, universal_class) = 0) | ( ~ (v1 =
% 72.22/10.25 | | | | | | | | 0) & power_class(all_38_0) = v0 &
% 72.22/10.25 | | | | | | | | member(all_96_0, v0) = v1 & $i(v0)))
% 72.22/10.25 | | | | | | | |
% 72.22/10.25 | | | | | | | | DELTA: instantiating (157) with fresh symbols all_308_0,
% 72.22/10.25 | | | | | | | | all_308_1, all_308_2 gives:
% 72.22/10.25 | | | | | | | | (158) (all_308_0 = 0 & member(all_96_0, universal_class) = 0)
% 72.22/10.25 | | | | | | | | | ( ~ (all_308_1 = 0) & power_class(all_38_0) =
% 72.22/10.25 | | | | | | | | all_308_2 & member(all_96_0, all_308_2) = all_308_1 &
% 72.22/10.25 | | | | | | | | $i(all_308_2))
% 72.22/10.25 | | | | | | | |
% 72.22/10.25 | | | | | | | | DELTA: instantiating (156) with fresh symbols all_311_0,
% 72.22/10.25 | | | | | | | | all_311_1, all_311_2 gives:
% 72.22/10.25 | | | | | | | | (159) (all_311_0 = 0 & power_class(all_38_0) = all_311_1 &
% 72.22/10.25 | | | | | | | | member(all_96_0, all_311_1) = 0 & $i(all_311_1)) | (
% 72.22/10.25 | | | | | | | | ~ (all_311_2 = 0) & member(all_96_0, universal_class)
% 72.22/10.25 | | | | | | | | = all_311_2)
% 72.22/10.25 | | | | | | | |
% 72.22/10.25 | | | | | | | | BETA: splitting (126) gives:
% 72.22/10.25 | | | | | | | |
% 72.22/10.25 | | | | | | | | Case 1:
% 72.22/10.25 | | | | | | | | |
% 72.22/10.25 | | | | | | | | | (160) all_168_0 = 0 & member(all_96_0, universal_class) = 0
% 72.22/10.25 | | | | | | | | |
% 72.22/10.25 | | | | | | | | | ALPHA: (160) implies:
% 72.22/10.25 | | | | | | | | | (161) member(all_96_0, universal_class) = 0
% 72.22/10.25 | | | | | | | | |
% 72.22/10.25 | | | | | | | | | BETA: splitting (159) gives:
% 72.22/10.25 | | | | | | | | |
% 72.22/10.25 | | | | | | | | | Case 1:
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | GROUND_INST: instantiating (2) with null_class, all_52_0,
% 72.22/10.25 | | | | | | | | | | universal_class, universal_class, all_161_1,
% 72.22/10.25 | | | | | | | | | | all_38_0, simplifying with (21), (22), (32), (45),
% 72.22/10.25 | | | | | | | | | | (133), (155) gives:
% 72.22/10.25 | | | | | | | | | | (162) member(all_52_0, universal_class) = 0 &
% 72.22/10.25 | | | | | | | | | | member(null_class, universal_class) = 0
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | ALPHA: (162) implies:
% 72.22/10.25 | | | | | | | | | | (163) member(null_class, universal_class) = 0
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | REF_CLOSE: (26), (29), (30), (163) are inconsistent by
% 72.22/10.25 | | | | | | | | | | sub-proof #1.
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | Case 2:
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | (164) ~ (all_311_2 = 0) & member(all_96_0,
% 72.22/10.25 | | | | | | | | | | universal_class) = all_311_2
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | ALPHA: (164) implies:
% 72.22/10.25 | | | | | | | | | | (165) ~ (all_311_2 = 0)
% 72.22/10.25 | | | | | | | | | | (166) member(all_96_0, universal_class) = all_311_2
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | GROUND_INST: instantiating (26) with 0, all_311_2,
% 72.22/10.25 | | | | | | | | | | universal_class, all_96_0, simplifying with (161),
% 72.22/10.25 | | | | | | | | | | (166) gives:
% 72.22/10.25 | | | | | | | | | | (167) all_311_2 = 0
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | REDUCE: (165), (167) imply:
% 72.22/10.25 | | | | | | | | | | (168) $false
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | CLOSE: (168) is inconsistent.
