%------------------------------------------------------------------------------ % File : ePrincess---1.0 % Problem : SET774+4 : TPTP v8.1.0. Released v2.2.0. % Transfm : none % Format : tptp:raw % Command : ePrincess-casc -timeout=%d %s % Computer : n022.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 00:21:59 EDT 2022 % Result : Theorem 4.61s 1.69s % Output : Proof 6.31s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.06/0.12 % Problem : SET774+4 : TPTP v8.1.0. Released v2.2.0. % 0.06/0.13 % Command : ePrincess-casc -timeout=%d %s % 0.13/0.34 % Computer : n022.cluster.edu % 0.13/0.34 % Model : x86_64 x86_64 % 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.13/0.34 % Memory : 8042.1875MB % 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.13/0.34 % CPULimit : 300 % 0.13/0.34 % WCLimit : 600 % 0.13/0.34 % DateTime : Sun Jul 10 10:41:25 EDT 2022 % 0.13/0.34 % CPUTime : % 0.19/0.58 ____ _ % 0.19/0.58 ___ / __ \_____(_)___ ________ __________ % 0.19/0.58 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/ % 0.19/0.58 / __/ ____/ / / / / / / /__/ __(__ |__ ) % 0.19/0.58 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/ % 0.19/0.58 % 0.19/0.58 A Theorem Prover for First-Order Logic % 0.19/0.58 (ePrincess v.1.0) % 0.19/0.58 % 0.19/0.58 (c) Philipp Rümmer, 2009-2015 % 0.19/0.58 (c) Peter Backeman, 2014-2015 % 0.19/0.58 (contributions by Angelo Brillout, Peter Baumgartner) % 0.19/0.58 Free software under GNU Lesser General Public License (LGPL). % 0.19/0.58 Bug reports to peter@backeman.se % 0.19/0.58 % 0.19/0.58 For more information, visit http://user.uu.se/~petba168/breu/ % 0.19/0.58 % 0.19/0.58 Loading /export/starexec/sandbox2/benchmark/theBenchmark.p ... % 0.73/0.63 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 1.75/0.95 Prover 0: Preprocessing ... % 2.49/1.21 Prover 0: Warning: ignoring some quantifiers % 2.57/1.23 Prover 0: Constructing countermodel ... % 3.27/1.39 Prover 0: gave up % 3.27/1.39 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all % 3.27/1.43 Prover 1: Preprocessing ... % 3.99/1.62 Prover 1: Constructing countermodel ... % 4.61/1.69 Prover 1: proved (299ms) % 4.61/1.69 % 4.61/1.69 No countermodel exists, formula is valid % 4.61/1.69 % SZS status Theorem for theBenchmark % 4.61/1.69 % 4.61/1.69 Generating proof ... found it (size 44) % 5.82/2.01 % 5.82/2.01 % SZS output start Proof for theBenchmark % 5.82/2.01 Assumed formulas after preprocessing and simplification: % 5.82/2.01 | (0) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ( ~ (v3 = 0) & pre_order(v2, v1) = v3 & pre_order(v2, v0) = 0 & subset(v1, v0) = 0 & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (pre_order(v4, v5) = 0) | ~ (apply(v4, v6, v8) = v9) | ~ (apply(v4, v6, v7) = 0) | ? [v10] : ? [v11] : ? [v12] : ? [v13] : (apply(v4, v7, v8) = v13 & member(v8, v5) = v12 & member(v7, v5) = v11 & member(v6, v5) = v10 & ( ~ (v13 = 0) | ~ (v12 = 0) | ~ (v11 = 0) | ~ (v10 = 0)))) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (equivalence_class(v6, v5, v4) = v8) | ~ (member(v7, v8) = v9) | ? [v10] : ? [v11] : (apply(v4, v6, v7) = v11 & member(v7, v5) = v10 & ( ~ (v11 = 0) | ~ (v10 = 0)))) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (equivalence(v5, v4) = 0) | ~ (apply(v5, v6, v8) = v9) | ~ (apply(v5, v6, v7) = 0) | ? [v10] : ? [v11] : ? [v12] : ? [v13] : (apply(v5, v7, v8) = v13 & member(v8, v4) = v12 & member(v7, v4) = v11 & member(v6, v4) = v10 & ( ~ (v13 = 0) | ~ (v12 = 0) | ~ (v11 = 0) | ~ (v10 = 0)))) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (product(v5) = v6) | ~ (member(v4, v7) = v8) | ~ (member(v4, v6) = 0) | ? [v9] : ( ~ (v9 = 