%------------------------------------------------------------------------------ % File : ePrincess---1.0 % Problem : GRA002+4 : TPTP v8.1.0. Bugfixed v3.2.0. % Transfm : none % Format : tptp:raw % Command : ePrincess-casc -timeout=%d %s % Computer : n023.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 : Sat Jul 16 07:15:31 EDT 2022 % Result : Theorem 4.32s 1.66s % Output : Proof 6.70s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.03/0.13 % Problem : GRA002+4 : TPTP v8.1.0. Bugfixed v3.2.0. % 0.03/0.13 % Command : ePrincess-casc -timeout=%d %s % 0.13/0.34 % Computer : n023.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 : Tue May 31 02:48:56 EDT 2022 % 0.13/0.35 % CPUTime : % 0.54/0.61 ____ _ % 0.54/0.61 ___ / __ \_____(_)___ ________ __________ % 0.54/0.61 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/ % 0.54/0.61 / __/ ____/ / / / / / / /__/ __(__ |__ ) % 0.54/0.61 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/ % 0.54/0.61 % 0.54/0.61 A Theorem Prover for First-Order Logic % 0.54/0.61 (ePrincess v.1.0) % 0.54/0.61 % 0.54/0.61 (c) Philipp Rümmer, 2009-2015 % 0.54/0.61 (c) Peter Backeman, 2014-2015 % 0.54/0.61 (contributions by Angelo Brillout, Peter Baumgartner) % 0.54/0.61 Free software under GNU Lesser General Public License (LGPL). % 0.54/0.61 Bug reports to peter@backeman.se % 0.54/0.61 % 0.54/0.61 For more information, visit http://user.uu.se/~petba168/breu/ % 0.54/0.61 % 0.54/0.61 Loading /export/starexec/sandbox2/benchmark/theBenchmark.p ... % 0.70/0.66 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 1.78/1.01 Prover 0: Preprocessing ... % 3.11/1.38 Prover 0: Warning: ignoring some quantifiers % 3.11/1.41 Prover 0: Constructing countermodel ... % 4.32/1.66 Prover 0: proved (997ms) % 4.32/1.66 % 4.32/1.66 No countermodel exists, formula is valid % 4.32/1.66 % SZS status Theorem for theBenchmark % 4.32/1.66 % 4.32/1.66 Generating proof ... Warning: ignoring some quantifiers % 6.31/2.08 found it (size 21) % 6.31/2.08 % 6.31/2.08 % SZS output start Proof for theBenchmark % 6.31/2.08 Assumed formulas after preprocessing and simplification: % 6.31/2.08 | (0) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ? [v5] : (minus(v4, n1) = v5 & number_of_in(triangles, graph) = v0 & length_of(v1) = v4 & shortest_path(v2, v3, v1) & complete & ~ less_or_equal(v5, v0) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : ! [v13] : ( ~ (tail_of(v13) = v11) | ~ (tail_of(v8) = v11) | ~ (head_of(v9) = v12) | ~ shortest_path(v6, v7, v10) | ~ precedes(v8, v9, v10) | ? [v14] : ( ~ (v14 = v12) & head_of(v13) = v14)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : ! [v13] : ( ~ (tail_of(v8) = v11) | ~ (head_of(v13) = v12) | ~ (head_of(v9) = v12) | ~ shortest_path(v6, v7, v10) | ~ precedes(v8, v9, v10) | ? [v14] : ( ~ (v14 = v11) & tail_of(v13) = v14)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : ( ~ (tail_of(v8) = v11) | ~ (head_of(v9) = v12) | ~ shortest_path(v6, v7, v10) | ~ precedes(v9, v8, v10) | ~ precedes(v8, v9, v10)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ( ~ (length_of(v10) = v11) | ~ (length_of(v8) = v9) | ~ shortest_path(v6, v7, v8) | ~ path(v6, v7, v10) | less_or_equal(v9, v11)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ( ~ precedes(v9, v8, v6) | ~ precedes(v7, v8, v6) | ~ sequential(v7, v9) | ~ sequential(v7, v8) | ~ path(v10, v11, v6)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ( ~ precedes(v9, v8, v6) | ~ sequential(v7, v9) | ~ on_path(v8, v6) | ~ on_path(v7, v6) | ~ path(v10, v11, v6) | precedes(v7, v8, v6)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (tail_of(v9) = v10) | ~ on_path(v9, v8) | ~ path(v6, v7, v8) | in_path(v10, v8)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (tail_of(v9) = v10) | ~ on_path(v9, v8) | ~ path(v6, v7, v8) | edge(v9)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (tail_of(v9) = v10) | ~ on_path(v9, v8) | ~ path(v6, v7, v8) | ? [v11] : (head_of(v9) = v11 & in_path(v11, v8))) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (head_of(v9) = v10) | ~ on_path(v9, v8) | ~ path(v6, v7, v8) | in_path(v10, v8)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (head_of(v9) = v10) | ~ on_path(v9, v8) | ~ path(v6, v7, v8) | edge(v9)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (head_of(v9) = v10) | ~ on_path(v9, v8) | ~ path(v6, v7, v8) | ? [v11] : (tail_of(v9) = v11 & in_path(v11, v8))) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ precedes(v7, v8, v6) | ~ path(v9, v10, v6) | sequential(v7, v8) | ? [v11] : (precedes(v11, v8, v6) & sequential(v7, v11))) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ precedes(v7, v8, v6) | ~ path(v9, v10, v6) | on_path(v8, v6)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ precedes(v7, v8, v6) | ~ path(v9, v10, v6) | on_path(v7, v6)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ sequential(v7, v8) | ~ on_path(v8, v6) | ~ on_path(v7, v6) | ~ path(v9, v10, v6) | precedes(v7, v8, v6)) & ? [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (path_cons(v9, empty) = v10) | ~ vertex(v8) | ~ vertex(v7) | ~ edge(v9) | path(v7, v8, v6) | ? [v11] : ? [v12] : (tail_of(v9) = v11 & head_of(v9) = v12 & ( ~ (v11 = v7) | ( ! [v13] : ( ~ (path_cons(v9, v13) = v6) | ~ path(v12, v8, v13)) & ( ~ (v12 = v8) | ~ (v10 = v6)))))) & ? [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (head_of(v9) = v10) | ~ vertex(v8) | ~ vertex(v7) | ~ edge(v9) | path(v7, v8, v6) | ? [v11] : ? [v12] : (path_cons(v9, empty) = v12 & tail_of(v9) = v11 & ( ~ (v11 = v7) | ( ! [v13] : ( ~ (path_cons(v9, v13) = v6) | ~ path(v10, v8, v13)) & ( ~ (v12 = v6) | ~ (v10 = v8)))))) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = v8 | ~ (tail_of(v7) = v9) | ~ (head_of(v6) = v8) | ~ sequential(v6, v7)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v7 = v6 | ~ (minus(v9, v8) = v7) | ~ (minus(v9, v8) = v6)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v7 = v6 | ~ (number_of_in(v9, v8) = v7) | ~ (number_of_in(v9, v8) = v6)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v7 = v6 | ~ (length_of(v8) = v9) | ~ path(v6, v7, v8) | shortest_path(v6, v7, v8) | ? [v10] : ? [v11] : (length_of(v10) = v11 & path(v6, v7, v10) & ~ less_or_equal(v9, v11))) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v7 = v6 | ~ (path_cons(v9, v8) = v7) | ~ (path_cons(v9, v8) = v6)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (number_of_in(triangles, v6) = v9) | ~ shortest_path(v7, v8, v6) | number_of_in(sequential_pairs, v6) = v9) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (number_of_in(triangles, v6) = v9) | ~ path(v7, v8, v6) | ? [v10] : ? [v11] : ((v10 = v9 & number_of_in(sequential_pairs, v6) = v9) | (sequential(v10, v11) & on_path(v11, v6) & on_path(v10, v6) & ! [v12] : ~ triangle(v10, v11, v12)))) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (number_of_in(sequential_pairs, v8) = v9) | ~ path(v6, v7, v8) | ? [v10] : (minus(v10, n1) = v9 & length_of(v8) = v10)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (number_of_in(sequential_pairs, v6) = v9) | ~ shortest_path(v7, v8, v6) | number_of_in(triangles, v6) = v9) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (number_of_in(sequential_pairs, v6) = v9) | ~ path(v7, v8, v6) | ? [v10] : ? [v11] : ((v10 = v9 & number_of_in(triangles, v6) = v9) | (sequential(v10, v11) & on_path(v11, v6) & on_path(v10, v6) & ! [v12] : ~ triangle(v10, v11, v12)))) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (number_of_in(edges, v8) = v9) | ~ path(v6, v7, v8) | length_of(v8) = v9) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (length_of(v8) = v9) | ~ shortest_path(v6, v7, v8) | path(v6, v7, v8)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (length_of(v8) = v9) | ~ path(v6, v7, v8) | number_of_in(edges, v8) = v9) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (length_of(v8) = v9) | ~ path(v6, v7, v8) | ? [v10] : (minus(v9, n1) = v10 & number_of_in(sequential_pairs, v8) = v10)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (tail_of(v7) = v9) | ~ (head_of(v6) = v8) | ~ sequential(v6, v7) | edge(v7)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (tail_of(v7) = v9) | ~ (head_of(v6) = v8) | ~ sequential(v6, v7) | edge(v6)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ in_path(v9, v8) | ~ path(v6, v7, v8) | vertex(v9)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ in_path(v9, v8) | ~ path(v6, v7, v8) | ? [v10] : ? [v11] : ? [v12] : (tail_of(v10) = v12 & head_of(v10) = v11 & on_path(v10, v8) & (v12 = v9 | v11 = v9))) & ? [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (tail_of(v9) = v7) | ~ vertex(v8) | ~ vertex(v7) | ~ edge(v9) | path(v7, v8, v6) | ? [v10] : ? [v11] : (path_cons(v9, empty) = v11 & head_of(v9) = v10 & ! [v12] : ( ~ (path_cons(v9, v12) = v6) | ~ path(v10, v8, v12)) & ( ~ (v11 = v6) | ~ (v10 = v8)))) & ! [v6] : ! [v7] : ! [v8] : (v7 = v6 | ~ (length_of(v8) = v7) | ~ (length_of(v8) = v6)) & ! [v6] : ! [v7] : ! [v8] : (v7 = v6 | ~ (tail_of(v8) = v7) | ~ (tail_of(v8) = v6)) & ! [v6] : ! [v7] : ! [v8] : (v7 = v6 | ~ (tail_of(v7) = v8) | ~ (head_of(v6) = v8) | ~ edge(v7) | ~ edge(v6) | sequential(v6, v7)) & ! [v6] : ! [v7] : ! [v8] : (v7 = v6 | ~ (head_of(v8) = v7) | ~ (head_of(v8) = v6)) & ! [v6] : ! [v7] : ! [v8] : ( ~ (number_of_in(v6, v7) = v8) | ? [v9] : (number_of_in(v6, graph) = v9 & less_or_equal(v8, v9))) & ! [v6] : ! [v7] : ! [v8] : ( ~ (length_of(v7) = v8) | ~ shortest_path(v6, v6, v7)) & ! [v6] : ! [v7] : ! [v8] : ( ~ (tail_of(v6) = v8) | ~ (head_of(v6) = v7) | ~ sequential(v6, v6)) & ! [v6] : ! [v7] : ! [v8] : ( ~ triangle(v6, v7, v8) | sequential(v8, v6)) & ! [v6] : ! [v7] : ! [v8] : ( ~ triangle(v6, v7, v8) | sequential(v7, v8)) & ! [v6] : ! [v7] : ! [v8] : ( ~ triangle(v6, v7, v8) | sequential(v6, v7)) & ! [v6] : ! [v7] : ! [v8] : ( ~ triangle(v6, v7, v8) | edge(v8)) & ! [v6] : ! [v7] : ! [v8] : ( ~ triangle(v6, v7, v8) | edge(v7)) & ! [v6] : ! [v7] : ! [v8] : ( ~ triangle(v6, v7, v8) | edge(v6)) & ! [v6] : ! [v7] : ! [v8] : ( ~ sequential(v8, v6) | ~ sequential(v7, v8) | ~ sequential(v6, v7) | ~ edge(v8) | ~ edge(v7) | ~ edge(v6) | triangle(v6, v7, v8)) & ! [v6] : ! [v7] : ! [v8] : ( ~ path(v6, v7, v8) | vertex(v7)) & ! [v6] : ! [v7] : ! [v8] : ( ~ path(v6, v7, v8) | vertex(v6)) & ! [v6] : ! [v7] : ! [v8] : ( ~ path(v6, v7, v8) | ? [v9] : ? [v10] : ? [v11] : ? [v12] : ? [v13] : (path_cons(v9, empty) = v11 & tail_of(v9) = v6 & head_of(v9) = v10 & edge(v9) & ( ~ (v11 = v8) | ~ (v10 = v7) | ! [v14] : ( ~ (path_cons(v9, v14) = v8) | ~ path(v7, v7, v14))) & ((v13 = v8 & path_cons(v9, v12) = v8 & path(v10, v7, v12)) | (v11 = v8 & v10 = v7)))) & ! [v6] : ! [v7] : (v7 = v6 | ~ vertex(v7) | ~ vertex(v6) | ? [v8] : ? [v9] : ? [v10] : (tail_of(v8) = v10 & head_of(v8) = v9 & edge(v8) & ((v10 = v7 & v9 = v6) | (v10 = v6 & v9 = v7)))) & ! [v6] : ! [v7] : ( ~ (tail_of(v6) = v7) | ~ edge(v6) | vertex(v7)) & ! [v6] : ! [v7] : ( ~ (tail_of(v6) = v7) | ~ edge(v6) | ? [v8] : ( ~ (v8 = v7) & head_of(v6) = v8)) & ! [v6] : ! [v7] : ( ~ (tail_of(v6) = v7) | ~ edge(v6) | ? [v8] : (head_of(v6) = v8 & vertex(v8))) & ! [v6] : ! [v7] : ( ~ (head_of(v6) = v7) | ~ edge(v6) | vertex(v7)) & ! [v6] : ! [v7] : ( ~ (head_of(v6) = v7) | ~ edge(v6) | ? [v8] : ( ~ (v8 = v7) & tail_of(v6) = v8)) & ! [v6] : ! [v7] : ( ~ (head_of(v6) = v7) | ~ edge(v6) | ? [v8] : (tail_of(v6) = v8 & vertex(v8)))) % 6.53/2.13 | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3, all_0_4_4, all_0_5_5 yields: % 6.53/2.13 | (1) minus(all_0_1_1, n1) = all_0_0_0 & number_of_in(triangles, graph) = all_0_5_5 & length_of(all_0_4_4) = all_0_1_1 & shortest_path(all_0_3_3, all_0_2_2, all_0_4_4) & complete & ~ less_or_equal(all_0_0_0, all_0_5_5) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (tail_of(v7) = v5) | ~ (tail_of(v2) = v5) | ~ (head_of(v3) = v6) | ~ shortest_path(v0, v1, v4) | ~ precedes(v2, v3, v4) | ? [v8] : ( ~ (v8 = v6) & head_of(v7) = v8)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (tail_of(v2) = v5) | ~ (head_of(v7) = v6) | ~ (head_of(v3) = v6) | ~ shortest_path(v0, v1, v4) | ~ precedes(v2, v3, v4) | ? [v8] : ( ~ (v8 = v5) & tail_of(v7) = v8)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (tail_of(v2) = v5) | ~ (head_of(v3) = v6) | ~ shortest_path(v0, v1, v4) | ~ precedes(v3, v2, v4) | ~ precedes(v2, v3, v4)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (length_of(v4) = v5) | ~ (length_of(v2) = v3) | ~ shortest_path(v0, v1, v2) | ~ path(v0, v1, v4) | less_or_equal(v3, v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ precedes(v3, v2, v0) | ~ precedes(v1, v2, v0) | ~ sequential(v1, v3) | ~ sequential(v1, v2) | ~ path(v4, v5, v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ precedes(v3, v2, v0) | ~ sequential(v1, v3) | ~ on_path(v2, v0) | ~ on_path(v1, v0) | ~ path(v4, v5, v0) | precedes(v1, v2, v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (tail_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | in_path(v4, v2)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (tail_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | edge(v3)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (tail_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | ? [v5] : (head_of(v3) = v5 & in_path(v5, v2))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (head_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | in_path(v4, v2)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (head_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | edge(v3)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (head_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | ? [v5] : (tail_of(v3) = v5 & in_path(v5, v2))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ precedes(v1, v2, v0) | ~ path(v3, v4, v0) | sequential(v1, v2) | ? [v5] : (precedes(v5, v2, v0) & sequential(v1, v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ precedes(v1, v2, v0) | ~ path(v3, v4, v0) | on_path(v2, v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ precedes(v1, v2, v0) | ~ path(v3, v4, v0) | on_path(v1, v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ sequential(v1, v2) | ~ on_path(v2, v0) | ~ on_path(v1, v0) | ~ path(v3, v4, v0) | precedes(v1, v2, v0)) & ? [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (path_cons(v3, empty) = v4) | ~ vertex(v2) | ~ vertex(v1) | ~ edge(v3) | path(v1, v2, v0) | ? [v5] : ? [v6] : (tail_of(v3) = v5 & head_of(v3) = v6 & ( ~ (v5 = v1) | ( ! [v7] : ( ~ (path_cons(v3, v7) = v0) | ~ path(v6, v2, v7)) & ( ~ (v6 = v2) | ~ (v4 = v0)))))) & ? [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (head_of(v3) = v4) | ~ vertex(v2) | ~ vertex(v1) | ~ edge(v3) | path(v1, v2, v0) | ? [v5] : ? [v6] : (path_cons(v3, empty) = v6 & tail_of(v3) = v5 & ( ~ (v5 = v1) | ( ! [v7] : ( ~ (path_cons(v3, v7) = v0) | ~ path(v4, v2, v7)) & ( ~ (v6 = v0) | ~ (v4 = v2)))))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = v2 | ~ (tail_of(v1) = v3) | ~ (head_of(v0) = v2) | ~ sequential(v0, v1)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (minus(v3, v2) = v1) | ~ (minus(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (number_of_in(v3, v2) = v1) | ~ (number_of_in(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (length_of(v2) = v3) | ~ path(v0, v1, v2) | shortest_path(v0, v1, v2) | ? [v4] : ? [v5] : (length_of(v4) = v5 & path(v0, v1, v4) & ~ less_or_equal(v3, v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (path_cons(v3, v2) = v1) | ~ (path_cons(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (number_of_in(triangles, v0) = v3) | ~ shortest_path(v1, v2, v0) | number_of_in(sequential_pairs, v0) = v3) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (number_of_in(triangles, v0) = v3) | ~ path(v1, v2, v0) | ? [v4] : ? [v5] : ((v4 = v3 & number_of_in(sequential_pairs, v0) = v3) | (sequential(v4, v5) & on_path(v5, v0) & on_path(v4, v0) & ! [v6] : ~ triangle(v4, v5, v6)))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (number_of_in(sequential_pairs, v2) = v3) | ~ path(v0, v1, v2) | ? [v4] : (minus(v4, n1) = v3 & length_of(v2) = v4)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (number_of_in(sequential_pairs, v0) = v3) | ~ shortest_path(v1, v2, v0) | number_of_in(triangles, v0) = v3) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (number_of_in(sequential_pairs, v0) = v3) | ~ path(v1, v2, v0) | ? [v4] : ? [v5] : ((v4 = v3 & number_of_in(triangles, v0) = v3) | (sequential(v4, v5) & on_path(v5, v0) & on_path(v4, v0) & ! [v6] : ~ triangle(v4, v5, v6)))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (number_of_in(edges, v2) = v3) | ~ path(v0, v1, v2) | length_of(v2) = v3) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (length_of(v2) = v3) | ~ shortest_path(v0, v1, v2) | path(v0, v1, v2)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (length_of(v2) = v3) | ~ path(v0, v1, v2) | number_of_in(edges, v2) = v3) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (length_of(v2) = v3) | ~ path(v0, v1, v2) | ? [v4] : (minus(v3, n1) = v4 & number_of_in(sequential_pairs, v2) = v4)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (tail_of(v1) = v3) | ~ (head_of(v0) = v2) | ~ sequential(v0, v1) | edge(v1)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (tail_of(v1) = v3) | ~ (head_of(v0) = v2) | ~ sequential(v0, v1) | edge(v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ in_path(v3, v2) | ~ path(v0, v1, v2) | vertex(v3)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ in_path(v3, v2) | ~ path(v0, v1, v2) | ? [v4] : ? [v5] : ? [v6] : (tail_of(v4) = v6 & head_of(v4) = v5 & on_path(v4, v2) & (v6 = v3 | v5 = v3))) & ? [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (tail_of(v3) = v1) | ~ vertex(v2) | ~ vertex(v1) | ~ edge(v3) | path(v1, v2, v0) | ? [v4] : ? [v5] : (path_cons(v3, empty) = v5 & head_of(v3) = v4 & ! [v6] : ( ~ (path_cons(v3, v6) = v0) | ~ path(v4, v2, v6)) & ( ~ (v5 = v0) | ~ (v4 = v2)))) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (length_of(v2) = v1) | ~ (length_of(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (tail_of(v2) = v1) | ~ (tail_of(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (tail_of(v1) = v2) | ~ (head_of(v0) = v2) | ~ edge(v1) | ~ edge(v0) | sequential(v0, v1)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (head_of(v2) = v1) | ~ (head_of(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (number_of_in(v0, v1) = v2) | ? [v3] : (number_of_in(v0, graph) = v3 & less_or_equal(v2, v3))) & ! [v0] : ! [v1] : ! [v2] : ( ~ (length_of(v1) = v2) | ~ shortest_path(v0, v0, v1)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (tail_of(v0) = v2) | ~ (head_of(v0) = v1) | ~ sequential(v0, v0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | sequential(v2, v0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | sequential(v1, v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | sequential(v0, v1)) & ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | edge(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | edge(v1)) & ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | edge(v0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ sequential(v2, v0) | ~ sequential(v1, v2) | ~ sequential(v0, v1) | ~ edge(v2) | ~ edge(v1) | ~ edge(v0) | triangle(v0, v1, v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ path(v0, v1, v2) | vertex(v1)) & ! [v0] : ! [v1] : ! [v2] : ( ~ path(v0, v1, v2) | vertex(v0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ path(v0, v1, v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : (path_cons(v3, empty) = v5 & tail_of(v3) = v0 & head_of(v3) = v4 & edge(v3) & ( ~ (v5 = v2) | ~ (v4 = v1) | ! [v8] : ( ~ (path_cons(v3, v8) = v2) | ~ path(v1, v1, v8))) & ((v7 = v2 & path_cons(v3, v6) = v2 & path(v4, v1, v6)) | (v5 = v2 & v4 = v1)))) & ! [v0] : ! [v1] : (v1 = v0 | ~ vertex(v1) | ~ vertex(v0) | ? [v2] : ? [v3] : ? [v4] : (tail_of(v2) = v4 & head_of(v2) = v3 & edge(v2) & ((v4 = v1 & v3 = v0) | (v4 = v0 & v3 = v1)))) & ! [v0] : ! [v1] : ( ~ (tail_of(v0) = v1) | ~ edge(v0) | vertex(v1)) & ! [v0] : ! [v1] : ( ~ (tail_of(v0) = v1) | ~ edge(v0) | ? [v2] : ( ~ (v2 = v1) & head_of(v0) = v2)) & ! [v0] : ! [v1] : ( ~ (tail_of(v0) = v1) | ~ edge(v0) | ? [v2] : (head_of(v0) = v2 & vertex(v2))) & ! [v0] : ! [v1] : ( ~ (head_of(v0) = v1) | ~ edge(v0) | vertex(v1)) & ! [v0] : ! [v1] : ( ~ (head_of(v0) = v1) | ~ edge(v0) | ? [v2] : ( ~ (v2 = v1) & tail_of(v0) = v2)) & ! [v0] : ! [v1] : ( ~ (head_of(v0) = v1) | ~ edge(v0) | ? [v2] : (tail_of(v0) = v2 & vertex(v2))) % 6.53/2.15 | % 6.53/2.15 | Applying alpha-rule on (1) yields: % 6.53/2.15 | (2) shortest_path(all_0_3_3, all_0_2_2, all_0_4_4) % 6.70/2.15 | (3) ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | edge(v2)) % 6.70/2.15 | (4) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (tail_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | edge(v3)) % 6.70/2.15 | (5) ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | sequential(v1, v2)) % 6.70/2.15 | (6) ! [v0] : ! [v1] : ! [v2] : ( ~ (number_of_in(v0, v1) = v2) | ? [v3] : (number_of_in(v0, graph) = v3 & less_or_equal(v2, v3))) % 6.70/2.15 | (7) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ precedes(v3, v2, v0) | ~ precedes(v1, v2, v0) | ~ sequential(v1, v3) | ~ sequential(v1, v2) | ~ path(v4, v5, v0)) % 6.70/2.15 | (8) minus(all_0_1_1, n1) = all_0_0_0 % 6.70/2.15 | (9) ! [v0] : ! [v1] : ! [v2] : ( ~ path(v0, v1, v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : (path_cons(v3, empty) = v5 & tail_of(v3) = v0 & head_of(v3) = v4 & edge(v3) & ( ~ (v5 = v2) | ~ (v4 = v1) | ! [v8] : ( ~ (path_cons(v3, v8) = v2) | ~ path(v1, v1, v8))) & ((v7 = v2 & path_cons(v3, v6) = v2 & path(v4, v1, v6)) | (v5 = v2 & v4 = v1)))) % 6.70/2.15 | (10) ! [v0] : ! [v1] : ( ~ (head_of(v0) = v1) | ~ edge(v0) | ? [v2] : (tail_of(v0) = v2 & vertex(v2))) % 6.70/2.15 | (11) ! [v0] : ! [v1] : ( ~ (tail_of(v0) = v1) | ~ edge(v0) | ? [v2] : (head_of(v0) = v2 & vertex(v2))) % 6.70/2.15 | (12) ! [v0] : ! [v1] : ! [v2] : ( ~ path(v0, v1, v2) | vertex(v0)) % 6.70/2.16 | (13) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (number_of_in(v3, v2) = v1) | ~ (number_of_in(v3, v2) = v0)) % 6.70/2.16 | (14) ! [v0] : ! [v1] : ! [v2] : ( ~ sequential(v2, v0) | ~ sequential(v1, v2) | ~ sequential(v0, v1) | ~ edge(v2) | ~ edge(v1) | ~ edge(v0) | triangle(v0, v1, v2)) % 6.70/2.16 | (15) ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | sequential(v0, v1)) % 6.70/2.16 | (16) ! [v0] : ! [v1] : ( ~ (head_of(v0) = v1) | ~ edge(v0) | vertex(v1)) % 6.70/2.16 | (17) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ in_path(v3, v2) | ~ path(v0, v1, v2) | ? [v4] : ? [v5] : ? [v6] : (tail_of(v4) = v6 & head_of(v4) = v5 & on_path(v4, v2) & (v6 = v3 | v5 = v3))) % 6.70/2.16 | (18) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (minus(v3, v2) = v1) | ~ (minus(v3, v2) = v0)) % 6.70/2.16 | (19) ? [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (head_of(v3) = v4) | ~ vertex(v2) | ~ vertex(v1) | ~ edge(v3) | path(v1, v2, v0) | ? [v5] : ? [v6] : (path_cons(v3, empty) = v6 & tail_of(v3) = v5 & ( ~ (v5 = v1) | ( ! [v7] : ( ~ (path_cons(v3, v7) = v0) | ~ path(v4, v2, v7)) & ( ~ (v6 = v0) | ~ (v4 = v2)))))) % 6.70/2.16 | (20) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (tail_of(v1) = v3) | ~ (head_of(v0) = v2) | ~ sequential(v0, v1) | edge(v1)) % 6.70/2.16 | (21) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (tail_of(v2) = v5) | ~ (head_of(v3) = v6) | ~ shortest_path(v0, v1, v4) | ~ precedes(v3, v2, v4) | ~ precedes(v2, v3, v4)) % 6.70/2.16 | (22) ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | edge(v1)) % 6.70/2.16 | (23) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (length_of(v2) = v3) | ~ shortest_path(v0, v1, v2) | path(v0, v1, v2)) % 6.70/2.16 | (24) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (length_of(v2) = v1) | ~ (length_of(v2) = v0)) % 6.70/2.16 | (25) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (length_of(v2) = v3) | ~ path(v0, v1, v2) | number_of_in(edges, v2) = v3) % 6.70/2.16 | (26) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ precedes(v1, v2, v0) | ~ path(v3, v4, v0) | on_path(v2, v0)) % 6.70/2.16 | (27) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (head_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | in_path(v4, v2)) % 6.70/2.16 | (28) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ precedes(v3, v2, v0) | ~ sequential(v1, v3) | ~ on_path(v2, v0) | ~ on_path(v1, v0) | ~ path(v4, v5, v0) | precedes(v1, v2, v0)) % 6.70/2.16 | (29) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (tail_of(v2) = v5) | ~ (head_of(v7) = v6) | ~ (head_of(v3) = v6) | ~ shortest_path(v0, v1, v4) | ~ precedes(v2, v3, v4) | ? [v8] : ( ~ (v8 = v5) & tail_of(v7) = v8)) % 6.70/2.16 | (30) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (head_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | edge(v3)) % 6.70/2.16 | (31) length_of(all_0_4_4) = all_0_1_1 % 6.70/2.16 | (32) ! [v0] : ! [v1] : (v1 = v0 | ~ vertex(v1) | ~ vertex(v0) | ? [v2] : ? [v3] : ? [v4] : (tail_of(v2) = v4 & head_of(v2) = v3 & edge(v2) & ((v4 = v1 & v3 = v0) | (v4 = v0 & v3 = v1)))) % 6.70/2.16 | (33) ! [v0] : ! [v1] : ( ~ (tail_of(v0) = v1) | ~ edge(v0) | vertex(v1)) % 6.70/2.16 | (34) ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | edge(v0)) % 6.70/2.16 | (35) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (length_of(v4) = v5) | ~ (length_of(v2) = v3) | ~ shortest_path(v0, v1, v2) | ~ path(v0, v1, v4) | less_or_equal(v3, v5)) % 6.70/2.16 | (36) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ precedes(v1, v2, v0) | ~ path(v3, v4, v0) | on_path(v1, v0)) % 6.70/2.16 | (37) ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | sequential(v2, v0)) % 6.70/2.17 | (38) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (number_of_in(sequential_pairs, v0) = v3) | ~ path(v1, v2, v0) | ? [v4] : ? [v5] : ((v4 = v3 & number_of_in(triangles, v0) = v3) | (sequential(v4, v5) & on_path(v5, v0) & on_path(v4, v0) & ! [v6] : ~ triangle(v4, v5, v6)))) % 6.70/2.17 | (39) ~ less_or_equal(all_0_0_0, all_0_5_5) % 