TSTP Solution File: SEV521+1 by ePrincess---1.0
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%------------------------------------------------------------------------------
% File : ePrincess---1.0
% Problem : SEV521+1 : TPTP v8.1.0. Released v7.3.0.
% Transfm : none
% Format : tptp:raw
% Command : ePrincess-casc -timeout=%d %s
% Computer : n011.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 : 600s
% DateTime : Tue Jul 19 16:56:45 EDT 2022
% Result : Theorem 7.51s 2.37s
% Output : Proof 9.29s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEV521+1 : TPTP v8.1.0. Released v7.3.0.
% 0.11/0.12 % Command : ePrincess-casc -timeout=%d %s
% 0.11/0.33 % Computer : n011.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 600
% 0.11/0.33 % DateTime : Tue Jun 28 16:09:16 EDT 2022
% 0.11/0.33 % CPUTime :
% 0.58/0.59 ____ _
% 0.58/0.59 ___ / __ \_____(_)___ ________ __________
% 0.58/0.59 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/
% 0.58/0.59 / __/ ____/ / / / / / / /__/ __(__ |__ )
% 0.58/0.59 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/
% 0.58/0.59
% 0.58/0.59 A Theorem Prover for First-Order Logic
% 0.58/0.60 (ePrincess v.1.0)
% 0.58/0.60
% 0.58/0.60 (c) Philipp Rümmer, 2009-2015
% 0.58/0.60 (c) Peter Backeman, 2014-2015
% 0.58/0.60 (contributions by Angelo Brillout, Peter Baumgartner)
% 0.58/0.60 Free software under GNU Lesser General Public License (LGPL).
% 0.58/0.60 Bug reports to peter@backeman.se
% 0.58/0.60
% 0.58/0.60 For more information, visit http://user.uu.se/~petba168/breu/
% 0.58/0.60
% 0.58/0.60 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 0.70/0.67 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all
% 1.62/0.99 Prover 0: Preprocessing ...
% 2.63/1.28 Prover 0: Warning: ignoring some quantifiers
% 2.63/1.31 Prover 0: Constructing countermodel ...
% 6.53/2.15 Prover 0: gave up
% 6.53/2.15 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all
% 6.53/2.19 Prover 1: Preprocessing ...
% 7.51/2.31 Prover 1: Warning: ignoring some quantifiers
% 7.51/2.32 Prover 1: Constructing countermodel ...
% 7.51/2.37 Prover 1: proved (222ms)
% 7.51/2.37
% 7.51/2.37 No countermodel exists, formula is valid
% 7.51/2.37 % SZS status Theorem for theBenchmark
% 7.51/2.37
% 7.51/2.37 Generating proof ... Warning: ignoring some quantifiers
% 8.84/2.67 found it (size 33)
% 8.84/2.67
% 8.84/2.67 % SZS output start Proof for theBenchmark
% 8.84/2.67 Assumed formulas after preprocessing and simplification:
% 8.84/2.67 | (0) ? [v0] : ? [v1] : ? [v2] : ( ~ (v2 = 0) & ~ (v0 = empty_set) & partition(v1, v0) = v2 & singleton(v0) = v1 & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (singleton(v4) = v8) | ~ (singleton(v3) = v6) | ~ (difference(v5, v8) = v9) | ~ (union(v7, v9) = v10) | ~ (union(v4, v6) = v7) | insertIntoMember(v3, v4, v5) = v10) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (pre_order(v3, v4) = 0) | ~ (apply(v3, v5, v7) = v8) | ~ (apply(v3, v5, v6) = 0) | ? [v9] : ? [v10] : ? [v11] : ? [v12] : (apply(v3, v6, v7) = v12 & member(v7, v4) = v11 & member(v6, v4) = v10 & member(v5, v4) = v9 & ( ~ (v12 = 0) | ~ (v11 = 0) | ~ (v10 = 0) | ~ (v9 = 0)))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (equivalence_class(v5, v4, v3) = v7) | ~ (member(v6, v7) = v8) | ? [v9] : ? [v10] : (apply(v3, v5, v6) = v10 & member(v6, v4) = v9 & ( ~ (v10 = 0) | ~ (v9 = 0)))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (equivalence(v4, v3) = 0) | ~ (apply(v4, v5, v7) = v8) | ~ (apply(v4, v5, v6) = 0) | ? [v9] : ? [v10] : ? [v11] : ? [v12] : (apply(v4, v6, v7) = v12 & member(v7, v3) = v11 & member(v6, v3) = v10 & member(v5, v3) = v9 & ( ~ (v12 = 0) | ~ (v11 = 0) | ~ (v10 = 0) | ~ (v9 = 0)))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (product(v4) = v5) | ~ (member(v3, v6) = v7) | ~ (member(v3, v5) = 0) | ? [v8] : ( ~ (v8 = 0) & member(v6, v4) = v8)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (difference(v5, v4) = v6) | ~ (member(v3, v6) = v7) | ? [v8] : ? [v9] : (member(v3, v5) = v8 & member(v3, v4) = v9 & ( ~ (v8 = 0) | v9 = 0))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (union(v4, v5) = v6) | ~ (member(v3, v6) = v7) | ? [v8] : ? [v9] : ( ~ (v9 = 0) & ~ (v8 = 0) & member(v3, v5) = v9 & member(v3, v4) = v8)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (intersection(v4, v5) = v6) | ~ (member(v3, v6) = v7) | ? [v8] : ? [v9] : (member(v3, v5) = v9 & member(v3, v4) = v8 & ( ~ (v9 = 0) | ~ (v8 = 0)))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v6 = v5 | ~ (partition(v3, v4) = 0) | ~ (member(v7, v5) = 0) | ~ (member(v6, v3) = 0) | ~ (member(v5, v3) = 0) | ? [v8] : ( ~ (v8 = 0) & member(v7, v6) = v8)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v6 = 0 | ~ (unaryUnion(v3) = v4) | ~ (member(v5, v7) = 0) | ~ (member(v5, v4) = v6) | ? [v8] : ( ~ (v8 = 0) & member(v7, v3) = v8)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v6 = 0 | ~ (sum(v4) = v5) | ~ (member(v3, v7) = 0) | ~ (member(v3, v5) = v6) | ? [v8] : ( ~ (v8 = 0) & member(v7, v4) = v8)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v4 = v3 | ~ (insertIntoMember(v7, v6, v5) = v4) | ~ (insertIntoMember(v7, v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v4 = v3 | ~ (equivalence_class(v7, v6, v5) = v4) | ~ (equivalence_class(v7, v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v4 = v3 | ~ (apply(v7, v6, v5) = v4) | ~ (apply(v7, v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (equivalence_class(v5, v4, v3) = v7) | ~ (member(v6, v7) = 0) | (apply(v3, v5, v6) = 0 & member(v6, v4) = 0)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (pre_order(v3, v4) = 0) | ~ (apply(v3, v5, v5) = v6) | ? [v7] : ( ~ (v7 = 0) & member(v5, v4) = v7)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (equivalence(v4, v3) = 0) | ~ (apply(v4, v5, v5) = v6) | ? [v7] : ( ~ (v7 = 0) & member(v5, v3) = v7)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (partition(v3, v4) = 0) | ~ (subset(v5, v4) = v6) | ? [v7] : ( ~ (v7 = 0) & member(v5, v3) = v7)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (product(v4) = v5) | ~ (member(v3, v5) = v6) | ? [v7] : ? [v8] : ( ~ (v8 = 0) & member(v7, v4) = 0 & member(v3, v7) = v8)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (unordered_pair(v4, v3) = v5) | ~ (member(v3, v5) = v6)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (unordered_pair(v3, v4) = v5) | ~ (member(v3, v5) = v6)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (power_set(v4) = v5) | ~ (member(v3, v5) = v6) | ? [v7] : ( ~ (v7 = 0) & subset(v3, v4) = v7)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v5 = v3 | v4 = v3 | ~ (unordered_pair(v4, v5) = v6) | ~ (member(v3, v6) = 0)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (pre_order(v6, v5) = v4) | ~ (pre_order(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (equivalence(v6, v5) = v4) | ~ (equivalence(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (partition(v6, v5) = v4) | ~ (partition(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (disjoint(v6, v5) = v4) | ~ (disjoint(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (unordered_pair(v6, v5) = v4) | ~ (unordered_pair(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (difference(v6, v5) = v4) | ~ (difference(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (union(v6, v5) = v4) | ~ (union(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (intersection(v6, v5) = v4) | ~ (intersection(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (equal_set(v6, v5) = v4) | ~ (equal_set(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (subset(v6, v5) = v4) | ~ (subset(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (member(v6, v5) = v4) | ~ (member(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (equivalence(v4, v3) = 0) | ~ (apply(v4, v5, v6) = 0) | ? [v7] : ? [v8] : ? [v9] : (apply(v4, v6, v5) = v9 & member(v6, v3) = v8 & member(v5, v3) = v7 & ( ~ (v8 = 0) | ~ (v7 = 0) | v9 = 0))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (difference(v5, v4) = v6) | ~ (member(v3, v6) = 0) | ? [v7] : ( ~ (v7 = 0) & member(v3, v5) = 0 & member(v3, v4) = v7)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (union(v4, v5) = v6) | ~ (member(v3, v6) = 0) | ? [v7] : ? [v8] : (member(v3, v5) = v8 & member(v3, v4) = v7 & (v8 = 0 | v7 = 0))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (intersection(v4, v5) = v6) | ~ (member(v3, v6) = 0) | (member(v3, v5) = 0 & member(v3, v4) = 0)) & ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (pre_order(v3, v4) = v5) | ? [v6] : ? [v7] : ? [v8] : ? [v9] : ? [v10] : ? [v11] : ? [v12] : ? [v13] : ? [v14] : ((v13 = 0 & v12 = 0 & v11 = 0 & v10 = 0 & v9 = 0 & ~ (v14 = 0) & apply(v3, v7, v8) = 0 & apply(v3, v6, v8) = v14 & apply(v3, v6, v7) = 0 & member(v8, v4) = 0 & member(v7, v4) = 0 & member(v6, v4) = 0) | (v7 = 0 & ~ (v8 = 0) & apply(v3, v6, v6) = v8 & member(v6, v4) = 0))) & ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (equivalence(v4, v3) = v5) | ? [v6] : ? [v7] : ? [v8] : ? [v9] : ? [v10] : ? [v11] : ? [v12] : ? [v13] : ? [v14] : ((v13 = 0 & v12 = 0 & v11 = 0 & v10 = 0 & v9 = 0 & ~ (v14 = 0) & apply(v4, v7, v8) = 0 & apply(v4, v6, v8) = v14 & apply(v4, v6, v7) = 0 & member(v8, v3) = 0 & member(v7, v3) = 0 & member(v6, v3) = 0) | (v10 = 0 & v9 = 0 & v8 = 0 & ~ (v11 = 0) & apply(v4, v7, v6) = v11 & apply(v4, v6, v7) = 0 & member(v7, v3) = 0 & member(v6, v3) = 0) | (v7 = 0 & ~ (v8 = 0) & apply(v4, v6, v6) = v8 & member(v6, v3) = 0))) & ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (partition(v3, v4) = v5) | ? [v6] : ? [v7] : ? [v8] : ? [v9] : ? [v10] : ? [v11] : ? [v12] : ((v12 = 0 & v11 = 0 & v9 = 0 & v8 = 0 & ~ (v7 = v6) & member(v10, v7) = 0 & member(v10, v6) = 0 & member(v7, v3) = 0 & member(v6, v3) = 0) | (v7 = 0 & ~ (v8 = 0) & subset(v6, v4) = v8 & member(v6, v3) = 0) | (v7 = 0 & member(v6, v4) = 0 & ! [v13] : ( ~ (member(v6, v13) = 0) | ? [v14] : ( ~ (v14 = 0) & member(v13, v3) = v14))))) & ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (disjoint(v3, v4) = v5) | ? [v6] : (member(v6, v4) = 0 & member(v6, v3) = 0)) & ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (singleton(v3) = v4) | ~ (member(v3, v4) = v5)) & ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (equal_set(v3, v4) = v5) | ? [v6] : ? [v7] : (subset(v4, v3) = v7 & subset(v3, v4) = v6 & ( ~ (v7 = 0) | ~ (v6 = 0)))) & ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (subset(v3, v4) = v5) | ? [v6] : ? [v7] : ( ~ (v7 = 0) & member(v6, v4) = v7 & member(v6, v3) = 0)) & ! [v3] : ! [v4] : ! [v5] : (v4 = v3 | ~ (unaryUnion(v5) = v4) | ~ (unaryUnion(v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : (v4 = v3 | ~ (non_overlapping(v5) = v4) | ~ (non_overlapping(v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : (v4 = v3 | ~ (product(v5) = v4) | ~ (product(v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : (v4 = v3 | ~ (sum(v5) = v4) | ~ (sum(v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : (v4 = v3 | ~ (singleton(v5) = v4) | ~ (singleton(v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : (v4 = v3 | ~ (singleton(v4) = v5) | ~ (member(v3, v5) = 0)) & ! [v3] : ! [v4] : ! [v5] : (v4 = v3 | ~ (power_set(v5) = v4) | ~ (power_set(v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : (v4 = 0 | ~ (non_overlapping(v3) = v4) | ~ (partition(v3, v5) = 0)) & ! [v3] : ! [v4] : ! [v5] : ( ~ (unaryUnion(v3) = v4) | ~ (member(v5, v4) = 0) | ? [v6] : (member(v6, v3) = 0 & member(v5, v6) = 0)) & ! [v3] : ! [v4] : ! [v5] : ( ~ (partition(v3, v4) = 0) | ~ (member(v5, v4) = 0) | ? [v6] : (member(v6, v3) = 0 & member(v5, v6) = 