0) & member(v7, v5) = v9)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (difference(v6, v5) = v7) | ~ (member(v4, v7) = v8) | ? [v9] : ? [v10] : (member(v4, v6) = v9 & member(v4, v5) = v10 & ( ~ (v9 = 0) | v10 = 0))) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (union(v5, v6) = v7) | ~ (member(v4, v7) = v8) | ? [v9] : ? [v10] : ( ~ (v10 = 0) & ~ (v9 = 0) & member(v4, v6) = v10 & member(v4, v5) = v9)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (intersection(v5, v6) = v7) | ~ (member(v4, v7) = v8) | ? [v9] : ? [v10] : (member(v4, v6) = v10 & member(v4, v5) = v9 & ( ~ (v10 = 0) | ~ (v9 = 0)))) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v7 = v6 | ~ (partition(v4, v5) = 0) | ~ (member(v8, v6) = 0) | ~ (member(v7, v4) = 0) | ~ (member(v6, v4) = 0) | ? [v9] : ( ~ (v9 = 0) & member(v8, v7) = v9)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v7 = 0 | ~ (sum(v5) = v6) | ~ (member(v4, v8) = 0) | ~ (member(v4, v6) = v7) | ? [v9] : ( ~ (v9 = 0) & member(v8, v5) = v9)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v5 = v4 | ~ (equivalence_class(v8, v7, v6) = v5) | ~ (equivalence_class(v8, v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v5 = v4 | ~ (apply(v8, v7, v6) = v5) | ~ (apply(v8, v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (equivalence_class(v6, v5, v4) = v8) | ~ (member(v7, v8) = 0) | (apply(v4, v6, v7) = 0 & member(v7, v5) = 0)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (pre_order(v4, v5) = 0) | ~ (apply(v4, v6, v6) = v7) | ? [v8] : ( ~ (v8 = 0) & member(v6, v5) = v8)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (equivalence(v5, v4) = 0) | ~ (apply(v5, v6, v6) = v7) | ? [v8] : ( ~ (v8 = 0) & member(v6, v4) = v8)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (partition(v4, v5) = 0) | ~ (subset(v6, v5) = v7) | ? [v8] : ( ~ (v8 = 0) & member(v6, v4) = v8)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (product(v5) = v6) | ~ (member(v4, v6) = v7) | ? [v8] : ? [v9] : ( ~ (v9 = 0) & member(v8, v5) = 0 & member(v4, v8) = v9)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (unordered_pair(v5, v4) = v6) | ~ (member(v4, v6) = v7)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (unordered_pair(v4, v5) = v6) | ~ (member(v4, v6) = v7)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (power_set(v5) = v6) | ~ (member(v4, v6) = v7) | ? [v8] : ( ~ (v8 = 0) & subset(v4, v5) = v8)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v6 = v4 | v5 = v4 | ~ (unordered_pair(v5, v6) = v7) | ~ (member(v4, v7) = 0)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (pre_order(v7, v6) = v5) | ~ (pre_order(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (equivalence(v7, v6) = v5) | ~ (equivalence(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (partition(v7, v6) = v5) | ~ (partition(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (disjoint(v7, v6) = v5) | ~ (disjoint(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (unordered_pair(v7, v6) = v5) | ~ (unordered_pair(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (difference(v7, v6) = v5) | ~ (difference(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (union(v7, v6) = v5) | ~ (union(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (intersection(v7, v6) = v5) | ~ (intersection(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (equal_set(v7, v6) = v5) | ~ (equal_set(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (subset(v7, v6) = v5) | ~ (subset(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (member(v7, v6) = v5) | ~ (member(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (equivalence(v5, v4) = 0) | ~ (apply(v5, v6, v7) = 0) | ? [v8] : ? [v9] : ? [v10] : (apply(v5, v7, v6) = v10 & member(v7, v4) = v9 & member(v6, v4) = v8 & ( ~ (v9 = 0) | ~ (v8 = 0) | v10 = 0))) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (difference(v6, v5) = v7) | ~ (member(v4, v7) = 0) | ? [v8] : ( ~ (v8 = 0) & member(v4, v6) = 0 & member(v4, v5) = v8)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (union(v5, v6) = v7) | ~ (member(v4, v7) = 0) | ? [v8] : ? [v9] : (member(v4, v6) = v9 & member(v4, v5) = v8 & (v9 = 0 | v8 = 0))) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (intersection(v5, v6) = v7) | ~ (member(v4, v7) = 0) | (member(v4, v6) = 0 & member(v4, v5) = 0)) & ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (pre_order(v4, v5) = v6) | ? [v7] : ? [v8] : ? [v9] : ? [v10] : ? [v11] : ? [v12] : ? [v13] : ? [v14] : ? [v15] : ((v14 = 0 & v13 = 0 & v12 = 0 & v11 = 0 & v10 = 0 & ~ (v15 = 0) & apply(v4, v8, v9) = 0 & apply(v4, v7, v9) = v15 & apply(v4, v7, v8) = 0 & member(v9, v5) = 0 & member(v8, v5) = 0 & member(v7, v5) = 0) | (v8 = 0 & ~ (v9 = 0) & apply(v4, v7, v7) = v9 & member(v7, v5) = 0))) & ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (equivalence(v5, v4) = v6) | ? [v7] : ? [v8] : ? [v9] : ? [v10] : ? [v11] : ? [v12] : ? [v13] : ? [v14] : ? [v15] : ((v14 = 0 & v13 = 0 & v12 = 0 & v11 = 0 & v10 = 0 & ~ (v15 = 0) & apply(v5, v8, v9) = 0 & apply(v5, v7, v9) = v15 & apply(v5, v7, v8) = 0 & member(v9, v4) = 0 & member(v8, v4) = 0 & member(v7, v4) = 0) | (v11 = 0 & v10 = 0 & v9 = 0 & ~ (v12 = 0) & apply(v5, v8, v7) = v12 & apply(v5, v7, v8) = 0 & member(v8, v4) = 0 & member(v7, v4) = 0) | (v8 = 0 & ~ (v9 = 0) & apply(v5, v7, v7) = v9 & member(v7, v4) = 0))) & ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (partition(v4, v5) = v6) | ? [v7] : ? [v8] : ? [v9] : ? [v10] : ? [v11] : ? [v12] : ? [v13] : ((v13 = 0 & v12 = 0 & v10 = 0 & v9 = 0 & ~ (v8 = v7) & member(v11, v8) = 0 & member(v11, v7) = 0 & member(v8, v4) = 0 & member(v7, v4) = 0) | (v8 = 0 & ~ (v9 = 0) & subset(v7, v5) = v9 & member(v7, v4) = 0) | (v8 = 0 & member(v7, v5) = 0 & ! [v14] : ( ~ (member(v7, v14) = 0) | ? [v15] : ( ~ (v15 = 0) & member(v14, v4) = v15))))) & ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (disjoint(v4, v5) = v6) | ? [v7] : (member(v7, v5) = 0 & member(v7, v4) = 0)) & ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (singleton(v4) = v5) | ~ (member(v4, v5) = v6)) & ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (equal_set(v4, v5) = v6) | ? [v7] : ? [v8] : (subset(v5, v4) = v8 & subset(v4, v5) = v7 & ( ~ (v8 = 0) | ~ (v7 = 0)))) & ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (subset(v4, v5) = v6) | ? [v7] : ? [v8] : ( ~ (v8 = 0) & member(v7, v5) = v8 & member(v7, v4) = 0)) & ! [v4] : ! [v5] : ! [v6] : (v5 = v4 | ~ (product(v6) = v5) | ~ (product(v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : (v5 = v4 | ~ (sum(v6) = v5) | ~ (sum(v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : (v5 = v4 | ~ (singleton(v6) = v5) | ~ (singleton(v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : (v5 = v4 | ~ (singleton(v5) = v6) | ~ (member(v4, v6) = 0)) & ! [v4] : ! [v5] : ! [v6] : (v5 = v4 | ~ (power_set(v6) = v5) | ~ (power_set(v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ( ~ (partition(v4, v5) = 0) | ~ (member(v6, v5) = 0) | ? [v7] : (member(v7, v4) = 0 & member(v6, v7) = 0)) & ! [v4] : ! [v5] : ! [v6] : ( ~ (disjoint(v4, v5) = 0) | ~ (member(v6, v4) = 0) | ? [v7] : ( ~ (v7 = 0) & member(v6, v5) = v7)) & ! [v4] : ! [v5] : ! [v6] : ( ~ (sum(v5) = v6) | ~ (member(v4, v6) = 0) | ? [v7] : (member(v7, v5) = 0 & member(v4, v7) = 0)) & ! [v4] : ! [v5] : ! [v6] : ( ~ (power_set(v5) = v6) | ~ (member(v4, v6) = 0) | subset(v4, v5) = 0) & ! [v4] : ! [v5] : ! [v6] : ( ~ (subset(v4, v5) = 