%------------------------------------------------------------------------------ % File : ePrincess---1.0 % Problem : GRA010+2 : TPTP v8.1.0. Bugfixed v3.2.0. % Transfm : none % Format : tptp:raw % Command : ePrincess-casc -timeout=%d %s % Computer : n027.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 07:15:32 EDT 2022 % Result : Theorem 3.98s 1.51s % Output : Proof 6.16s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.11 % Problem : GRA010+2 : TPTP v8.1.0. Bugfixed v3.2.0. % 0.12/0.12 % Command : ePrincess-casc -timeout=%d %s % 0.13/0.33 % Computer : n027.cluster.edu % 0.13/0.33 % Model : x86_64 x86_64 % 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.13/0.33 % Memory : 8042.1875MB % 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.13/0.33 % CPULimit : 300 % 0.13/0.33 % WCLimit : 600 % 0.13/0.33 % DateTime : Mon May 30 22:54:46 EDT 2022 % 0.13/0.33 % CPUTime : % 0.19/0.56 ____ _ % 0.19/0.56 ___ / __ \_____(_)___ ________ __________ % 0.19/0.56 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/ % 0.19/0.56 / __/ ____/ / / / / / / /__/ __(__ |__ ) % 0.19/0.56 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/ % 0.19/0.56 % 0.19/0.56 A Theorem Prover for First-Order Logic % 0.19/0.56 (ePrincess v.1.0) % 0.19/0.56 % 0.19/0.56 (c) Philipp Rümmer, 2009-2015 % 0.19/0.56 (c) Peter Backeman, 2014-2015 % 0.19/0.56 (contributions by Angelo Brillout, Peter Baumgartner) % 0.19/0.56 Free software under GNU Lesser General Public License (LGPL). % 0.19/0.56 Bug reports to peter@backeman.se % 0.19/0.56 % 0.19/0.56 For more information, visit http://user.uu.se/~petba168/breu/ % 0.19/0.56 % 0.19/0.56 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ... % 0.65/0.61 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 1.80/0.93 Prover 0: Preprocessing ... % 2.88/1.26 Prover 0: Warning: ignoring some quantifiers % 2.88/1.29 Prover 0: Constructing countermodel ... % 3.87/1.50 Prover 0: proved (892ms) % 3.98/1.51 % 3.98/1.51 No countermodel exists, formula is valid % 3.98/1.51 % SZS status Theorem for theBenchmark % 3.98/1.51 % 3.98/1.51 Generating proof ... Warning: ignoring some quantifiers % 5.79/1.93 found it (size 14) % 5.79/1.93 % 5.79/1.93 % SZS output start Proof for theBenchmark % 5.79/1.93 Assumed formulas after preprocessing and simplification: % 5.79/1.93 | (0) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ( ~ (v4 = v3) & number_of_in(triangles, v0) = v4 & number_of_in(sequential_pairs, v0) = v3 & path(v1, v2, v0) & complete & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : ( ~ (tail_of(v12) = v10) | ~ (tail_of(v7) = v10) | ~ (head_of(v8) = v11) | ~ shortest_path(v5, v6, v9) | ~ precedes(v7, v8, v9) | ? [v13] : ( ~ (v13 = v11) & head_of(v12) = v13)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : ( ~ (tail_of(v7) = v10) | ~ (head_of(v12) = v11) | ~ (head_of(v8) = v11) | ~ shortest_path(v5, v6, v9) | ~ precedes(v7, v8, v9) | ? [v13] : ( ~ (v13 = v10) & tail_of(v12) = v13)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ( ~ (tail_of(v7) = v10) | ~ (head_of(v8) = v11) | ~ shortest_path(v5, v6, v9) | ~ precedes(v8, v7, v9) | ~ precedes(v7, v8, v9)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (length_of(v9) = v10) | ~ (length_of(v7) = v8) | ~ shortest_path(v5, v6, v7) | ~ path(v5, v6, v9) | less_or_equal(v8, v10)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ precedes(v8, v7, v5) | ~ precedes(v6, v7, v5) | ~ sequential(v6, v8) | ~ sequential(v6, v7) | ~ path(v9, v10, v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ precedes(v8, v7, v5) | ~ sequential(v6, v8) | ~ on_path(v7, v5) | ~ on_path(v6, v5) | ~ path(v9, v10, v5) | precedes(v6, v7, v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (tail_of(v8) = v9) | ~ on_path(v8, v7) | ~ path(v5, v6, v7) | in_path(v9, v7)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (tail_of(v8) = v9) | ~ on_path(v8, v7) | ~ path(v5, v6, v7) | edge(v8)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (tail_of(v8) = v9) | ~ on_path(v8, v7) | ~ path(v5, v6, v7) | ? [v10] : (head_of(v8) = v10 & in_path(v10, v7))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (head_of(v8) = v9) | ~ on_path(v8, v7) | ~ path(v5, v6, v7) | in_path(v9, v7)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (head_of(v8) = v9) | ~ on_path(v8, v7) | ~ path(v5, v6, v7) | edge(v8)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (head_of(v8) = v9) | ~ on_path(v8, v7) | ~ path(v5, v6, v7) | ? [v10] : (tail_of(v8) = v10 & in_path(v10, v7))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ shortest_path(v5, v6, v9) | ~ precedes(v7, v8, v9) | ~ sequential(v7, v8) | ? [v10] : triangle(v7, v8, v10)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ precedes(v6, v7, v5) | ~ path(v8, v9, v5) | sequential(v6, v7) | ? [v10] : (precedes(v10, v7, v5) & sequential(v6, v10))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ precedes(v6, v7, v5) | ~ path(v8, v9, v5) | on_path(v7, v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ precedes(v6, v7, v5) | ~ path(v8, v9, v5) | on_path(v6, v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ sequential(v6, v7) | ~ on_path(v7, v5) | ~ on_path(v6, v5) | ~ path(v8, v9, v5) | precedes(v6, v7, v5)) & ? [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (path_cons(v8, empty) = v9) | ~ vertex(v7) | ~ vertex(v6) | ~ edge(v8) | path(v6, v7, v5) | ? [v10] : ? [v11] : (tail_of(v8) = v10 & head_of(v8) = v11 & ( ~ (v10 = v6) | ( ! [v12] : ( ~ (path_cons(v8, v12) = v5) | ~ path(v11, v7, v12)) & ( ~ (v11 = v7) | ~ (v9 = v5)))))) & ? [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (head_of(v8) = v9) | ~ vertex(v7) | ~ vertex(v6) | ~ edge(v8) | path(v6, v7, v5) | ? [v10] : ? [v11] : (path_cons(v8, empty) = v11 & tail_of(v8) = v10 & ( ~ (v10 = v6) | ( ! [v12] : ( ~ (path_cons(v8, v12) = v5) | ~ path(v9, v7, v12)) & ( ~ (v11 = v5) | ~ (v9 = v7)))))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = v7 | ~ (tail_of(v6) = v8) | ~ (head_of(v5) = v7) | ~ sequential(v5, v6)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (minus(v8, v7) = v6) | ~ (minus(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (number_of_in(v8, v7) = v6) | ~ (number_of_in(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (length_of(v7) = v8) | ~ path(v5, v6, v7) | shortest_path(v5, v6, v7) | ? [v9] : ? [v10] : (length_of(v9) = v10 & path(v5, v6, v9) & ~ less_or_equal(v8, v10))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (path_cons(v8, v7) = v6) | ~ (path_cons(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (number_of_in(triangles, v5) = v8) | ~ path(v6, v7, v5) | ? [v9] : ? [v10] : ((v9 = v8 & number_of_in(sequential_pairs, v5) = v8) | (sequential(v9, v10) & on_path(v10, v5) & on_path(v9, v5) & ! [v11] : ~ triangle(v9, v10, v11)))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (number_of_in(sequential_pairs, v7) = v8) | ~ path(v5, v6, v7) | ? [v9] : (minus(v9, n1) = v8 & length_of(v7) = v9)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (number_of_in(sequential_pairs, v5) = v8) | ~ path(v6, v7, v5) | ? [v9] : ? [v10] : ((v9 = v8 & number_of_in(triangles, v5) = v8) | (sequential(v9, v10) & on_path(v10, v5) & on_path(v9, v5) & ! [v11] : ~ triangle(v9, v10, v11)))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (number_of_in(edges, v7) = v8) | ~ path(v5, v6, v7) | length_of(v7) = v8) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (length_of(v7) = v8) | ~ shortest_path(v5, v6, v7) | path(v5, v6, v7)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (length_of(v7) = v8) | ~ path(v5, v6, v7) | number_of_in(edges, v7) = v8) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (length_of(v7) = v8) | ~ path(v5, v6, v7) | ? [v9] : (minus(v8, n1) = v9 & number_of_in(sequential_pairs, v7) = v9)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (tail_of(v6) = v8) | ~ (head_of(v5) = v7) | ~ sequential(v5, v6) | edge(v6)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (tail_of(v6) = v8) | ~ (head_of(v5) = v7) | ~ sequential(v5, v6) | edge(v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ in_path(v8, v7) | ~ path(v5, v6, v7) | vertex(v8)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ in_path(v8, v7) | ~ path(v5, v6, v7) | ? [v9] : ? [v10] : ? [v11] : (tail_of(v9) = v11 & head_of(v9) = v10 & on_path(v9, v7) & (v11 = v8 | v10 = v8))) & ? [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (tail_of(v8) = v6) | ~ vertex(v7) | ~ vertex(v6) | ~ edge(v8) | path(v6, v7, v5) | ? [v9] : ? [v10] : (path_cons(v8, empty) = v10 & head_of(v8) = v9 & ! [v11] : ( ~ (path_cons(v8, v11) = v5) | ~ path(v9, v7, v11)) & ( ~ (v10 = v5) | ~ (v9 = v7)))) & ! [v5] : ! [v6] : ! [v7] : (v6 = v5 | ~ (length_of(v7) = v6) | ~ (length_of(v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : (v6 = v5 | ~ (tail_of(v7) = v6) | ~ (tail_of(v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : (v6 = v5 | ~ (tail_of(v6) = v7) | ~ (head_of(v5) = v7) | ~ edge(v6) | ~ edge(v5) | sequential(v5, v6)) & ! [v5] : ! [v6] : ! [v7] : (v6 = v5 | ~ (head_of(v7) = v6) | ~ (head_of(v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ( ~ (number_of_in(v5, v6) = v7) | ? [v8] : (number_of_in(v5, graph) = v8 & less_or_equal(v7, v8))) & ! [v5] : ! [v6] : ! [v7] : ( ~ (length_of(v6) = v7) | ~ shortest_path(v5, v5, v6)) & ! [v5] : ! [v6] : ! [v7] : ( ~ (tail_of(v5) = v7) | ~ (head_of(v5) = v6) | ~ sequential(v5, v5)) & ! [v5] : ! [v6] : ! [v7] : ( ~ triangle(v5, v6, v7) | sequential(v7, v5)) & ! [v5] : ! [v6] : ! [v7] : ( ~ triangle(v5, v6, v7) | sequential(v6, v7)) & ! [v5] : ! [v6] : ! [v7] : ( ~ triangle(v5, v6, v7) | sequential(v5, v6)) & ! [v5] : ! [v6] : ! [v7] : ( ~ triangle(v5, v6, v7) | edge(v7)) & ! [v5] : ! [v6] : ! [v7] : ( ~ triangle(v5, v6, v7) | edge(v6)) & ! [v5] : ! [v6] : ! [v7] : ( ~ triangle(v5, v6, v7) | edge(v5)) & ! [v5] : ! [v6] : ! [v7] : ( ~ sequential(v7, v5) | ~ sequential(v6, v7) | ~ sequential(v5, v6) | ~ edge(v7) | ~ edge(v6) | ~ edge(v5) | triangle(v5, v6, v7)) & ! [v5] : ! [v6] : ! [v7] : ( ~ path(v5, v6, v7) | vertex(v6)) & ! [v5] : ! [v6] : ! [v7] : ( ~ path(v5, v6, v7) | vertex(v5)) & ! [v5] : ! [v6] : ! [v7] : ( ~ path(v5, v6, v7) | ? [v8] : ? [v9] : ? [v10] : ? [v11] : ? [v12] : (path_cons(v8, empty) = v10 & tail_of(v8) = v5 & head_of(v8) = v9 & edge(v8) & ( ~ (v10 = v7) | ~ (v9 = v6) | ! [v13] : ( ~ (path_cons(v8, v13) = v7) | ~ path(v6, v6, v13))) & ((v12 = v7 & path_cons(v8, v11) = v7 & path(v9, v6, v11)) | (v10 = v7 & v9 = v6)))) & ! [v5] : ! [v6] : (v6 = v5 | ~ vertex(v6) | ~ vertex(v5) | ? [v7] : ? [v8] : ? [v9] : (tail_of(v7) = v9 & head_of(v7) = v8 & edge(v7) & ((v9 = v6 & v8 = v5) | (v9 = v5 & v8 = v6)))) & ! [v5] : ! [v6] : ( ~ (tail_of(v5) = v6) | ~ edge(v5) | vertex(v6)) & ! [v5] : ! [v6] : ( ~ (tail_of(v5) = v6) | ~ edge(v5) | ? [v7] : ( ~ (v7 = v6) & head_of(v5) = v7)) & ! [v5] : ! [v6] : ( ~ (tail_of(v5) = v6) | ~ edge(v5) | ? [v7] : (head_of(v5) = v7 & vertex(v7))) & ! [v5] : ! [v6] : ( ~ (head_of(v5) = v6) | ~ edge(v5) | vertex(v6)) & ! [v5] : ! [v6] : ( ~ (head_of(v5) = v6) | ~ edge(v5) | ? [v7] : ( ~ (v7 = v6) & tail_of(v5) = v7)) & ! [v5] : ! [v6] : ( ~ (head_of(v5) = v6) | ~ edge(v5) | ? [v7] : (tail_of(v5) = v7 & vertex(v7))) & ! [v5] : ! [v6] : ( ~ sequential(v5, v6) | ~ on_path(v6, v0) | ~ on_path(v5, v0) | ? [v7] : triangle(v5, v6, v7))) % 5.79/1.98 | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3, all_0_4_4 yields: % 5.79/1.98 | (1) ~ (all_0_0_0 = all_0_1_1) & number_of_in(triangles, all_0_4_4) = all_0_0_0 & number_of_in(sequential_pairs, all_0_4_4) = all_0_1_1 & path(all_0_3_3, all_0_2_2, all_0_4_4) & complete & ! [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] : ( ~ shortest_path(v0, v1, v4) | ~ precedes(v2, v3, v4) | ~ sequential(v2, v3) | ? [v5] : triangle(v2, v3, v5)) & ! [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) | ~ 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) | ~ 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))) & ! [v0] : ! [v1] : ( ~ sequential(v0, v1) | ~ on_path(v1, all_0_4_4) | ~ on_path(v0, all_0_4_4) | ? [v2] : triangle(v0, v1, v2)) % 6.16/2.00 | % 6.16/2.00 | Applying alpha-rule on (1) yields: % 6.16/2.00 | (2) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (tail_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | edge(v3)) % 6.16/2.00 | (3) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (tail_of(v1) = v2) | ~ (head_of(v0) = v2) | ~ edge(v1) | ~ edge(v0) | sequential(v0, v1)) % 6.16/2.01 | (4) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (tail_of(v1) = v3) | ~ (head_of(v0) = v2) | ~ sequential(v0, v1) | edge(v0)) % 6.16/2.01 | (5) ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | edge(v2)) % 6.16/2.01 | (6) ! [v0] : ! [v1] : ( ~ (tail_of(v0) = v1) | ~ edge(v0) | ? [v2] : ( ~ (v2 = v1) & head_of(v0) = v2)) % 6.16/2.01 | (7) ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | sequential(v1, v2)) % 