0)) & ! [v3] : ! [v4] : ! [v5] : ( ~ (disjoint(v3, v4) = 0) | ~ (member(v5, v3) = 0) | ? [v6] : ( ~ (v6 = 0) & member(v5, v4) = v6)) & ! [v3] : ! [v4] : ! [v5] : ( ~ (sum(v4) = v5) | ~ (member(v3, v5) = 0) | ? [v6] : (member(v6, v4) = 0 & member(v3, v6) = 0)) & ! [v3] : ! [v4] : ! [v5] : ( ~ (power_set(v4) = v5) | ~ (member(v3, v5) = 0) | subset(v3, v4) = 0) & ! [v3] : ! [v4] : ! [v5] : ( ~ (subset(v3, v4) = 0) | ~ (member(v5, v3) = 0) | member(v5, v4) = 0) & ? [v3] : ! [v4] : ! [v5] : (v5 = v3 | ~ (unaryUnion(v4) = v5) | ? [v6] : ? [v7] : ? [v8] : ? [v9] : ? [v10] : (member(v6, v3) = v7 & ( ~ (v7 = 0) | ! [v11] : ( ~ (member(v6, v11) = 0) | ? [v12] : ( ~ (v12 = 0) & member(v11, v4) = v12))) & (v7 = 0 | (v10 = 0 & v9 = 0 & member(v8, v4) = 0 & member(v6, v8) = 0)))) & ! [v3] : ! [v4] : ( ~ (equal_set(v3, v4) = 0) | (subset(v4, v3) = 0 & subset(v3, v4) = 0)) & ! [v3] : ( ~ (non_overlapping(v3) = 0) | ? [v4] : partition(v3, v4) = 0) & ! [v3] : ~ (member(v3, empty_set) = 0))
% 9.29/2.72 | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2 yields:
% 9.29/2.72 | (1) ~ (all_0_0_0 = 0) & ~ (all_0_2_2 = empty_set) & partition(all_0_1_1, all_0_2_2) = all_0_0_0 & singleton(all_0_2_2) = all_0_1_1 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (singleton(v1) = v5) | ~ (singleton(v0) = v3) | ~ (difference(v2, v5) = v6) | ~ (union(v4, v6) = v7) | ~ (union(v1, v3) = v4) | insertIntoMember(v0, v1, v2) = v7) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (pre_order(v0, v1) = 0) | ~ (apply(v0, v2, v4) = v5) | ~ (apply(v0, v2, v3) = 0) | ? [v6] : ? [v7] : ? [v8] : ? [v9] : (apply(v0, v3, v4) = v9 & member(v4, v1) = v8 & member(v3, v1) = v7 & member(v2, v1) = v6 & ( ~ (v9 = 0) | ~ (v8 = 0) | ~ (v7 = 0) | ~ (v6 = 0)))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (equivalence_class(v2, v1, v0) = v4) | ~ (member(v3, v4) = v5) | ? [v6] : ? [v7] : (apply(v0, v2, v3) = v7 & member(v3, v1) = v6 & ( ~ (v7 = 0) | ~ (v6 = 0)))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (equivalence(v1, v0) = 0) | ~ (apply(v1, v2, v4) = v5) | ~ (apply(v1, v2, v3) = 0) | ? [v6] : ? [v7] : ? [v8] : ? [v9] : (apply(v1, v3, v4) = v9 & member(v4, v0) = v8 & member(v3, v0) = v7 & member(v2, v0) = v6 & ( ~ (v9 = 0) | ~ (v8 = 0) | ~ (v7 = 0) | ~ (v6 = 0)))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | ~ (product(v1) = v2) | ~ (member(v0, v3) = v4) | ~ (member(v0, v2) = 0) | ? [v5] : ( ~ (v5 = 0) & member(v3, v1) = v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | ~ (difference(v2, v1) = v3) | ~ (member(v0, v3) = v4) | ? [v5] : ? [v6] : (member(v0, v2) = v5 & member(v0, v1) = v6 & ( ~ (v5 = 0) | v6 = 0))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | ~ (union(v1, v2) = v3) | ~ (member(v0, v3) = v4) | ? [v5] : ? [v6] : ( ~ (v6 = 0) & ~ (v5 = 0) & member(v0, v2) = v6 & member(v0, v1) = v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | ~ (intersection(v1, v2) = v3) | ~ (member(v0, v3) = v4) | ? [v5] : ? [v6] : (member(v0, v2) = v6 & member(v0, v1) = v5 & ( ~ (v6 = 0) | ~ (v5 = 0)))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v3 = v2 | ~ (partition(v0, v1) = 0) | ~ (member(v4, v2) = 0) | ~ (member(v3, v0) = 0) | ~ (member(v2, v0) = 0) | ? [v5] : ( ~ (v5 = 0) & member(v4, v3) = v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v3 = 0 | ~ (unaryUnion(v0) = v1) | ~ (member(v2, v4) = 0) | ~ (member(v2, v1) = v3) | ? [v5] : ( ~ (v5 = 0) & member(v4, v0) = v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v3 = 0 | ~ (sum(v1) = v2) | ~ (member(v0, v4) = 0) | ~ (member(v0, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & member(v4, v1) = v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (insertIntoMember(v4, v3, v2) = v1) | ~ (insertIntoMember(v4, v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (equivalence_class(v4, v3, v2) = v1) | ~ (equivalence_class(v4, v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (apply(v4, v3, v2) = v1) | ~ (apply(v4, v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (equivalence_class(v2, v1, v0) = v4) | ~ (member(v3, v4) = 0) | (apply(v0, v2, v3) = 0 & member(v3, v1) = 0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (pre_order(v0, v1) = 0) | ~ (apply(v0, v2, v2) = v3) | ? [v4] : ( ~ (v4 = 0) & member(v2, v1) = v4)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (equivalence(v1, v0) = 0) | ~ (apply(v1, v2, v2) = v3) | ? [v4] : ( ~ (v4 = 0) & member(v2, v0) = v4)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (partition(v0, v1) = 0) | ~ (subset(v2, v1) = v3) | ? [v4] : ( ~ (v4 = 0) & member(v2, v0) = v4)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (product(v1) = v2) | ~ (member(v0, v2) = v3) | ? [v4] : ? [v5] : ( ~ (v5 = 0) & member(v4, v1) = 0 & member(v0, v4) = v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (unordered_pair(v1, v0) = v2) | ~ (member(v0, v2) = v3)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (unordered_pair(v0, v1) = v2) | ~ (member(v0, v2) = v3)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (power_set(v1) = v2) | ~ (member(v0, v2) = v3) | ? [v4] : ( ~ (v4 = 0) & subset(v0, v1) = v4)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v2 = v0 | v1 = v0 | ~ (unordered_pair(v1, v2) = v3) | ~ (member(v0, v3) = 0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (pre_order(v3, v2) = v1) | ~ (pre_order(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equivalence(v3, v2) = v1) | ~ (equivalence(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (partition(v3, v2) = v1) | ~ (partition(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (disjoint(v3, v2) = v1) | ~ (disjoint(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (unordered_pair(v3, v2) = v1) | ~ (unordered_pair(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (difference(v3, v2) = v1) | ~ (difference(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (union(v3, v2) = v1) | ~ (union(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (intersection(v3, v2) = v1) | ~ (intersection(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equal_set(v3, v2) = v1) | ~ (equal_set(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (subset(v3, v2) = v1) | ~ (subset(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (member(v3, v2) = v1) | ~ (member(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (equivalence(v1, v0) = 0) | ~ (apply(v1, v2, v3) = 0) | ? [v4] : ? [v5] : ? [v6] : (apply(v1, v3, v2) = v6 & member(v3, v0) = v5 & member(v2, v0) = v4 & ( ~ (v5 = 0) | ~ (v4 = 0) | v6 = 0))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (difference(v2, v1) = v3) | ~ (member(v0, v3) = 0) | ? [v4] : ( ~ (v4 = 0) & member(v0, v2) = 0 & member(v0, v1) = v4)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (union(v1, v2) = v3) | ~ (member(v0, v3) = 0) | ? [v4] : ? [v5] : (member(v0, v2) = v5 & member(v0, v1) = v4 & (v5 = 0 | v4 = 0))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (intersection(v1, v2) = v3) | ~ (member(v0, v3) = 0) | (member(v0, v2) = 0 & member(v0, v1) = 0)) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (pre_order(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : ? [v9] : ? [v10] : ? [v11] : ((v10 = 0 & v9 = 0 & v8 = 0 & v7 = 0 & v6 = 0 & ~ (v11 = 0) & apply(v0, v4, v5) = 0 & apply(v0, v3, v5) = v11 & apply(v0, v3, v4) = 0 & member(v5, v1) = 0 & member(v4, v1) = 0 & member(v3, v1) = 0) | (v4 = 0 & ~ (v5 = 0) & apply(v0, v3, v3) = v5 & member(v3, v1) = 0))) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (equivalence(v1, v0) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : ? [v9] : ? [v10] : ? [v11] : ((v10 = 0 & v9 = 0 & v8 = 0 & v7 = 0 & v6 = 0 & ~ (v11 = 0) & apply(v1, v4, v5) = 0 & apply(v1, v3, v5) = v11 & apply(v1, v3, v4) = 0 & member(v5, v0) = 0 & member(v4, v0) = 0 & member(v3, v0) = 0) | (v7 = 0 & v6 = 0 & v5 = 0 & ~ (v8 = 0) & apply(v1, v4, v3) = v8 & apply(v1, v3, v4) = 0 & member(v4, v0) = 0 & member(v3, v0) = 0) | (v4 = 0 & ~ (v5 = 0) & apply(v1, v3, v3) = v5 & member(v3, v0) = 0))) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (partition(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : ? [v9] : ((v9 = 0 & v8 = 0 & v6 = 0 & v5 = 0 & ~ (v4 = v3) & member(v7, v4) = 0 & member(v7, v3) = 0 & member(v4, v0) = 0 & member(v3, v0) = 0) | (v4 = 0 & ~ (v5 = 0) & subset(v3, v1) = v5 & member(v3, v0) = 0) | (v4 = 0 & member(v3, v1) = 0 & ! [v10] : ( ~ (member(v3, v10) = 0) | ? [v11] : ( ~ (v11 = 0) & member(v10, v0) = v11))))) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (disjoint(v0, v1) = v2) | ? [v3] : (member(v3, v1) = 0 & member(v3, v0) = 0)) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (singleton(v0) = v1) | ~ (member(v0, v1) = v2)) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (equal_set(v0, v1) = v2) | ? [v3] : ? [v4] : (subset(v1, v0) = v4 & subset(v0, v1) = v3 & ( ~ (v4 = 0) | ~ (v3 = 0)))) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (subset(v0, v1) = v2) | ? [v3] : ? [v4] : ( ~ (v4 = 0) & member(v3, v1) = v4 & member(v3, v0) = 0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (unaryUnion(v2) = v1) | ~ (unaryUnion(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (non_overlapping(v2) = v1) | ~ (non_overlapping(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (product(v2) = v1) | ~ (product(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (sum(v2) = v1) | ~ (sum(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (singleton(v2) = v1) | ~ (singleton(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (singleton(v1) = v2) | ~ (member(v0, v2) = 0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (power_set(v2) = v1) | ~ (power_set(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = 0 | ~ (non_overlapping(v0) = v1) | ~ (partition(v0, v2) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (unaryUnion(v0) = v1) | ~ (member(v2, v1) = 0) | ? [v3] : (member(v3, v0) = 0 & member(v2, v3) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (partition(v0, v1) = 0) | ~ (member(v2, v1) = 0) | ? [v3] : (member(v3, v0) = 0 & member(v2, v3) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (disjoint(v0, v1) = 0) | ~ (member(v2, v0) = 0) | ? [v3] : ( ~ (v3 = 0) & member(v2, v1) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (sum(v1) = v2) | ~ (member(v0, v2) = 0) | ? [v3] : (member(v3, v1) = 0 & member(v0, v3) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (power_set(v1) = v2) | ~ (member(v0, v2) = 0) | subset(v0, v1) = 0) & ! [v0] : ! [v1] : ! [v2] : ( ~ (subset(v0, v1) = 0) | ~ (member(v2, v0) = 0) | member(v2, v1) = 0) & ? [v0] : ! [v1] : ! [v2] : (v2 = v0 | ~ (unaryUnion(v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : (member(v3, v0) = v4 & ( ~ (v4 = 0) | ! [v8] : ( ~ (member(v3, v8) = 0) | ? [v9] : ( ~ (v9 = 0) & member(v8, v1) = v9))) & (v4 = 0 | (v7 = 0 & v6 = 0 & member(v5, v1) = 0 & member(v3, v5) = 0)))) & ! [v0] : ! [v1] : ( ~ (equal_set(v0, v1) = 0) | (subset(v1, v0) = 0 & subset(v0, v1) = 0)) & ! [v0] : ( ~ (non_overlapping(v0) = 0) | ? [v1] : partition(v0, v1) = 0) & ! [v0] : ~ (member(v0, empty_set) = 0)
% 9.29/2.75 |
% 9.29/2.75 | Applying alpha-rule on (1) yields:
% 9.29/2.75 | (2) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (member(v3, v2) = v1) | ~ (member(v3, v2) = v0))
% 9.29/2.75 | (3) ! [v0] : ! [v1] : ! [v2] : ( ~ (partition(v0, v1) = 0) | ~ (member(v2, v1) = 0) | ? [v3] : (member(v3, v0) = 0 & member(v2, v3) = 0))
% 9.29/2.75 | (4) ! [v0] : ( ~ (non_overlapping(v0) = 0) | ? [v1] : partition(v0, v1) = 0)
% 9.29/2.75 | (5) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (unordered_pair(v0, v1) = v2) | ~ (member(v0, v2) = v3))