6.70/2.17 | (40) ! [v0] : ! [v1] : ( ~ (head_of(v0) = v1) | ~ edge(v0) | ? [v2] : ( ~ (v2 = v1) & tail_of(v0) = v2)) % 6.70/2.17 | (41) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = v2 | ~ (tail_of(v1) = v3) | ~ (head_of(v0) = v2) | ~ sequential(v0, v1)) % 6.70/2.17 | (42) number_of_in(triangles, graph) = all_0_5_5 % 6.70/2.17 | (43) ! [v0] : ! [v1] : ! [v2] : ( ~ (tail_of(v0) = v2) | ~ (head_of(v0) = v1) | ~ sequential(v0, v0)) % 6.70/2.17 | (44) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (number_of_in(edges, v2) = v3) | ~ path(v0, v1, v2) | length_of(v2) = v3) % 6.70/2.17 | (45) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (tail_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | in_path(v4, v2)) % 6.70/2.17 | (46) complete % 6.70/2.17 | (47) ! [v0] : ! [v1] : ( ~ (tail_of(v0) = v1) | ~ edge(v0) | ? [v2] : ( ~ (v2 = v1) & head_of(v0) = v2)) % 6.70/2.17 | (48) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (tail_of(v7) = v5) | ~ (tail_of(v2) = v5) | ~ (head_of(v3) = v6) | ~ shortest_path(v0, v1, v4) | ~ precedes(v2, v3, v4) | ? [v8] : ( ~ (v8 = v6) & head_of(v7) = v8)) % 6.70/2.17 | (49) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ sequential(v1, v2) | ~ on_path(v2, v0) | ~ on_path(v1, v0) | ~ path(v3, v4, v0) | precedes(v1, v2, v0)) % 6.70/2.17 | (50) ! [v0] : ! [v1] : ! [v2] : ( ~ path(v0, v1, v2) | vertex(v1)) % 6.70/2.17 | (51) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (path_cons(v3, v2) = v1) | ~ (path_cons(v3, v2) = v0)) % 6.70/2.17 | (52) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (length_of(v2) = v3) | ~ path(v0, v1, v2) | ? [v4] : (minus(v3, n1) = v4 & number_of_in(sequential_pairs, v2) = v4)) % 6.70/2.17 | (53) ? [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (tail_of(v3) = v1) | ~ vertex(v2) | ~ vertex(v1) | ~ edge(v3) | path(v1, v2, v0) | ? [v4] : ? [v5] : (path_cons(v3, empty) = v5 & head_of(v3) = v4 & ! [v6] : ( ~ (path_cons(v3, v6) = v0) | ~ path(v4, v2, v6)) & ( ~ (v5 = v0) | ~ (v4 = v2)))) % 6.70/2.17 | (54) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (head_of(v2) = v1) | ~ (head_of(v2) = v0)) % 6.70/2.17 | (55) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (tail_of(v1) = v3) | ~ (head_of(v0) = v2) | ~ sequential(v0, v1) | edge(v0)) % 6.70/2.17 | (56) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (tail_of(v1) = v2) | ~ (head_of(v0) = v2) | ~ edge(v1) | ~ edge(v0) | sequential(v0, v1)) % 6.70/2.17 | (57) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ precedes(v1, v2, v0) | ~ path(v3, v4, v0) | sequential(v1, v2) | ? [v5] : (precedes(v5, v2, v0) & sequential(v1, v5))) % 6.70/2.17 | (58) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (length_of(v2) = v3) | ~ path(v0, v1, v2) | shortest_path(v0, v1, v2) | ? [v4] : ? [v5] : (length_of(v4) = v5 & path(v0, v1, v4) & ~ less_or_equal(v3, v5))) % 6.70/2.17 | (59) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (number_of_in(sequential_pairs, v0) = v3) | ~ shortest_path(v1, v2, v0) | number_of_in(triangles, v0) = v3) % 6.70/2.17 | (60) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (number_of_in(triangles, v0) = v3) | ~ shortest_path(v1, v2, v0) | number_of_in(sequential_pairs, v0) = v3) % 6.70/2.17 | (61) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (tail_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | ? [v5] : (head_of(v3) = v5 & in_path(v5, v2))) % 6.70/2.17 | (62) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (head_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | ? [v5] : (tail_of(v3) = v5 & in_path(v5, v2))) % 6.70/2.18 | (63) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ in_path(v3, v2) | ~ path(v0, v1, v2) | vertex(v3)) % 6.70/2.18 | (64) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (number_of_in(triangles, v0) = v3) | ~ path(v1, v2, v0) | ? [v4] : ? [v5] : ((v4 = v3 & number_of_in(sequential_pairs, v0) = v3) | (sequential(v4, v5) & on_path(v5, v0) & on_path(v4, v0) & ! [v6] : ~ triangle(v4, v5, v6)))) % 6.70/2.18 | (65) ! [v0] : ! [v1] : ! [v2] : ( ~ (length_of(v1) = v2) | ~ shortest_path(v0, v0, v1)) % 6.70/2.18 | (66) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (tail_of(v2) = v1) | ~ (tail_of(v2) = v0)) % 6.70/2.18 | (67) ? [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (path_cons(v3, empty) = v4) | ~ vertex(v2) | ~ vertex(v1) | ~ edge(v3) | path(v1, v2, v0) | ? [v5] : ? [v6] : (tail_of(v3) = v5 & head_of(v3) = v6 & ( ~ (v5 = v1) | ( ! [v7] : ( ~ (path_cons(v3, v7) = v0) | ~ path(v6, v2, v7)) & ( ~ (v6 = v2) | ~ (v4 = v0)))))) % 6.70/2.18 | (68) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (number_of_in(sequential_pairs, v2) = v3) | ~ path(v0, v1, v2) | ? [v4] : (minus(v4, n1) = v3 & length_of(v2) = v4)) % 6.70/2.18 | % 6.70/2.18 | Instantiating formula (6) with all_0_5_5, graph, triangles and discharging atoms number_of_in(triangles, graph) = all_0_5_5, yields: % 6.70/2.18 | (69) ? [v0] : (number_of_in(triangles, graph) = v0 & less_or_equal(all_0_5_5, v0)) % 6.70/2.18 | % 6.70/2.18 | Instantiating formula (23) with all_0_1_1, all_0_4_4, all_0_2_2, all_0_3_3 and discharging atoms length_of(all_0_4_4) = all_0_1_1, shortest_path(all_0_3_3, all_0_2_2, all_0_4_4), yields: % 6.70/2.18 | (70) path(all_0_3_3, all_0_2_2, all_0_4_4) % 6.70/2.18 | % 6.70/2.18 | Instantiating (69) with all_15_0_9 yields: % 6.70/2.18 | (71) number_of_in(triangles, graph) = all_15_0_9 & less_or_equal(all_0_5_5, all_15_0_9) % 6.70/2.18 | % 6.70/2.18 | Applying alpha-rule on (71) yields: % 6.70/2.18 | (72) number_of_in(triangles, graph) = all_15_0_9 % 6.70/2.18 | (73) less_or_equal(all_0_5_5, all_15_0_9) % 6.70/2.18 | % 6.70/2.18 | Instantiating formula (13) with triangles, graph, all_15_0_9, all_0_5_5 and discharging atoms number_of_in(triangles, graph) = all_15_0_9, number_of_in(triangles, graph) = all_0_5_5, yields: % 6.70/2.18 | (74) all_15_0_9 = all_0_5_5 % 6.70/2.18 | % 6.70/2.18 | From (74) and (72) follows: % 6.70/2.18 | (42) number_of_in(triangles, graph) = all_0_5_5 % 6.70/2.18 | % 6.70/2.18 | Instantiating formula (52) with all_0_1_1, all_0_4_4, all_0_2_2, all_0_3_3 and discharging atoms length_of(all_0_4_4) = all_0_1_1, path(all_0_3_3, all_0_2_2, all_0_4_4), yields: % 6.70/2.18 | (76) ? [v0] : (minus(all_0_1_1, n1) = v0 & number_of_in(sequential_pairs, all_0_4_4) = v0) % 6.70/2.18 | % 6.70/2.18 | Instantiating (76) with all_27_0_10 yields: % 6.70/2.18 | (77) minus(all_0_1_1, n1) = all_27_0_10 & number_of_in(sequential_pairs, all_0_4_4) = all_27_0_10 % 6.70/2.18 | % 6.70/2.18 | Applying alpha-rule on (77) yields: % 6.70/2.18 | (78) minus(all_0_1_1, n1) = all_27_0_10 % 6.70/2.18 | (79) number_of_in(sequential_pairs, all_0_4_4) = all_27_0_10 % 6.70/2.18 | % 6.70/2.18 | Instantiating formula (18) with all_0_1_1, n1, all_27_0_10, all_0_0_0 and discharging atoms minus(all_0_1_1, n1) = all_27_0_10, minus(all_0_1_1, n1) = all_0_0_0, yields: % 6.70/2.19 | (80) all_27_0_10 = all_0_0_0 % 6.70/2.19 | % 6.70/2.19 | From (80) and (79) follows: % 6.70/2.19 | (81) number_of_in(sequential_pairs, all_0_4_4) = all_0_0_0 % 6.70/2.19 | % 6.70/2.19 | Instantiating formula (59) with all_0_0_0, all_0_2_2, all_0_3_3, all_0_4_4 and discharging atoms number_of_in(sequential_pairs, all_0_4_4) = all_0_0_0, shortest_path(all_0_3_3, all_0_2_2, all_0_4_4), yields: % 6.70/2.19 | (82) number_of_in(triangles, all_0_4_4) = all_0_0_0 % 6.70/2.19 | % 6.70/2.19 | Instantiating formula (6) with all_0_0_0, all_0_4_4, triangles and discharging atoms number_of_in(triangles, all_0_4_4) = all_0_0_0, yields: % 6.70/2.19 | (83) ? [v0] : (number_of_in(triangles, graph) = v0 & less_or_equal(all_0_0_0, v0)) % 6.70/2.19 | % 6.70/2.19 | Instantiating (83) with all_65_0_25 yields: % 6.70/2.19 | (84) number_of_in(triangles, graph) = all_65_0_25 & less_or_equal(all_0_0_0, all_65_0_25) % 6.70/2.19 | % 6.70/2.19 | Applying alpha-rule on (84) yields: % 6.70/2.19 | (85) number_of_in(triangles, graph) = all_65_0_25 % 6.70/2.19 | (86) less_or_equal(all_0_0_0, all_65_0_25) % 6.70/2.19 | % 6.70/2.19 | Instantiating formula (13) with triangles, graph, all_65_0_25, all_0_5_5 and discharging atoms number_of_in(triangles, graph) = all_65_0_25, number_of_in(triangles, graph) = all_0_5_5, yields: % 6.70/2.19 | (87) all_65_0_25 = all_0_5_5 % 6.70/2.19 | % 6.70/2.19 | From (87) and (86) follows: % 6.70/2.19 | (88) less_or_equal(all_0_0_0, all_0_5_5) % 6.70/2.19 | % 6.70/2.19 | Using (88) and (39) yields: % 6.70/2.19 | (89) $false % 6.70/2.19 | % 6.70/2.19 |-The branch is then unsatisfiable % 6.70/2.19 % SZS output end Proof for theBenchmark % 6.70/2.19 % 6.70/2.19 1570ms %------------------------------------------------------------------------------