%------------------------------------------------------------------------------ % File : ePrincess---1.0 % Problem : GEO170+3 : TPTP v8.1.0. Released v4.0.0. % 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 03:48:07 EDT 2022 % Result : Theorem 14.44s 4.01s % Output : Proof 15.98s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.12 % Problem : GEO170+3 : TPTP v8.1.0. Released v4.0.0. % 0.07/0.12 % 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 : Sat Jun 18 15:59:08 EDT 2022 % 0.13/0.33 % CPUTime : % 0.19/0.57 ____ _ % 0.19/0.57 ___ / __ \_____(_)___ ________ __________ % 0.19/0.57 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/ % 0.19/0.57 / __/ ____/ / / / / / / /__/ __(__ |__ ) % 0.19/0.57 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/ % 0.19/0.57 % 0.19/0.57 A Theorem Prover for First-Order Logic % 0.19/0.57 (ePrincess v.1.0) % 0.19/0.57 % 0.19/0.57 (c) Philipp Rümmer, 2009-2015 % 0.19/0.57 (c) Peter Backeman, 2014-2015 % 0.19/0.57 (contributions by Angelo Brillout, Peter Baumgartner) % 0.19/0.57 Free software under GNU Lesser General Public License (LGPL). % 0.19/0.57 Bug reports to peter@backeman.se % 0.19/0.57 % 0.19/0.57 For more information, visit http://user.uu.se/~petba168/breu/ % 0.19/0.57 % 0.19/0.57 Loading /export/starexec/sandbox2/benchmark/theBenchmark.p ... % 0.71/0.62 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 1.79/0.99 Prover 0: Preprocessing ... % 2.54/1.19 Prover 0: Warning: ignoring some quantifiers % 2.58/1.22 Prover 0: Constructing countermodel ... % 13.08/3.77 Prover 0: gave up % 13.08/3.77 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all % 13.52/3.81 Prover 1: Preprocessing ... % 13.92/3.96 Prover 1: Constructing countermodel ... % 14.44/4.01 Prover 1: proved (239ms) % 14.44/4.01 % 14.44/4.01 No countermodel exists, formula is valid % 14.44/4.01 % SZS status Theorem for theBenchmark % 14.44/4.01 % 14.44/4.01 Generating proof ... found it (size 52) % 15.67/4.28 % 15.67/4.28 % SZS output start Proof for theBenchmark % 15.67/4.28 Assumed formulas after preprocessing and simplification: % 15.67/4.28 | (0) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ( ~ (v4 = 0) & incident_point_and_line(v1, v2) = 0 & incident_point_and_line(v0, v2) = 0 & equal_lines(v2, v3) = v4 & line_connecting(v0, v1) = v3 & distinct_points(v0, v1) = 0 & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | v9 = 0 | ~ (unorthogonal_lines(v6, v8) = v10) | ~ (apart_point_and_line(v5, v6) = v9) | ~ (distinct_lines(v6, v7) = 0) | ? [v11] : ? [v12] : (unorthogonal_lines(v7, v8) = v12 & apart_point_and_line(v5, v7) = v11 & (v12 = 0 | v11 = 0))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | v8 = 0 | ~ (unorthogonal_lines(v5, v7) = v9) | ~ (unorthogonal_lines(v5, v6) = v8) | ? [v10] : ( ~ (v10 = 0) & convergent_lines(v6, v7) = v10)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (unorthogonal_lines(v5, v7) = v9) | ~ (unorthogonal_lines(v5, v6) = v8) | ? [v10] : ? [v11] : ? [v12] : ? [v13] : (unorthogonal_lines(v6, v7) = v13 & convergent_lines(v6, v7) = v12 & convergent_lines(v5, v7) = v11 & convergent_lines(v5, v6) = v10 & ( ~ (v13 = 0) | ~ (v12 = 0) | (v11 = 0 & v9 = 0) | (v10 = 0 & v8 = 0)))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (apart_point_and_line(v5, v6) = v8) | ~ (distinct_lines(v6, v7) = 0) | ? [v9] : ? [v10] : (apart_point_and_line(v5, v7) = v9 & convergent_lines(v6, v7) = v10 & (v10 = 0 | v9 = 0))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (apart_point_and_line(v5, v6) = 0) | ~ (distinct_lines(v6, v7) = v8) | apart_point_and_line(v5, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (apart_point_and_line(v5, v6) = 0) | ~ (distinct_points(v5, v7) = v8) | apart_point_and_line(v7, v6) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (convergent_lines(v5, v7) = v8) | ~ (convergent_lines(v5, v6) = 