% 9.29/2.75 | (6) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (pre_order(v0, v1) = 0) | ~ (apply(v0, v2, v4) = v5) | ~ (apply(v0, v2, v3) = 0) | ? [v6] : ? [v7] : ? [v8] : ? [v9] : (apply(v0, v3, v4) = v9 & member(v4, v1) = v8 & member(v3, v1) = v7 & member(v2, v1) = v6 & ( ~ (v9 = 0) | ~ (v8 = 0) | ~ (v7 = 0) | ~ (v6 = 0))))
% 9.29/2.75 | (7) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | ~ (union(v1, v2) = v3) | ~ (member(v0, v3) = v4) | ? [v5] : ? [v6] : ( ~ (v6 = 0) & ~ (v5 = 0) & member(v0, v2) = v6 & member(v0, v1) = v5))
% 9.29/2.75 | (8) ! [v0] : ! [v1] : ! [v2] : ( ~ (sum(v1) = v2) | ~ (member(v0, v2) = 0) | ? [v3] : (member(v3, v1) = 0 & member(v0, v3) = 0))
% 9.29/2.75 | (9) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (equivalence_class(v2, v1, v0) = v4) | ~ (member(v3, v4) = v5) | ? [v6] : ? [v7] : (apply(v0, v2, v3) = v7 & member(v3, v1) = v6 & ( ~ (v7 = 0) | ~ (v6 = 0))))
% 9.29/2.75 | (10) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v2 = v0 | v1 = v0 | ~ (unordered_pair(v1, v2) = v3) | ~ (member(v0, v3) = 0))
% 9.29/2.75 | (11) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (singleton(v1) = v5) | ~ (singleton(v0) = v3) | ~ (difference(v2, v5) = v6) | ~ (union(v4, v6) = v7) | ~ (union(v1, v3) = v4) | insertIntoMember(v0, v1, v2) = v7)
% 9.29/2.75 | (12) ~ (all_0_0_0 = 0)
% 9.29/2.75 | (13) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (pre_order(v3, v2) = v1) | ~ (pre_order(v3, v2) = v0))
% 9.29/2.75 | (14) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (product(v2) = v1) | ~ (product(v2) = v0))
% 9.29/2.75 | (15) ? [v0] : ! [v1] : ! [v2] : (v2 = v0 | ~ (unaryUnion(v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : (member(v3, v0) = v4 & ( ~ (v4 = 0) | ! [v8] : ( ~ (member(v3, v8) = 0) | ? [v9] : ( ~ (v9 = 0) & member(v8, v1) = v9))) & (v4 = 0 | (v7 = 0 & v6 = 0 & member(v5, v1) = 0 & member(v3, v5) = 0))))
% 9.29/2.75 | (16) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (union(v3, v2) = v1) | ~ (union(v3, v2) = v0))
% 9.29/2.75 | (17) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (unaryUnion(v2) = v1) | ~ (unaryUnion(v2) = v0))
% 9.29/2.75 | (18) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equal_set(v3, v2) = v1) | ~ (equal_set(v3, v2) = v0))
% 9.29/2.75 | (19) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (singleton(v2) = v1) | ~ (singleton(v2) = v0))
% 9.29/2.75 | (20) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (unordered_pair(v1, v0) = v2) | ~ (member(v0, v2) = v3))
% 9.29/2.75 | (21) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (intersection(v3, v2) = v1) | ~ (intersection(v3, v2) = v0))
% 9.29/2.75 | (22) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v3 = 0 | ~ (sum(v1) = v2) | ~ (member(v0, v4) = 0) | ~ (member(v0, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & member(v4, v1) = v5))
% 9.29/2.75 | (23) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (product(v1) = v2) | ~ (member(v0, v2) = v3) | ? [v4] : ? [v5] : ( ~ (v5 = 0) & member(v4, v1) = 0 & member(v0, v4) = v5))
% 9.29/2.75 | (24) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (singleton(v0) = v1) | ~ (member(v0, v1) = v2))
% 9.29/2.76 | (25) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | ~ (difference(v2, v1) = v3) | ~ (member(v0, v3) = v4) | ? [v5] : ? [v6] : (member(v0, v2) = v5 & member(v0, v1) = v6 & ( ~ (v5 = 0) | v6 = 0)))
% 9.29/2.76 | (26) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (intersection(v1, v2) = v3) | ~ (member(v0, v3) = 0) | (member(v0, v2) = 0 & member(v0, v1) = 0))
% 9.29/2.76 | (27) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (subset(v3, v2) = v1) | ~ (subset(v3, v2) = v0))
% 9.29/2.76 | (28) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (partition(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : ? [v9] : ((v9 = 0 & v8 = 0 & v6 = 0 & v5 = 0 & ~ (v4 = v3) & member(v7, v4) = 0 & member(v7, v3) = 0 & member(v4, v0) = 0 & member(v3, v0) = 0) | (v4 = 0 & ~ (v5 = 0) & subset(v3, v1) = v5 & member(v3, v0) = 0) | (v4 = 0 & member(v3, v1) = 0 & ! [v10] : ( ~ (member(v3, v10) = 0) | ? [v11] : ( ~ (v11 = 0) & member(v10, v0) = v11)))))
% 9.29/2.76 | (29) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (equivalence(v1, v0) = 0) | ~ (apply(v1, v2, v2) = v3) | ? [v4] : ( ~ (v4 = 0) & member(v2, v0) = v4))
% 9.29/2.76 | (30) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (subset(v0, v1) = v2) | ? [v3] : ? [v4] : ( ~ (v4 = 0) & member(v3, v1) = v4 & member(v3, v0) = 0))
% 9.29/2.76 | (31) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (partition(v0, v1) = 0) | ~ (subset(v2, v1) = v3) | ? [v4] : ( ~ (v4 = 0) & member(v2, v0) = v4))
% 9.29/2.76 | (32) ! [v0] : ! [v1] : ! [v2] : ( ~ (disjoint(v0, v1) = 0) | ~ (member(v2, v0) = 0) | ? [v3] : ( ~ (v3 = 0) & member(v2, v1) = v3))
% 9.29/2.76 | (33) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equivalence(v3, v2) = v1) | ~ (equivalence(v3, v2) = v0))
% 9.29/2.76 | (34) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (equivalence_class(v2, v1, v0) = v4) | ~ (member(v3, v4) = 0) | (apply(v0, v2, v3) = 0 & member(v3, v1) = 0))
% 9.29/2.76 | (35) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (equivalence(v1, v0) = 0) | ~ (apply(v1, v2, v3) = 0) | ? [v4] : ? [v5] : ? [v6] : (apply(v1, v3, v2) = v6 & member(v3, v0) = v5 & member(v2, v0) = v4 & ( ~ (v5 = 0) | ~ (v4 = 0) | v6 = 0)))
% 9.29/2.76 | (36) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (partition(v3, v2) = v1) | ~ (partition(v3, v2) = v0))
% 9.29/2.76 | (37) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (pre_order(v0, v1) = 0) | ~ (apply(v0, v2, v2) = v3) | ? [v4] : ( ~ (v4 = 0) & member(v2, v1) = v4))
% 9.29/2.76 | (38) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (power_set(v2) = v1) | ~ (power_set(v2) = v0))