0) | ~ (member(v6, v4) = 0) | member(v6, v5) = 0) & ! [v4] : ! [v5] : ( ~ (equal_set(v4, v5) = 0) | (subset(v5, v4) = 0 & subset(v4, v5) = 0)) & ! [v4] : ~ (member(v4, empty_set) = 0)) % 6.20/2.06 | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3 yields: % 6.20/2.06 | (1) ~ (all_0_0_0 = 0) & pre_order(all_0_1_1, all_0_2_2) = all_0_0_0 & pre_order(all_0_1_1, all_0_3_3) = 0 & subset(all_0_2_2, all_0_3_3) = 0 & ! [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 | ~ (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 | ~ (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 | ~ (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] : ( ~ (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] : ( ~ (equal_set(v0, v1) = 0) | (subset(v1, v0) = 0 & subset(v0, v1) = 0)) & ! [v0] : ~ (member(v0, empty_set) = 0) % 6.20/2.08 | % 6.20/2.08 | Applying alpha-rule on (1) yields: % 6.20/2.08 | (2) ! [v0] : ! [v1] : ( ~ (equal_set(v0, v1) = 0) | (subset(v1, v0) = 0 & subset(v0, v1) = 0)) % 6.20/2.08 | (3) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (difference(v3, v2) = v1) | ~ (difference(v3, v2) = v0)) % 6.20/2.08 | (4) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (difference(v2, v1) = v3) | ~ (member(v0, v3) = 0) | ? [v4] : ( ~ (v4 = 0) & member(v0, v2) = 0 & member(v0, v1) = v4)) % 6.20/2.08 | (5) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (disjoint(v0, v1) = v2) | ? [v3] : (member(v3, v1) = 0 & member(v3, v0) = 0)) % 6.20/2.08 | (6) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (subset(v0, v1) = v2) | ? [v3] : ? [v4] : ( ~ (v4 = 0) & member(v3, v1) = v4 & member(v3, v0) = 0)) % 6.20/2.08 | (7) ! [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)) % 6.31/2.08 | (8) pre_order(all_0_1_1, all_0_2_2) = all_0_0_0 % 6.31/2.08 | (9) ! [v0] : ! [v1] : ! [v2] : ( ~ (power_set(v1) = v2) | ~ (member(v0, v2) = 0) | subset(v0, v1) = 0) % 6.31/2.08 | (10) ! [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)) % 6.31/2.08 | (11) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (singleton(v0) = v1) | ~ (member(v0, v1) = v2)) % 6.31/2.08 | (12) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (unordered_pair(v0, v1) = v2) | ~ (member(v0, v2) = v3)) % 6.31/2.08 | (13) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (disjoint(v3, v2) = v1) | ~ (disjoint(v3, v2) = v0)) % 6.31/2.08 | (14) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (unordered_pair(v3, v2) = v1) | ~ (unordered_pair(v3, v2) = v0)) % 6.31/2.08 | (15) ~ (all_0_0_0 = 0) % 6.31/2.08 | (16) ! [v0] : ! [v1] : ! [v2] : ( ~ (partition(v0, v1) = 0) | ~ (member(v2, v1) = 0) | ? [v3] : (member(v3, v0) = 0 & member(v2, v3) = 0)) % 6.31/2.08 | (17) ! [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))) % 6.31/2.09 | (18) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (union(v3, v2) = v1) | ~ (union(v3, v2) = v0)) % 6.31/2.09 | (19) ! [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))) % 6.31/2.09 | (20) ! [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)))) % 6.31/2.09 | (21) ! [v0] : ! [v1] : ! [v2] : ( ~ (disjoint(v0, v1) = 0) | ~ (member(v2, v0) = 0) | ? [v3] : ( ~ (v3 = 0) & member(v2, v1) = v3)) % 6.31/2.09 | (22) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (power_set(v1) = v2) | ~ (member(v0, v2) = v3) | ? [v4] : ( ~ (v4 = 0) & subset(v0, v1) = v4)) % 6.31/2.09 | (23) pre_order(all_0_1_1, all_0_3_3) = 0 % 6.31/2.09 | (24) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (subset(v3, v2) = v1) | ~ (subset(v3, v2) = v0)) % 6.31/2.09 | (25) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (power_set(v2) = v1) | ~ (power_set(v2) = v0)) % 6.31/2.09 | (26) ! [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)))) % 