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | End of split
% 72.22/10.25 | | | | | | | | |
% 72.22/10.25 | | | | | | | | Case 2:
% 72.22/10.25 | | | | | | | | |
% 72.22/10.25 | | | | | | | | | (169) ~ (all_168_1 = 0) & power_class(identity_relation) =
% 72.22/10.25 | | | | | | | | | all_168_2 & member(all_96_0, all_168_2) = all_168_1 &
% 72.22/10.25 | | | | | | | | | $i(all_168_2)
% 72.22/10.25 | | | | | | | | |
% 72.22/10.25 | | | | | | | | | ALPHA: (169) implies:
% 72.22/10.25 | | | | | | | | | (170) ~ (all_168_1 = 0)
% 72.22/10.25 | | | | | | | | | (171) member(all_96_0, all_168_2) = all_168_1
% 72.22/10.25 | | | | | | | | | (172) power_class(identity_relation) = all_168_2
% 72.22/10.25 | | | | | | | | |
% 72.22/10.25 | | | | | | | | | BETA: splitting (127) gives:
% 72.22/10.25 | | | | | | | | |
% 72.22/10.25 | | | | | | | | | Case 1:
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | (173) all_176_0 = 0 & power_class(identity_relation) =
% 72.22/10.25 | | | | | | | | | | all_176_1 & member(all_96_0, all_176_1) = 0 &
% 72.22/10.25 | | | | | | | | | | $i(all_176_1)
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | ALPHA: (173) implies:
% 72.22/10.25 | | | | | | | | | | (174) member(all_96_0, all_176_1) = 0
% 72.22/10.25 | | | | | | | | | | (175) power_class(identity_relation) = all_176_1
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | GROUND_INST: instantiating (25) with all_168_2, all_176_1,
% 72.22/10.25 | | | | | | | | | | identity_relation, simplifying with (172), (175)
% 72.22/10.25 | | | | | | | | | | gives:
% 72.22/10.25 | | | | | | | | | | (176) all_176_1 = all_168_2
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | REDUCE: (174), (176) imply:
% 72.22/10.25 | | | | | | | | | | (177) member(all_96_0, all_168_2) = 0
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | GROUND_INST: instantiating (26) with all_168_1, 0, all_168_2,
% 72.22/10.25 | | | | | | | | | | all_96_0, simplifying with (171), (177) gives:
% 72.22/10.25 | | | | | | | | | | (178) all_168_1 = 0
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | REDUCE: (170), (178) imply:
% 72.22/10.25 | | | | | | | | | | (179) $false
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | CLOSE: (179) is inconsistent.
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | Case 2:
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | (180) ~ (all_176_2 = 0) & member(all_96_0,
% 72.22/10.25 | | | | | | | | | | universal_class) = all_176_2
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | ALPHA: (180) implies:
% 72.22/10.25 | | | | | | | | | | (181) ~ (all_176_2 = 0)
% 72.22/10.25 | | | | | | | | | | (182) member(all_96_0, universal_class) = all_176_2
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | BETA: splitting (158) gives:
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | Case 1:
% 72.22/10.25 | | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | | (183) all_308_0 = 0 & member(all_96_0, universal_class)
% 72.22/10.25 | | | | | | | | | | | = 0
% 72.22/10.25 | | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | | ALPHA: (183) implies:
% 72.22/10.25 | | | | | | | | | | | (184) member(all_96_0, universal_class) = 0
% 72.22/10.25 | | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | | GROUND_INST: instantiating (26) with 0, all_176_2,
% 72.22/10.25 | | | | | | | | | | | universal_class, all_96_0, simplifying with (182),
% 72.22/10.25 | | | | | | | | | | | (184) gives:
% 72.22/10.25 | | | | | | | | | | | (185) all_176_2 = 0
% 72.22/10.25 | | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | | REDUCE: (181), (185) imply:
% 72.22/10.25 | | | | | | | | | | | (186) $false
% 72.22/10.25 | | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | | CLOSE: (186) is inconsistent.