% 9.29/2.76 | (39) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (equivalence(v1, v0) = 0) | ~ (apply(v1, v2, v4) = v5) | ~ (apply(v1, v2, v3) = 0) | ? [v6] : ? [v7] : ? [v8] : ? [v9] : (apply(v1, v3, v4) = v9 & member(v4, v0) = v8 & member(v3, v0) = v7 & member(v2, v0) = v6 & ( ~ (v9 = 0) | ~ (v8 = 0) | ~ (v7 = 0) | ~ (v6 = 0))))
% 9.29/2.76 | (40) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (disjoint(v0, v1) = v2) | ? [v3] : (member(v3, v1) = 0 & member(v3, v0) = 0))
% 9.29/2.76 | (41) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (insertIntoMember(v4, v3, v2) = v1) | ~ (insertIntoMember(v4, v3, v2) = v0))
% 9.29/2.76 | (42) ! [v0] : ! [v1] : ! [v2] : (v1 = 0 | ~ (non_overlapping(v0) = v1) | ~ (partition(v0, v2) = 0))
% 9.29/2.76 | (43) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (pre_order(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : ? [v9] : ? [v10] : ? [v11] : ((v10 = 0 & v9 = 0 & v8 = 0 & v7 = 0 & v6 = 0 & ~ (v11 = 0) & apply(v0, v4, v5) = 0 & apply(v0, v3, v5) = v11 & apply(v0, v3, v4) = 0 & member(v5, v1) = 0 & member(v4, v1) = 0 & member(v3, v1) = 0) | (v4 = 0 & ~ (v5 = 0) & apply(v0, v3, v3) = v5 & member(v3, v1) = 0)))
% 9.29/2.76 | (44) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (equivalence(v1, v0) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : ? [v9] : ? [v10] : ? [v11] : ((v10 = 0 & v9 = 0 & v8 = 0 & v7 = 0 & v6 = 0 & ~ (v11 = 0) & apply(v1, v4, v5) = 0 & apply(v1, v3, v5) = v11 & apply(v1, v3, v4) = 0 & member(v5, v0) = 0 & member(v4, v0) = 0 & member(v3, v0) = 0) | (v7 = 0 & v6 = 0 & v5 = 0 & ~ (v8 = 0) & apply(v1, v4, v3) = v8 & apply(v1, v3, v4) = 0 & member(v4, v0) = 0 & member(v3, v0) = 0) | (v4 = 0 & ~ (v5 = 0) & apply(v1, v3, v3) = v5 & member(v3, v0) = 0)))
% 9.29/2.76 | (45) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (disjoint(v3, v2) = v1) | ~ (disjoint(v3, v2) = v0))
% 9.29/2.76 | (46) ! [v0] : ~ (member(v0, empty_set) = 0)
% 9.29/2.76 | (47) singleton(all_0_2_2) = all_0_1_1
% 9.29/2.76 | (48) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (power_set(v1) = v2) | ~ (member(v0, v2) = v3) | ? [v4] : ( ~ (v4 = 0) & subset(v0, v1) = v4))
% 9.29/2.76 | (49) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | ~ (intersection(v1, v2) = v3) | ~ (member(v0, v3) = v4) | ? [v5] : ? [v6] : (member(v0, v2) = v6 & member(v0, v1) = v5 & ( ~ (v6 = 0) | ~ (v5 = 0))))
% 9.29/2.76 | (50) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (non_overlapping(v2) = v1) | ~ (non_overlapping(v2) = v0))
% 9.29/2.76 | (51) ! [v0] : ! [v1] : ! [v2] : ( ~ (power_set(v1) = v2) | ~ (member(v0, v2) = 0) | subset(v0, v1) = 0)
% 9.29/2.76 | (52) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (sum(v2) = v1) | ~ (sum(v2) = v0))
% 9.29/2.76 | (53) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | ~ (product(v1) = v2) | ~ (member(v0, v3) = v4) | ~ (member(v0, v2) = 0) | ? [v5] : ( ~ (v5 = 0) & member(v3, v1) = v5))
% 9.29/2.77 | (54) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (difference(v3, v2) = v1) | ~ (difference(v3, v2) = v0))
% 9.29/2.77 | (55) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (difference(v2, v1) = v3) | ~ (member(v0, v3) = 0) | ? [v4] : ( ~ (v4 = 0) & member(v0, v2) = 0 & member(v0, v1) = v4))
% 9.29/2.77 | (56) ! [v0] : ! [v1] : ( ~ (equal_set(v0, v1) = 0) | (subset(v1, v0) = 0 & subset(v0, v1) = 0))
% 9.29/2.77 | (57) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (singleton(v1) = v2) | ~ (member(v0, v2) = 0))
% 9.29/2.77 | (58) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (equal_set(v0, v1) = v2) | ? [v3] : ? [v4] : (subset(v1, v0) = v4 & subset(v0, v1) = v3 & ( ~ (v4 = 0) | ~ (v3 = 0))))
% 9.29/2.77 | (59) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v3 = 0 | ~ (unaryUnion(v0) = v1) | ~ (member(v2, v4) = 0) | ~ (member(v2, v1) = v3) | ? [v5] : ( ~ (v5 = 0) & member(v4, v0) = v5))
% 9.29/2.77 | (60) partition(all_0_1_1, all_0_2_2) = all_0_0_0
% 9.29/2.77 | (61) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (equivalence_class(v4, v3, v2) = v1) | ~ (equivalence_class(v4, v3, v2) = v0))
% 9.29/2.77 | (62) ! [v0] : ! [v1] : ! [v2] : ( ~ (subset(v0, v1) = 0) | ~ (member(v2, v0) = 0) | member(v2, v1) = 0)
% 9.29/2.77 | (63) ~ (all_0_2_2 = empty_set)
% 9.29/2.77 | (64) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v3 = v2 | ~ (partition(v0, v1) = 0) | ~ (member(v4, v2) = 0) | ~ (member(v3, v0) = 0) | ~ (member(v2, v0) = 0) | ? [v5] : ( ~ (v5 = 0) & member(v4, v3) = v5))
% 9.29/2.77 | (65) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (apply(v4, v3, v2) = v1) | ~ (apply(v4, v3, v2) = v0))
% 9.29/2.77 | (66) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (unordered_pair(v3, v2) = v1) | ~ (unordered_pair(v3, v2) = v0))
% 9.29/2.77 | (67) ! [v0] : ! [v1] : ! [v2] : ( ~ (unaryUnion(v0) = v1) | ~ (member(v2, v1) = 0) | ? [v3] : (member(v3, v0) = 0 & member(v2, v3) = 0))
% 9.29/2.77 | (68) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (union(v1, v2) = v3) | ~ (member(v0, v3) = 0) | ? [v4] : ? [v5] : (member(v0, v2) = v5 & member(v0, v1) = v4 & (v5 = 0 | v4 = 0)))
% 9.29/2.77 |
% 9.29/2.77 | Instantiating formula (28) with all_0_0_0, all_0_2_2, all_0_1_1 and discharging atoms partition(all_0_1_1, all_0_2_2) = all_0_0_0, yields:
% 9.29/2.77 | (69) all_0_0_0 = 0 | ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ? [v5] : ? [v6] : ((v6 = 0 & v5 = 0 & v3 = 0 & v2 = 0 & ~ (v1 = v0) & member(v4, v1) = 0 & member(v4, v0) = 0 & member(v1, all_0_1_1) = 0 & member(v0, all_0_1_1) = 0) | (v1 = 0 & ~ (v2 = 0) & subset(v0, all_0_2_2) = v2 & member(v0, all_0_1_1) = 0) | (v1 = 0 & member(v0, all_0_2_2) = 0 & ! [v7] : ( ~ (member(v0, v7) = 0) | ? [v8] : ( ~ (v8 = 0) & member(v7, all_0_1_1) = v8))))
% 9.29/2.77 |
% 9.29/2.77 +-Applying beta-rule and splitting (69), into two cases.