6.16/2.01 | (8) number_of_in(triangles, all_0_4_4) = all_0_0_0 % 6.16/2.01 | (9) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ precedes(v3, v2, v0) | ~ precedes(v1, v2, v0) | ~ sequential(v1, v3) | ~ sequential(v1, v2) | ~ path(v4, v5, v0)) % 6.16/2.01 | (10) ! [v0] : ! [v1] : ! [v2] : ( ~ (length_of(v1) = v2) | ~ shortest_path(v0, v0, v1)) % 6.16/2.01 | (11) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (minus(v3, v2) = v1) | ~ (minus(v3, v2) = v0)) % 6.16/2.01 | (12) ! [v0] : ! [v1] : ! [v2] : ( ~ path(v0, v1, v2) | vertex(v0)) % 6.16/2.01 | (13) ! [v0] : ! [v1] : ! [v2] : ( ~ sequential(v2, v0) | ~ sequential(v1, v2) | ~ sequential(v0, v1) | ~ edge(v2) | ~ edge(v1) | ~ edge(v0) | triangle(v0, v1, v2)) % 6.16/2.01 | (14) ? [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)))))) % 6.16/2.01 | (15) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ shortest_path(v0, v1, v4) | ~ precedes(v2, v3, v4) | ~ sequential(v2, v3) | ? [v5] : triangle(v2, v3, v5)) % 6.16/2.01 | (16) ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | sequential(v0, v1)) % 6.16/2.01 | (17) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (path_cons(v3, v2) = v1) | ~ (path_cons(v3, v2) = v0)) % 6.16/2.01 | (18) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (tail_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | in_path(v4, v2)) % 6.16/2.01 | (19) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (length_of(v2) = v3) | ~ path(v0, v1, v2) | number_of_in(edges, v2) = v3) % 6.16/2.01 | (20) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (length_of(v2) = v3) | ~ path(v0, v1, v2) | ? [v4] : (minus(v3, n1) = v4 & number_of_in(sequential_pairs, v2) = v4)) % 6.16/2.01 | (21) ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | edge(v1)) % 6.16/2.01 | (22) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (number_of_in(edges, v2) = v3) | ~ path(v0, v1, v2) | length_of(v2) = v3) % 6.16/2.01 | (23) ! [v0] : ! [v1] : ! [v2] : ( ~ (number_of_in(v0, v1) = v2) | ? [v3] : (number_of_in(v0, graph) = v3 & less_or_equal(v2, v3))) % 6.16/2.01 | (24) ! [v0] : ! [v1] : ( ~ sequential(v0, v1) | ~ on_path(v1, all_0_4_4) | ~ on_path(v0, all_0_4_4) | ? [v2] : triangle(v0, v1, v2)) % 6.16/2.01 | (25) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (tail_of(v1) = v3) | ~ (head_of(v0) = v2) | ~ sequential(v0, v1) | edge(v1)) % 6.16/2.01 | (26) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (number_of_in(sequential_pairs, v2) = v3) | ~ path(v0, v1, v2) | ? [v4] : (minus(v4, n1) = v3 & length_of(v2) = v4)) % 6.16/2.01 | (27) ! [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))) % 6.16/2.01 | (28) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ precedes(v1, v2, v0) | ~ path(v3, v4, v0) | on_path(v2, v0)) % 6.16/2.01 | (29) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (length_of(v2) = v3) | ~ shortest_path(v0, v1, v2) | path(v0, v1, v2)) % 6.16/2.01 | (30) ! [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))) % 6.16/2.01 | (31) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (number_of_in(v3, v2) = v1) | ~ (number_of_in(v3, v2) = v0)) % 6.16/2.01 | (32) ! [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)) % 