6.31/2.09 | (27) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (product(v2) = v1) | ~ (product(v2) = v0)) % 6.31/2.09 | (28) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (intersection(v3, v2) = v1) | ~ (intersection(v3, v2) = v0)) % 6.31/2.09 | (29) subset(all_0_2_2, all_0_3_3) = 0 % 6.31/2.09 | (30) ! [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)) % 6.31/2.09 | (31) ! [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))) % 6.31/2.09 | (32) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (sum(v2) = v1) | ~ (sum(v2) = v0)) % 6.31/2.09 | (33) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (unordered_pair(v1, v0) = v2) | ~ (member(v0, v2) = v3)) % 6.31/2.09 | (34) ! [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)))) % 6.31/2.09 | (35) ! [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)) % 6.31/2.10 | (36) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (equal_set(v0, v1) = v2) | ? [v3] : ? [v4] : (subset(v1, v0) = v4 & subset(v0, v1) = v3 & ( ~ (v4 = 0) | ~ (v3 = 0)))) % 6.31/2.10 | (37) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (equivalence(v1, v0) = 0) | ~ (apply(v1, v2, v2) = v3) | ? [v4] : ( ~ (v4 = 0) & member(v2, v0) = v4)) % 6.31/2.10 | (38) ! [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))) % 6.31/2.10 | (39) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (partition(v3, v2) = v1) | ~ (partition(v3, v2) = v0)) % 6.31/2.10 | (40) ! [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)) % 6.31/2.10 | (41) ! [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)))) % 6.31/2.10 | (42) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (singleton(v2) = v1) | ~ (singleton(v2) = v0)) % 6.31/2.10 | (43) ! [v0] : ~ (member(v0, empty_set) = 0) % 6.31/2.10 | (44) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equivalence(v3, v2) = v1) | ~ (equivalence(v3, v2) = v0)) % 6.31/2.10 | (45) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equal_set(v3, v2) = v1) | ~ (equal_set(v3, v2) = v0)) % 6.31/2.10 | (46) ! [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))))) % 6.31/2.10 | (47) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (equivalence_class(v2, v1, v0) = v4) | ~ (member(v3, v4) = 0) | (apply(v0, v2, v3) = 0 & member(v3, v1) = 0)) % 6.31/2.10 | (48) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (singleton(v1) = v2) | ~ (member(v0, v2) = 0)) % 6.31/2.10 | (49) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (apply(v4, v3, v2) = v1) | ~ (apply(v4, v3, v2) = v0)) % 6.31/2.10 | (50) ! [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))) % 6.31/2.10 | (51) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (pre_order(v0, v1) = 0) | ~ (apply(v0, v2, v2) = v3) | ? [v4] : ( ~ (v4 = 0) & member(v2, v1) = v4)) % 6.31/2.10 | (52) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (intersection(v1, v2) = v3) | ~ (member(v0, v3) = 0) | (member(v0, v2) = 0 & member(v0, v1) = 0)) % 6.31/2.10 | (53) ! [v0] : ! [v1] : ! [v2] : ( ~ (sum(v1) = v2) | ~ (member(v0, v2) = 0) | ? [v3] : (member(v3, v1) = 0 & member(v0, v3) = 0)) % 6.31/2.10 | (54) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v2 = v0 | v1 = v0 | ~ (unordered_pair(v1, v2) = v3) | ~ (member(v0, v3) = 0)) % 6.31/2.10 | (55) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (equivalence_class(v4, v3, v2) = v1) | ~ (equivalence_class(v4, v3, v2) = v0)) % 6.31/2.10 | (56) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (member(v3, v2) = v1) | ~ (member(v3, v2) = v0)) % 6.31/2.11 | (57) ! [v0] : ! [v1] : ! [v2] : ( ~ (subset(v0, v1) = 0) | ~ (member(v2, v0) = 0) | member(v2, v1) = 0) % 6.31/2.11 | (58) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (pre_order(v3, v2) = v1) | ~ (pre_order(v3, v2) = v0)) % 6.31/2.11 | (59) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (partition(v0, v1) = 0) | ~ (subset(v2, v1) = v3) | ? [v4] : ( ~ (v4 = 0) & member(v2, v0) = v4)) % 6.31/2.11 | % 6.31/2.11 | Instantiating formula (19) with all_0_0_0, all_0_2_2, all_0_1_1 and discharging atoms pre_order(all_0_1_1, all_0_2_2) = all_0_0_0, yields: % 6.31/2.11 | (60) all_0_0_0 = 0 | ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : ((v7 = 0 & v6 = 0 & v5 = 0 & v4 = 0 & v3 = 0 & ~ (v8 = 0) & apply(all_0_1_1, v1, v2) = 0 & apply(all_0_1_1, v0, v2) = v8 & apply(all_0_1_1, v0, v1) = 0 & member(v2, all_0_2_2) = 0 & member(v1, all_0_2_2) = 0 & member(v0, all_0_2_2) = 0) | (v1 = 0 & ~ (v2 = 0) & apply(all_0_1_1, v0, v0) = v2 & member(v0, all_0_2_2) = 0)) % 6.31/2.11 | % 6.31/2.11 +-Applying beta-rule and splitting (60), into two cases. % 6.31/2.11 |-Branch one: % 6.31/2.11 | (61) all_0_0_0 = 0 % 6.31/2.11 | % 6.31/2.11 | Equations (61) can reduce 15 to: % 6.31/2.11 | (62) $false % 6.31/2.11 | % 6.31/2.11 |-The branch is then unsatisfiable % 6.31/2.11 |-Branch two: % 6.31/2.11 | (15) ~ (all_0_0_0 = 0) % 6.31/2.11 | (64) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : ((v7 = 0 & v6 = 0 & v5 = 0 & v4 = 0 & v3 = 0 & ~ (v8 = 0) & apply(all_0_1_1, v1, v2) = 0 & apply(all_0_1_1, v0, v2) = v8 & apply(all_0_1_1, v0, v1) = 0 & member(v2, all_0_2_2) = 0 & member(v1, all_0_2_2) = 0 & member(v0, all_0_2_2) = 0) | (v1 = 0 & ~ (v2 = 0) & apply(all_0_1_1, v0, v0) = v2 & member(v0, all_0_2_2) = 0)) % 6.31/2.11 | % 6.31/2.11 | Instantiating (64) with all_14_0_4, all_14_1_5, all_14_2_6, all_14_3_7, all_14_4_8, all_14_5_9, all_14_6_10, all_14_7_11, all_14_8_12 yields: % 6.31/2.11 | (65) (all_14_1_5 = 0 & all_14_2_6 = 0 & all_14_3_7 = 0 & all_14_4_8 = 0 & all_14_5_9 = 0 & ~ (all_14_0_4 = 0) & apply(all_0_1_1, all_14_7_11, all_14_6_10) = 0 & apply(all_0_1_1, all_14_8_12, all_14_6_10) = all_14_0_4 & apply(all_0_1_1, all_14_8_12, all_14_7_11) = 0 & member(all_14_6_10, all_0_2_2) = 0 & member(all_14_7_11, all_0_2_2) = 0 & member(all_14_8_12, all_0_2_2) = 0) | (all_14_7_11 = 0 & ~ (all_14_6_10 = 0) & apply(all_0_1_1, all_14_8_12, all_14_8_12) = all_14_6_10 & member(all_14_8_12, all_0_2_2) = 0) % 6.31/2.11 | % 6.31/2.11 +-Applying beta-rule and splitting (65), into two cases. % 6.31/2.11 |-Branch one: % 6.31/2.11 | (66) all_14_1_5 = 0 & all_14_2_6 = 0 & all_14_3_7 = 0 & all_14_4_8 = 0 & all_14_5_9 = 0 & ~ (all_14_0_4 = 0) & apply(all_0_1_1, all_14_7_11, all_14_6_10) = 0 & apply(all_0_1_1, all_14_8_12, all_14_6_10) = all_14_0_4 & apply(all_0_1_1, all_14_8_12, all_14_7_11) = 0 & member(all_14_6_10, all_0_2_2) = 0 & member(all_14_7_11, all_0_2_2) = 0 & member(all_14_8_12, all_0_2_2) = 0 % 6.31/2.11 | % 6.31/2.11 | Applying alpha-rule on (66) yields: % 6.31/2.11 | (67) all_14_3_7 = 0 % 6.31/2.11 | (68) all_14_2_6 = 0 % 6.31/2.11 | (69) member(all_14_6_10, all_0_2_2) = 0 % 6.31/2.11 | (70) all_14_5_9 = 0 % 6.31/2.11 | (71) all_14_1_5 = 0 % 6.31/2.11 | (72) member(all_14_7_11, all_0_2_2) = 0 % 6.31/2.11 | (73) ~ (all_14_0_4 = 0) % 6.31/2.11 | (74) apply(all_0_1_1, all_14_8_12, all_14_7_11) = 0 % 6.31/2.11 | (75) all_14_4_8 = 0 % 6.31/2.11 | (76) apply(all_0_1_1, all_14_8_12, all_14_6_10) = all_14_0_4 % 6.31/2.11 | (77) apply(all_0_1_1, all_14_7_11, all_14_6_10) = 0 % 6.31/2.11 | (78) member(all_14_8_12, all_0_2_2) = 0 % 6.31/2.11 | % 6.31/2.11 | Instantiating formula (26) with all_14_0_4, all_14_6_10, all_14_7_11, all_14_8_12, all_0_3_3, all_0_1_1 and discharging atoms pre_order(all_0_1_1, all_0_3_3) = 0, apply(all_0_1_1, all_14_8_12, all_14_6_10) = all_14_0_4, apply(all_0_1_1, all_14_8_12, all_14_7_11) = 