% 9.29/2.77 |-Branch one:
% 9.29/2.77 | (70) all_0_0_0 = 0
% 9.29/2.77 |
% 9.29/2.77 | Equations (70) can reduce 12 to:
% 9.29/2.77 | (71) $false
% 9.29/2.77 |
% 9.29/2.77 |-The branch is then unsatisfiable
% 9.29/2.77 |-Branch two:
% 9.29/2.77 | (12) ~ (all_0_0_0 = 0)
% 9.29/2.77 | (73) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ? [v5] : ? [v6] : ((v6 = 0 & v5 = 0 & v3 = 0 & v2 = 0 & ~ (v1 = v0) & member(v4, v1) = 0 & member(v4, v0) = 0 & member(v1, all_0_1_1) = 0 & member(v0, all_0_1_1) = 0) | (v1 = 0 & ~ (v2 = 0) & subset(v0, all_0_2_2) = v2 & member(v0, all_0_1_1) = 0) | (v1 = 0 & member(v0, all_0_2_2) = 0 & ! [v7] : ( ~ (member(v0, v7) = 0) | ? [v8] : ( ~ (v8 = 0) & member(v7, all_0_1_1) = v8))))
% 9.29/2.77 |
% 9.29/2.77 | Instantiating (73) with all_12_0_4, all_12_1_5, all_12_2_6, all_12_3_7, all_12_4_8, all_12_5_9, all_12_6_10 yields:
% 9.29/2.77 | (74) (all_12_0_4 = 0 & all_12_1_5 = 0 & all_12_3_7 = 0 & all_12_4_8 = 0 & ~ (all_12_5_9 = all_12_6_10) & member(all_12_2_6, all_12_5_9) = 0 & member(all_12_2_6, all_12_6_10) = 0 & member(all_12_5_9, all_0_1_1) = 0 & member(all_12_6_10, all_0_1_1) = 0) | (all_12_5_9 = 0 & ~ (all_12_4_8 = 0) & subset(all_12_6_10, all_0_2_2) = all_12_4_8 & member(all_12_6_10, all_0_1_1) = 0) | (all_12_5_9 = 0 & member(all_12_6_10, all_0_2_2) = 0 & ! [v0] : ( ~ (member(all_12_6_10, v0) = 0) | ? [v1] : ( ~ (v1 = 0) & member(v0, all_0_1_1) = v1)))
% 9.29/2.77 |
% 9.29/2.77 +-Applying beta-rule and splitting (74), into two cases.
% 9.29/2.77 |-Branch one:
% 9.29/2.77 | (75) (all_12_0_4 = 0 & all_12_1_5 = 0 & all_12_3_7 = 0 & all_12_4_8 = 0 & ~ (all_12_5_9 = all_12_6_10) & member(all_12_2_6, all_12_5_9) = 0 & member(all_12_2_6, all_12_6_10) = 0 & member(all_12_5_9, all_0_1_1) = 0 & member(all_12_6_10, all_0_1_1) = 0) | (all_12_5_9 = 0 & ~ (all_12_4_8 = 0) & subset(all_12_6_10, all_0_2_2) = all_12_4_8 & member(all_12_6_10, all_0_1_1) = 0)
% 9.29/2.77 |
% 9.29/2.77 +-Applying beta-rule and splitting (75), into two cases.