6.16/2.02 | (33) ! [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)))) % 6.16/2.02 | (34) ! [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)) % 6.16/2.02 | (35) ! [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)))) % 6.16/2.02 | (36) ! [v0] : ! [v1] : ( ~ (head_of(v0) = v1) | ~ edge(v0) | ? [v2] : (tail_of(v0) = v2 & vertex(v2))) % 6.16/2.02 | (37) ! [v0] : ! [v1] : ( ~ (tail_of(v0) = v1) | ~ edge(v0) | ? [v2] : (head_of(v0) = v2 & vertex(v2))) % 6.16/2.02 | (38) ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | edge(v0)) % 6.16/2.02 | (39) ! [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)) % 6.16/2.02 | (40) number_of_in(sequential_pairs, all_0_4_4) = all_0_1_1 % 6.16/2.02 | (41) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (tail_of(v2) = v1) | ~ (tail_of(v2) = v0)) % 6.16/2.02 | (42) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (head_of(v2) = v1) | ~ (head_of(v2) = v0)) % 6.16/2.02 | (43) ! [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)))) % 6.16/2.02 | (44) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ precedes(v1, v2, v0) | ~ path(v3, v4, v0) | on_path(v1, v0)) % 6.16/2.02 | (45) ! [v0] : ! [v1] : ! [v2] : ( ~ triangle(v0, v1, v2) | sequential(v2, v0)) % 6.16/2.02 | (46) complete % 6.16/2.02 | (47) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ sequential(v1, v2) | ~ on_path(v2, v0) | ~ on_path(v1, v0) | ~ path(v3, v4, v0) | precedes(v1, v2, v0)) % 6.16/2.02 | (48) ! [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))) % 6.16/2.02 | (49) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (length_of(v2) = v1) | ~ (length_of(v2) = v0)) % 6.16/2.02 | (50) ! [v0] : ! [v1] : ! [v2] : ( ~ (tail_of(v0) = v2) | ~ (head_of(v0) = v1) | ~ sequential(v0, v0)) % 6.16/2.02 | (51) ! [v0] : ! [v1] : ! [v2] : ( ~ path(v0, v1, v2) | vertex(v1)) % 6.16/2.02 | (52) ! [v0] : ! [v1] : ( ~ (head_of(v0) = v1) | ~ edge(v0) | vertex(v1)) % 6.16/2.02 | (53) ! [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)) % 6.16/2.02 | (54) ? [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)))))) % 6.16/2.02 | (55) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = v2 | ~ (tail_of(v1) = v3) | ~ (head_of(v0) = v2) | ~ sequential(v0, v1)) % 6.16/2.03 | (56) ? [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)))) % 6.16/2.03 | (57) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (head_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | in_path(v4, v2)) % 6.16/2.03 | (58) ! [v0] : ! [v1] : ( ~ (tail_of(v0) = v1) | ~ edge(v0) | vertex(v1)) % 6.16/2.03 | (59) ! [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)) % 6.16/2.03 | (60) ! [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))) % 6.16/2.03 | (61) ! [v0] : ! [v1] : ( ~ (head_of(v0) = v1) | ~ edge(v0) | ? [v2] : ( ~ (v2 = v1) & tail_of(v0) = v2)) % 6.16/2.03 | (62) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ precedes(v1, v2, v0) | ~ path(v3, v4, v0) | sequential(v1, v2) | ? [v5] : (precedes(v5, v2, v0) & sequential(v1, v5))) % 6.16/2.03 | (63) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (head_of(v3) = v4) | ~ on_path(v3, v2) | ~ path(v0, v1, v2) | edge(v3)) % 6.16/2.03 | (64) ! [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)))) % 6.16/2.03 | (65) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ in_path(v3, v2) | ~ path(v0, v1, v2) | vertex(v3)) % 6.16/2.03 | (66) path(all_0_3_3, all_0_2_2, all_0_4_4) % 6.16/2.03 | (67) ~ (all_0_0_0 = all_0_1_1) % 6.16/2.03 | % 6.16/2.03 | Instantiating formula (43) with all_0_0_0, all_0_2_2, all_0_3_3, all_0_4_4 and discharging atoms number_of_in(triangles, all_0_4_4) = all_0_0_0, path(all_0_3_3, all_0_2_2, all_0_4_4), yields: % 6.16/2.03 | (68) ? [v0] : ? [v1] : ((v0 = all_0_0_0 & number_of_in(sequential_pairs, all_0_4_4) = all_0_0_0) | (sequential(v0, v1) & on_path(v1, all_0_4_4) & on_path(v0, all_0_4_4) & ! [v2] : ~ triangle(v0, v1, v2))) % 6.16/2.03 | % 6.16/2.03 | Instantiating (68) with all_23_0_16, all_23_1_17 yields: % 6.16/2.03 | (69) (all_23_1_17 = all_0_0_0 & number_of_in(sequential_pairs, all_0_4_4) = all_0_0_0) | (sequential(all_23_1_17, all_23_0_16) & on_path(all_23_0_16, all_0_4_4) & on_path(all_23_1_17, all_0_4_4) & ! [v0] : ~ triangle(all_23_1_17, all_23_0_16, v0)) % 6.16/2.03 | % 6.16/2.03 +-Applying beta-rule and splitting (69), into two cases. % 6.16/2.03 |-Branch one: % 6.16/2.03 | (70) all_23_1_17 = all_0_0_0 & number_of_in(sequential_pairs, all_0_4_4) = all_0_0_0 % 6.16/2.03 | % 6.16/2.03 | Applying alpha-rule on (70) yields: % 6.16/2.03 | (71) all_23_1_17 = all_0_0_0 % 6.16/2.03 | (72) number_of_in(sequential_pairs, all_0_4_4) = all_0_0_0 % 6.16/2.03 | % 6.16/2.03 | Instantiating formula (31) with sequential_pairs, all_0_4_4, all_0_0_0, all_0_1_1 and discharging atoms number_of_in(sequential_pairs, all_0_4_4) = all_0_0_0, number_of_in(sequential_pairs, all_0_4_4) = all_0_1_1, yields: % 6.16/2.03 | (73) all_0_0_0 = all_0_1_1 % 6.16/2.03 | % 6.16/2.03 | Equations (73) can reduce 67 to: % 6.16/2.03 | (74) $false % 6.16/2.04 | % 6.16/2.04 |-The branch is then unsatisfiable % 6.16/2.04 |-Branch two: % 6.16/2.04 | (75) sequential(all_23_1_17, all_23_0_16) & on_path(all_23_0_16, all_0_4_4) & on_path(all_23_1_17, all_0_4_4) & ! [v0] : ~ triangle(all_23_1_17, all_23_0_16, v0) % 6.16/2.04 | % 6.16/2.04 | Applying alpha-rule on (75) yields: % 6.16/2.04 | (76) sequential(all_23_1_17, all_23_0_16) % 6.16/2.04 | (77) on_path(all_23_0_16, all_0_4_4) % 6.16/2.04 | (78) on_path(all_23_1_17, all_0_4_4) % 6.16/2.04 | (79) ! [v0] : ~ triangle(all_23_1_17, all_23_0_16, v0) % 6.16/2.04 | % 6.16/2.04 | Instantiating formula (24) with all_23_0_16, all_23_1_17 and discharging atoms sequential(all_23_1_17, all_23_0_16), on_path(all_23_0_16, all_0_4_4), on_path(all_23_1_17, all_0_4_4), yields: % 6.16/2.04 | (80) ? [v0] : triangle(all_23_1_17, all_23_0_16, v0) % 6.16/2.04 | % 6.16/2.04 | Instantiating (80) with all_53_0_26 yields: % 6.16/2.04 | (81) triangle(all_23_1_17, all_23_0_16, all_53_0_26) % 6.16/2.04 | % 6.16/2.04 | Instantiating formula (79) with all_53_0_26 and discharging atoms triangle(all_23_1_17, all_23_0_16, all_53_0_26), yields: % 6.16/2.04 | (82) $false % 6.16/2.04 | % 6.16/2.04 |-The branch is then unsatisfiable % 6.16/2.04 % SZS output end Proof for theBenchmark % 6.16/2.04 % 6.16/2.04 1463ms %------------------------------------------------------------------------------