0, yields: % 6.31/2.11 | (79) all_14_0_4 = 0 | ? [v0] : ? [v1] : ? [v2] : ? [v3] : (apply(all_0_1_1, all_14_7_11, all_14_6_10) = v3 & member(all_14_6_10, all_0_3_3) = v2 & member(all_14_7_11, all_0_3_3) = v1 & member(all_14_8_12, all_0_3_3) = v0 & ( ~ (v3 = 0) | ~ (v2 = 0) | ~ (v1 = 0) | ~ (v0 = 0))) % 6.31/2.11 | % 6.31/2.11 | Instantiating formula (57) with all_14_6_10, all_0_3_3, all_0_2_2 and discharging atoms subset(all_0_2_2, all_0_3_3) = 0, member(all_14_6_10, all_0_2_2) = 0, yields: % 6.31/2.11 | (80) member(all_14_6_10, all_0_3_3) = 0 % 6.31/2.11 | % 6.31/2.11 | Instantiating formula (57) with all_14_7_11, all_0_3_3, all_0_2_2 and discharging atoms subset(all_0_2_2, all_0_3_3) = 0, member(all_14_7_11, all_0_2_2) = 0, yields: % 6.31/2.11 | (81) member(all_14_7_11, all_0_3_3) = 0 % 6.31/2.11 | % 6.31/2.11 | Instantiating formula (57) with all_14_8_12, all_0_3_3, all_0_2_2 and discharging atoms subset(all_0_2_2, all_0_3_3) = 0, member(all_14_8_12, all_0_2_2) = 0, yields: % 6.31/2.11 | (82) member(all_14_8_12, all_0_3_3) = 0 % 6.31/2.11 | % 6.31/2.11 +-Applying beta-rule and splitting (79), into two cases. % 6.31/2.11 |-Branch one: % 6.31/2.11 | (83) all_14_0_4 = 0 % 6.31/2.11 | % 6.31/2.11 | Equations (83) can reduce 73 to: % 6.31/2.11 | (62) $false % 6.31/2.11 | % 6.31/2.11 |-The branch is then unsatisfiable % 6.31/2.11 |-Branch two: % 6.31/2.11 | (73) ~ (all_14_0_4 = 0) % 6.31/2.11 | (86) ? [v0] : ? [v1] : ? [v2] : ? [v3] : (apply(all_0_1_1, all_14_7_11, all_14_6_10) = v3 & member(all_14_6_10, all_0_3_3) = v2 & member(all_14_7_11, all_0_3_3) = v1 & member(all_14_8_12, all_0_3_3) = v0 & ( ~ (v3 = 0) | ~ (v2 = 0) | ~ (v1 = 0) | ~ (v0 = 0))) % 6.31/2.12 | % 6.31/2.12 | Instantiating (86) with all_48_0_13, all_48_1_14, all_48_2_15, all_48_3_16 yields: % 6.31/2.12 | (87) apply(all_0_1_1, all_14_7_11, all_14_6_10) = all_48_0_13 & member(all_14_6_10, all_0_3_3) = all_48_1_14 & member(all_14_7_11, all_0_3_3) = all_48_2_15 & member(all_14_8_12, all_0_3_3) = all_48_3_16 & ( ~ (all_48_0_13 = 0) | ~ (all_48_1_14 = 0) | ~ (all_48_2_15 = 0) | ~ (all_48_3_16 = 0)) % 6.31/2.12 | % 6.31/2.12 | Applying alpha-rule on (87) yields: % 6.31/2.12 | (88) member(all_14_8_12, all_0_3_3) = all_48_3_16 % 6.31/2.12 | (89) member(all_14_6_10, all_0_3_3) = all_48_1_14 % 6.31/2.12 | (90) apply(all_0_1_1, all_14_7_11, all_14_6_10) = all_48_0_13 % 6.31/2.12 | (91) member(all_14_7_11, all_0_3_3) = all_48_2_15 % 6.31/2.12 | (92) ~ (all_48_0_13 = 0) | ~ (all_48_1_14 = 0) | ~ (all_48_2_15 = 0) | ~ (all_48_3_16 = 0) % 6.31/2.12 | % 6.31/2.12 | Instantiating formula (49) with all_0_1_1, all_14_7_11, all_14_6_10, all_48_0_13, 0 and discharging atoms apply(all_0_1_1, all_14_7_11, all_14_6_10) = all_48_0_13, apply(all_0_1_1, all_14_7_11, all_14_6_10) = 0, yields: % 6.31/2.12 | (93) all_48_0_13 = 0 % 6.31/2.12 | % 6.31/2.12 | Instantiating formula (56) with all_14_6_10, all_0_3_3, 0, all_48_1_14 and discharging atoms member(all_14_6_10, all_0_3_3) = all_48_1_14, member(all_14_6_10, all_0_3_3) = 0, yields: % 6.31/2.12 | (94) all_48_1_14 = 0 % 6.31/2.12 | % 6.31/2.12 | Instantiating formula (56) with all_14_7_11, all_0_3_3, all_48_2_15, 0 and discharging atoms member(all_14_7_11, all_0_3_3) = all_48_2_15, member(all_14_7_11, all_0_3_3) = 0, yields: % 6.31/2.12 | (95) all_48_2_15 = 0 % 6.31/2.12 | % 6.31/2.12 | Instantiating formula (56) with all_14_8_12, all_0_3_3, all_48_3_16, 0 and discharging atoms member(all_14_8_12, all_0_3_3) = all_48_3_16, member(all_14_8_12, all_0_3_3) = 0, yields: % 6.31/2.12 | (96) all_48_3_16 = 0 % 6.31/2.12 | % 6.31/2.12 +-Applying beta-rule and splitting (92), into two cases. % 6.31/2.12 |-Branch one: % 6.31/2.12 | (97) ~ (all_48_0_13 = 0) % 6.31/2.12 | % 6.31/2.12 | Equations (93) can reduce 97 to: % 6.31/2.12 | (62) $false % 6.31/2.12 | % 6.31/2.12 |-The branch is then unsatisfiable % 6.31/2.12 |-Branch two: % 6.31/2.12 | (93) all_48_0_13 = 0 % 6.31/2.12 | (100) ~ (all_48_1_14 = 0) | ~ (all_48_2_15 = 0) | ~ (all_48_3_16 = 0) % 6.31/2.12 | % 6.31/2.12 +-Applying beta-rule and splitting (100), into two cases. % 6.31/2.12 |-Branch one: % 6.31/2.12 | (101) ~ (all_48_1_14 = 0) % 6.31/2.12 | % 6.31/2.12 | Equations (94) can reduce 101 to: % 6.31/2.12 | (62) $false % 6.31/2.12 | % 6.31/2.12 |-The branch is then unsatisfiable % 6.31/2.12 |-Branch two: % 6.31/2.12 | (94) all_48_1_14 = 0 % 6.31/2.12 | (104) ~ (all_48_2_15 = 0) | ~ (all_48_3_16 = 0) % 6.31/2.12 | % 6.31/2.12 +-Applying beta-rule and splitting (104), into two cases. % 6.31/2.12 |-Branch one: % 6.31/2.12 | (105) ~ (all_48_2_15 = 0) % 6.31/2.12 | % 6.31/2.12 | Equations (95) can reduce 105 to: % 6.31/2.12 | (62) $false % 6.31/2.12 | % 6.31/2.12 |-The branch is then unsatisfiable % 6.31/2.12 |-Branch two: % 6.31/2.12 | (95) all_48_2_15 = 0 % 6.31/2.12 | (108) ~ (all_48_3_16 = 0) % 6.31/2.12 | % 6.31/2.12 | Equations (96) can reduce 108 to: % 6.31/2.12 | (62) $false % 6.31/2.12 | % 6.31/2.12 |-The branch is then unsatisfiable % 6.31/2.12 |-Branch two: % 6.31/2.12 | (110) all_14_7_11 = 0 & ~ (all_14_6_10 = 0) & apply(all_0_1_1, all_14_8_12, all_14_8_12) = all_14_6_10 & member(all_14_8_12, all_0_2_2) = 0 % 6.31/2.12 | % 6.31/2.12 | Applying alpha-rule on (110) yields: % 6.31/2.12 | (111) all_14_7_11 = 0 % 6.31/2.12 | (112) ~ (all_14_6_10 = 0) % 6.31/2.12 | (113) apply(all_0_1_1, all_14_8_12, all_14_8_12) = all_14_6_10 % 6.31/2.12 | (78) member(all_14_8_12, all_0_2_2) = 0 % 6.31/2.12 | % 6.31/2.12 | Instantiating formula (51) with all_14_6_10, all_14_8_12, all_0_3_3, all_0_1_1 and discharging atoms pre_order(all_0_1_1, all_0_3_3) = 0, apply(all_0_1_1, all_14_8_12, all_14_8_12) = all_14_6_10, yields: % 6.31/2.12 | (115) all_14_6_10 = 0 | ? [v0] : ( ~ (v0 = 0) & member(all_14_8_12, all_0_3_3) = v0) % 6.31/2.12 | % 6.31/2.12 | Instantiating formula (57) with all_14_8_12, all_0_3_3, all_0_2_2 and discharging atoms subset(all_0_2_2, all_0_3_3) = 0, member(all_14_8_12, all_0_2_2) = 0, yields: % 6.31/2.12 | (82) member(all_14_8_12, all_0_3_3) = 0 % 6.31/2.12 | % 6.31/2.12 +-Applying beta-rule and splitting (115), into two cases. % 6.31/2.12 |-Branch one: % 6.31/2.12 | (117) all_14_6_10 = 0 % 6.31/2.12 | % 6.31/2.12 | Equations (117) can reduce 112 to: % 6.31/2.12 | (62) $false % 6.31/2.12 | % 6.31/2.12 |-The branch is then unsatisfiable % 6.31/2.12 |-Branch two: % 6.31/2.12 | (112) ~ (all_14_6_10 = 0) % 6.31/2.12 | (120) ? [v0] : ( ~ (v0 = 0) & member(all_14_8_12, all_0_3_3) = v0) % 6.31/2.12 | % 6.31/2.12 | Instantiating (120) with all_44_0_17 yields: % 6.31/2.12 | (121) ~ (all_44_0_17 = 0) & member(all_14_8_12, all_0_3_3) = all_44_0_17 % 6.31/2.12 | % 6.31/2.12 | Applying alpha-rule on (121) yields: % 6.31/2.12 | (122) ~ (all_44_0_17 = 0) % 6.31/2.12 | (123) member(all_14_8_12, all_0_3_3) = all_44_0_17 % 6.31/2.12 | % 6.31/2.12 | Instantiating formula (56) with all_14_8_12, all_0_3_3, 0, all_44_0_17 and discharging atoms member(all_14_8_12, all_0_3_3) = all_44_0_17, member(all_14_8_12, all_0_3_3) = 0, yields: % 6.31/2.12 | (124) all_44_0_17 = 0 % 6.31/2.12 | % 6.31/2.12 | Equations (124) can reduce 122 to: % 6.31/2.12 | (62) $false % 6.31/2.12 | % 6.31/2.12 |-The branch is then unsatisfiable % 6.31/2.12 % SZS output end Proof for theBenchmark % 6.31/2.12 % 6.31/2.12 1529ms %------------------------------------------------------------------------------