% 9.29/2.77 |-Branch one:
% 9.29/2.77 | (76) all_12_0_4 = 0 & all_12_1_5 = 0 & all_12_3_7 = 0 & all_12_4_8 = 0 & ~ (all_12_5_9 = all_12_6_10) & member(all_12_2_6, all_12_5_9) = 0 & member(all_12_2_6, all_12_6_10) = 0 & member(all_12_5_9, all_0_1_1) = 0 & member(all_12_6_10, all_0_1_1) = 0
% 9.29/2.77 |
% 9.29/2.77 | Applying alpha-rule on (76) yields:
% 9.29/2.77 | (77) all_12_4_8 = 0
% 9.29/2.77 | (78) all_12_1_5 = 0
% 9.29/2.77 | (79) ~ (all_12_5_9 = all_12_6_10)
% 9.29/2.77 | (80) all_12_3_7 = 0
% 9.29/2.77 | (81) all_12_0_4 = 0
% 9.29/2.77 | (82) member(all_12_2_6, all_12_6_10) = 0
% 9.29/2.77 | (83) member(all_12_2_6, all_12_5_9) = 0
% 9.29/2.77 | (84) member(all_12_5_9, all_0_1_1) = 0
% 9.29/2.77 | (85) member(all_12_6_10, all_0_1_1) = 0
% 9.29/2.77 |
% 9.29/2.77 | Instantiating formula (57) with all_0_1_1, all_0_2_2, all_12_5_9 and discharging atoms singleton(all_0_2_2) = all_0_1_1, member(all_12_5_9, all_0_1_1) = 0, yields:
% 9.29/2.78 | (86) all_12_5_9 = all_0_2_2
% 9.29/2.78 |
% 9.29/2.78 | Instantiating formula (57) with all_0_1_1, all_0_2_2, all_12_6_10 and discharging atoms singleton(all_0_2_2) = all_0_1_1, member(all_12_6_10, all_0_1_1) = 0, yields:
% 9.29/2.78 | (87) all_12_6_10 = all_0_2_2
% 9.29/2.78 |
% 9.29/2.78 | Equations (86,87) can reduce 79 to:
% 9.29/2.78 | (71) $false
% 9.29/2.78 |
% 9.29/2.78 |-The branch is then unsatisfiable
% 9.29/2.78 |-Branch two:
% 9.29/2.78 | (89) all_12_5_9 = 0 & ~ (all_12_4_8 = 0) & subset(all_12_6_10, all_0_2_2) = all_12_4_8 & member(all_12_6_10, all_0_1_1) = 0
% 9.29/2.78 |
% 9.29/2.78 | Applying alpha-rule on (89) yields:
% 9.29/2.78 | (90) all_12_5_9 = 0
% 9.29/2.78 | (91) ~ (all_12_4_8 = 0)
% 9.29/2.78 | (92) subset(all_12_6_10, all_0_2_2) = all_12_4_8
% 9.29/2.78 | (85) member(all_12_6_10, all_0_1_1) = 0
% 9.29/2.78 |
% 9.29/2.78 | Instantiating formula (57) with all_0_1_1, all_0_2_2, all_12_6_10 and discharging atoms singleton(all_0_2_2) = all_0_1_1, member(all_12_6_10, all_0_1_1) = 0, yields:
% 9.29/2.78 | (87) all_12_6_10 = all_0_2_2
% 9.29/2.78 |
% 9.29/2.78 | From (87) and (92) follows:
% 9.29/2.78 | (95) subset(all_0_2_2, all_0_2_2) = all_12_4_8
% 9.29/2.78 |
% 9.29/2.78 | Instantiating formula (30) with all_12_4_8, all_0_2_2, all_0_2_2 and discharging atoms subset(all_0_2_2, all_0_2_2) = all_12_4_8, yields:
% 9.29/2.78 | (96) all_12_4_8 = 0 | ? [v0] : ? [v1] : ( ~ (v1 = 0) & member(v0, all_0_2_2) = v1 & member(v0, all_0_2_2) = 0)
% 9.29/2.78 |
% 9.29/2.78 +-Applying beta-rule and splitting (96), into two cases.
% 9.29/2.78 |-Branch one:
% 9.29/2.78 | (77) all_12_4_8 = 0
% 9.29/2.78 |
% 9.29/2.78 | Equations (77) can reduce 91 to:
% 9.29/2.78 | (71) $false
% 9.29/2.78 |
% 9.29/2.78 |-The branch is then unsatisfiable
% 9.29/2.78 |-Branch two:
% 9.29/2.78 | (91) ~ (all_12_4_8 = 0)
% 9.29/2.78 | (100) ? [v0] : ? [v1] : ( ~ (v1 = 0) & member(v0, all_0_2_2) = v1 & member(v0, all_0_2_2) = 0)
% 9.29/2.78 |
% 9.29/2.78 | Instantiating (100) with all_35_0_11, all_35_1_12 yields:
% 9.29/2.78 | (101) ~ (all_35_0_11 = 0) & member(all_35_1_12, all_0_2_2) = all_35_0_11 & member(all_35_1_12, all_0_2_2) = 0
% 9.29/2.78 |
% 9.29/2.78 | Applying alpha-rule on (101) yields:
% 9.29/2.78 | (102) ~ (all_35_0_11 = 0)
% 9.29/2.78 | (103) member(all_35_1_12, all_0_2_2) = all_35_0_11
% 9.29/2.78 | (104) member(all_35_1_12, all_0_2_2) = 0
% 9.29/2.78 |
% 9.29/2.78 | Instantiating formula (2) with all_35_1_12, all_0_2_2, 0, all_35_0_11 and discharging atoms member(all_35_1_12, all_0_2_2) = all_35_0_11, member(all_35_1_12, all_0_2_2) = 0, yields:
% 9.29/2.78 | (105) all_35_0_11 = 0
% 9.29/2.78 |
% 9.29/2.78 | Equations (105) can reduce 102 to:
% 9.29/2.78 | (71) $false
% 9.29/2.78 |
% 9.29/2.78 |-The branch is then unsatisfiable
% 9.29/2.78 |-Branch two:
% 9.29/2.78 | (107) all_12_5_9 = 0 & member(all_12_6_10, all_0_2_2) = 0 & ! [v0] : ( ~ (member(all_12_6_10, v0) = 0) | ? [v1] : ( ~ (v1 = 0) & member(v0, all_0_1_1) = v1))
% 9.29/2.78 |
% 9.29/2.78 | Applying alpha-rule on (107) yields:
% 9.29/2.78 | (90) all_12_5_9 = 0
% 9.29/2.78 | (109) member(all_12_6_10, all_0_2_2) = 0
% 9.29/2.78 | (110) ! [v0] : ( ~ (member(all_12_6_10, v0) = 0) | ? [v1] : ( ~ (v1 = 0) & member(v0, all_0_1_1) = v1))
% 9.29/2.78 |
% 9.29/2.78 | Instantiating formula (110) with all_0_2_2 and discharging atoms member(all_12_6_10, all_0_2_2) = 0, yields:
% 9.29/2.78 | (111) ? [v0] : ( ~ (v0 = 0) & member(all_0_2_2, all_0_1_1) = v0)
% 9.29/2.78 |
% 9.29/2.78 | Instantiating (111) with all_26_0_13 yields:
% 9.29/2.78 | (112) ~ (all_26_0_13 = 0) & member(all_0_2_2, all_0_1_1) = all_26_0_13
% 9.29/2.78 |
% 9.29/2.78 | Applying alpha-rule on (112) yields:
% 9.29/2.78 | (113) ~ (all_26_0_13 = 0)
% 9.29/2.78 | (114) member(all_0_2_2, all_0_1_1) = all_26_0_13
% 9.29/2.78 |
% 9.29/2.78 | Instantiating formula (24) with all_26_0_13, all_0_1_1, all_0_2_2 and discharging atoms singleton(all_0_2_2) = all_0_1_1, member(all_0_2_2, all_0_1_1) = all_26_0_13, yields:
% 9.29/2.78 | (115) all_26_0_13 = 0
% 9.29/2.78 |
% 9.29/2.78 | Equations (115) can reduce 113 to:
% 9.29/2.78 | (71) $false
% 9.29/2.78 |
% 9.29/2.78 |-The branch is then unsatisfiable
% 9.29/2.78 % SZS output end Proof for theBenchmark
% 9.29/2.78
% 9.29/2.78 2168ms
%------------------------------------------------------------------------------