% 72.22/10.25 | | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | Case 2:
% 72.22/10.25 | | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | | GROUND_INST: instantiating (2) with null_class, all_52_0,
% 72.22/10.25 | | | | | | | | | | | universal_class, universal_class, all_161_1,
% 72.22/10.25 | | | | | | | | | | | all_38_0, simplifying with (21), (22), (32), (45),
% 72.22/10.25 | | | | | | | | | | | (133), (155) gives:
% 72.22/10.25 | | | | | | | | | | | (187) member(all_52_0, universal_class) = 0 &
% 72.22/10.25 | | | | | | | | | | | member(null_class, universal_class) = 0
% 72.22/10.25 | | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | | ALPHA: (187) implies:
% 72.22/10.25 | | | | | | | | | | | (188) member(null_class, universal_class) = 0
% 72.22/10.25 | | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | | REF_CLOSE: (26), (29), (30), (188) are inconsistent by
% 72.22/10.25 | | | | | | | | | | | sub-proof #1.
% 72.22/10.25 | | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | | End of split
% 72.22/10.25 | | | | | | | | | |
% 72.22/10.25 | | | | | | | | | End of split
% 72.22/10.25 | | | | | | | | |
% 72.22/10.25 | | | | | | | | End of split
% 72.22/10.25 | | | | | | | |
% 72.22/10.25 | | | | | | | End of split
% 72.22/10.25 | | | | | | |
% 72.22/10.25 | | | | | | End of split
% 72.22/10.25 | | | | | |
% 72.22/10.25 | | | | | End of split
% 72.22/10.25 | | | | |
% 72.22/10.25 | | | | Case 2:
% 72.22/10.25 | | | | |
% 72.22/10.25 | | | | | (189) ~ (all_161_2 = 0) & member(all_52_0, universal_class) =
% 72.22/10.25 | | | | | all_161_2
% 72.22/10.25 | | | | |
% 72.22/10.25 | | | | | ALPHA: (189) implies:
% 72.22/10.25 | | | | | (190) ~ (all_161_2 = 0)
% 72.22/10.25 | | | | | (191) member(all_52_0, universal_class) = all_161_2
% 72.22/10.25 | | | | |
% 72.22/10.25 | | | | | GROUND_INST: instantiating (26) with 0, all_161_2, universal_class,
% 72.22/10.25 | | | | | all_52_0, simplifying with (46), (191) gives:
% 72.22/10.25 | | | | | (192) all_161_2 = 0
% 72.22/10.25 | | | | |
% 72.22/10.25 | | | | | REDUCE: (190), (192) imply:
% 72.22/10.25 | | | | | (193) $false
% 72.22/10.25 | | | | |
% 72.22/10.25 | | | | | CLOSE: (193) is inconsistent.
% 72.22/10.25 | | | | |
% 72.22/10.25 | | | | End of split
% 72.22/10.25 | | | |
% 72.22/10.25 | | | End of split
% 72.22/10.25 | | |
% 72.22/10.25 | | End of split
% 72.22/10.25 | |
% 72.22/10.25 | End of split
% 72.22/10.25 |
% 72.22/10.25 End of proof
% 72.22/10.25
% 72.22/10.25 Sub-proof #1 shows that the following formulas are inconsistent:
% 72.22/10.25 ----------------------------------------------------------------
% 72.22/10.25 (1) ! [v0: MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: $i] :
% 72.22/10.25 ! [v3: $i] : (v1 = v0 | ~ (member(v3, v2) = v1) | ~ (member(v3, v2) =
% 72.22/10.25 v0))
% 72.22/10.25 (2) member(null_class, universal_class) = all_36_0
% 72.22/10.25 (3) member(null_class, universal_class) = 0
% 72.22/10.25 (4) ~ (all_36_0 = 0)
% 72.22/10.25
% 72.22/10.25 Begin of proof
% 72.22/10.25 |
% 72.22/10.25 | GROUND_INST: instantiating (1) with all_36_0, 0, universal_class, null_class,
% 72.22/10.25 | simplifying with (2), (3) gives:
% 72.22/10.25 | (5) all_36_0 = 0
% 72.22/10.25 |
% 72.22/10.25 | REDUCE: (4), (5) imply:
% 72.22/10.25 | (6) $false
% 72.22/10.25 |
% 72.22/10.25 | CLOSE: (6) is inconsistent.
% 72.22/10.25 |
% 72.22/10.25 End of proof
% 72.22/10.25 % SZS output end Proof for theBenchmark
% 72.22/10.25
% 72.22/10.25 9622ms
%------------------------------------------------------------------------------