0) | convergent_lines(v6, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (convergent_lines(v5, v6) = 0) | ~ (distinct_lines(v6, v7) = v8) | convergent_lines(v5, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (distinct_lines(v5, v7) = v8) | ~ (distinct_lines(v5, v6) = 0) | distinct_lines(v6, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (distinct_points(v5, v7) = v8) | ~ (distinct_points(v5, v6) = 0) | distinct_points(v6, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (orthogonal_lines(v8, v7) = v6) | ~ (orthogonal_lines(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (incident_point_and_line(v8, v7) = v6) | ~ (incident_point_and_line(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (parallel_lines(v8, v7) = v6) | ~ (parallel_lines(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (equal_lines(v8, v7) = v6) | ~ (equal_lines(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (equal_points(v8, v7) = v6) | ~ (equal_points(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (orthogonal_through_point(v8, v7) = v6) | ~ (orthogonal_through_point(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (unorthogonal_lines(v8, v7) = v6) | ~ (unorthogonal_lines(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (parallel_through_point(v8, v7) = v6) | ~ (parallel_through_point(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (intersection_point(v8, v7) = v6) | ~ (intersection_point(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (line_connecting(v8, v7) = v6) | ~ (line_connecting(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (apart_point_and_line(v8, v7) = v6) | ~ (apart_point_and_line(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (convergent_lines(v8, v7) = v6) | ~ (convergent_lines(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (distinct_lines(v8, v7) = v6) | ~ (distinct_lines(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (distinct_points(v8, v7) = v6) | ~ (distinct_points(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (unorthogonal_lines(v5, v7) = v8) | ~ (unorthogonal_lines(v5, v6) = 0) | ? [v9] : ? [v10] : ? [v11] : ? [v12] : (unorthogonal_lines(v6, v7) = v12 & convergent_lines(v6, v7) = v11 & convergent_lines(v5, v7) = v10 & convergent_lines(v5, v6) = v9 & ( ~ (v9 = 0) | (v12 = 0 & v11 = 0) | (v10 = 0 & v8 = 0)))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (distinct_lines(v7, v8) = 0) | ~ (distinct_points(v5, v6) = 0) | ? [v9] : ? [v10] : ? [v11] : ? [v12] : (apart_point_and_line(v6, v8) = v12 & apart_point_and_line(v6, v7) = v11 & apart_point_and_line(v5, v8) = v10 & apart_point_and_line(v5, v7) = v9 & (v12 = 0 | v11 = 0 | v10 = 0 | v9 = 0))) & ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (orthogonal_lines(v5, v6) = v7) | unorthogonal_lines(v5, v6) = 0) & ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (incident_point_and_line(v5, v6) = v7) | apart_point_and_line(v5, v6) = 0) & ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (parallel_lines(v5, v6) = v7) | convergent_lines(v5, v6) = 0) & ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (equal_lines(v5, v6) = v7) | distinct_lines(v5, v6) = 0) & ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (equal_points(v5, v6) = v7) | distinct_points(v5, v6) = 0) & ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (unorthogonal_lines(v5, v6) = v7) | convergent_lines(v5, v6) = 0) & ! [v5] : ! [v6] : ! [v7] : (v6 = v5 | ~ (point(v7) = v6) | ~ (point(v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : (v6 = v5 | ~ (line(v7) = v6) | ~ (line(v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ( ~ (orthogonal_through_point(v6, v5) = v7) | ~ (unorthogonal_lines(v7, v6) = 0)) & ! [v5] : ! [v6] : ! [v7] : ( ~ (orthogonal_through_point(v6, v5) = v7) | ~ (apart_point_and_line(v5, v7) = 0)) & ! [v5] : ! [v6] : ! [v7] : ( ~ (orthogonal_through_point(v5, v6) = v7) | ? [v8] : ? [v9] : ? [v10] : (point(v6) = v9 & line(v7) = v10 & line(v5) = v8 & ( ~ (v9 = 0) | ~ (v8 = 0) | v10 = 0))) & ! [v5] : ! [v6] : ! [v7] : ( ~ (parallel_through_point(v6, v5) = v7) | ~ (apart_point_and_line(v5, v7) = 0)) & ! [v5] : ! [v6] : ! [v7] : ( ~ (parallel_through_point(v6, v5) = v7) | ~ (convergent_lines(v7, v6) = 0)) & ! [v5] : ! [v6] : ! [v7] : ( ~ (parallel_through_point(v5, v6) = v7) | ? [v8] : ? [v9] : ? [v10] : (point(v6) = v9 & line(v7) = v10 & line(v5) = v8 & ( ~ (v9 = 0) | ~ (v8 = 0) | v10 = 0))) & ! [v5] : ! [v6] : ! [v7] : ( ~ (intersection_point(v5, v6) = v7) | ~ (apart_point_and_line(v7, v6) = 0) | ? [v8] : ( ~ (v8 = 0) & convergent_lines(v5, v6) = v8)) & ! [v5] : ! [v6] : ! [v7] : ( ~ (intersection_point(v5, v6) = v7) | ~ (apart_point_and_line(v7, v5) = 0) | ? [v8] : ( ~ (v8 = 0) & convergent_lines(v5, v6) = v8)) & ! [v5] : ! [v6] : ! [v7] : ( ~ (intersection_point(v5, v6) = v7) | ? [v8] : ? [v9] : ? [v10] : ? [v11] : (point(v7) = v11 & line(v6) = v9 & line(v5) = v8 & convergent_lines(v5, v6) = v10 & ( ~ (v10 = 0) | ~ (v9 = 0) | ~ (v8 = 0) | v11 = 0))) & ! [v5] : ! [v6] : ! [v7] : ( ~ (line_connecting(v5, v6) = v7) | ~ (apart_point_and_line(v6, v7) = 0) | ? [v8] : ( ~ (v8 = 0) & distinct_points(v5, v6) = v8)) & ! [v5] : ! [v6] : ! [v7] : ( ~ (line_connecting(v5, v6) = v7) | ~ (apart_point_and_line(v5, v7) = 0) | ? [v8] : ( ~ (v8 = 0) & distinct_points(v5, v6) = v8)) & ! [v5] : ! [v6] : ! [v7] : ( ~ (line_connecting(v5, v6) = v7) | ? [v8] : ? [v9] : ? [v10] : ? [v11] : (point(v6) = v9 & point(v5) = v8 & line(v7) = v11 & distinct_points(v5, v6) = v10 & ( ~ (v10 = 0) | ~ (v9 = 0) | ~ (v8 = 0) | v11 = 0))) & ! [v5] : ! [v6] : ( ~ (orthogonal_lines(v5, v6) = 0) | ? [v7] : ( ~ (v7 = 0) & unorthogonal_lines(v5, v6) = v7)) & ! [v5] : ! [v6] : ( ~ (incident_point_and_line(v5, v6) = 0) | ? [v7] : ( ~ (v7 = 0) & apart_point_and_line(v5, v6) = v7)) & ! [v5] : ! [v6] : ( ~ (parallel_lines(v5, v6) = 0) | ? [v7] : ( ~ (v7 = 0) & convergent_lines(v5, v6) = v7)) & ! [v5] : ! [v6] : ( ~ (equal_lines(v5, v6) = 0) | ? [v7] : ( ~ (v7 = 0) & distinct_lines(v5, v6) = v7)) & ! [v5] : ! [v6] : ( ~ (equal_points(v5, v6) = 0) | ? [v7] : ( ~ (v7 = 0) & distinct_points(v5, v6) = v7)) & ! [v5] : ! [v6] : ( ~ (distinct_lines(v5, v6) = 0) | convergent_lines(v5, v6) = 0) & ! [v5] : ~ (convergent_lines(v5, v5) = 0) & ! [v5] : ~ (distinct_lines(v5, v5) = 0) & ! [v5] : ~ (distinct_points(v5, v5) = 0)) % 15.77/4.31 | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3, all_0_4_4 yields: % 15.77/4.31 | (1) ~ (all_0_0_0 = 0) & incident_point_and_line(all_0_3_3, all_0_2_2) = 0 & incident_point_and_line(all_0_4_4, all_0_2_2) = 0 & equal_lines(all_0_2_2, all_0_1_1) = all_0_0_0 & line_connecting(all_0_4_4, all_0_3_3) = all_0_1_1 & distinct_points(all_0_4_4, all_0_3_3) = 0 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (unorthogonal_lines(v1, v3) = v5) | ~ (apart_point_and_line(v0, v1) = v4) | ~ (distinct_lines(v1, v2) = 0) | ? [v6] : ? [v7] : (unorthogonal_lines(v2, v3) = v7 & apart_point_and_line(v0, v2) = v6 & (v7 = 0 | v6 = 0))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (unorthogonal_lines(v0, v2) = v4) | ~ (unorthogonal_lines(v0, v1) = v3) | ? [v5] : ( ~ (v5 = 0) & convergent_lines(v1, v2) = v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (unorthogonal_lines(v0, v2) = v4) | ~ (unorthogonal_lines(v0, v1) = v3) | ? [v5] : ? [v6] : ? [v7] : ? [v8] : (unorthogonal_lines(v1, v2) = v8 & convergent_lines(v1, v2) = v7 & convergent_lines(v0, v2) = v6 & convergent_lines(v0, v1) = v5 & ( ~ (v8 = 0) | ~ (v7 = 0) | (v6 = 0 & v4 = 0) | (v5 = 0 & v3 = 0)))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v1) = v3) | ~ (distinct_lines(v1, v2) = 0) | ? [v4] : ? [v5] : (apart_point_and_line(v0, v2) = v4 & convergent_lines(v1, v2) = v5 & (v5 = 0 | v4 = 0))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v1) = 0) | ~ (distinct_lines(v1, v2) = v3) | apart_point_and_line(v0, v2) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v1) = 0) | ~ (distinct_points(v0, v2) = v3) | apart_point_and_line(v2, v1) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v1, v2) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v1) = 0) | ~ (distinct_lines(v1, v2) = v3) | convergent_lines(v0, v2) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v0, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v0, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v1, v2) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (orthogonal_lines(v3, v2) = v1) | ~ (orthogonal_lines(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (incident_point_and_line(v3, v2) = v1) | ~ (incident_point_and_line(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (parallel_lines(v3, v2) = v1) | ~ (parallel_lines(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equal_lines(v3, v2) = v1) | ~ (equal_lines(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equal_points(v3, v2) = v1) | ~ (equal_points(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (orthogonal_through_point(v3, v2) = v1) | ~ (orthogonal_through_point(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (unorthogonal_lines(v3, v2) = v1) | ~ (unorthogonal_lines(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (parallel_through_point(v3, v2) = v1) | ~ (parallel_through_point(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (intersection_point(v3, v2) = v1) | ~ (intersection_point(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (line_connecting(v3, v2) = v1) | ~ (line_connecting(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (apart_point_and_line(v3, v2) = v1) | ~ (apart_point_and_line(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (convergent_lines(v3, v2) = v1) | ~ (convergent_lines(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_lines(v3, v2) = v1) | ~ (distinct_lines(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_points(v3, v2) = v1) | ~ (distinct_points(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v0, v2) = v3) | ~ (unorthogonal_lines(v0, v1) = 0) | ? [v4] : ? [v5] : ? [v6] : ? [v7] : (unorthogonal_lines(v1, v2) = v7 & convergent_lines(v1, v2) = v6 & convergent_lines(v0, v2) = v5 & convergent_lines(v0, v1) = v4 & ( ~ (v4 = 0) | (v7 = 0 & v6 = 0) | (v5 = 0 & v3 = 0)))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (distinct_lines(v2, v3) = 0) | ~ (distinct_points(v0, v1) = 0) | ? [v4] : ? [v5] : ? [v6] : ? [v7] : (apart_point_and_line(v1, v3) = v7 & apart_point_and_line(v1, v2) = v6 & apart_point_and_line(v0, v3) = v5 & apart_point_and_line(v0, v2) = v4 & (v7 = 0 | v6 = 0 | v5 = 0 | v4 = 0))) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (orthogonal_lines(v0, v1) = v2) | unorthogonal_lines(v0, v1) = 0) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (incident_point_and_line(v0, v1) = v2) | apart_point_and_line(v0, v1) = 0) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (parallel_lines(v0, v1) = v2) | convergent_lines(v0, v1) = 0) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (equal_lines(v0, v1) = v2) | distinct_lines(v0, v1) = 0) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (equal_points(v0, v1) = v2) | distinct_points(v0, v1) = 0) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (unorthogonal_lines(v0, v1) = v2) | convergent_lines(v0, v1) = 0) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (point(v2) = v1) | ~ (point(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (line(v2) = v1) | ~ (line(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) | ~ (unorthogonal_lines(v2, v1) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) | ~ (apart_point_and_line(v0, v2) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (orthogonal_through_point(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (point(v1) = v4 & line(v2) = v5 & line(v0) = v3 & ( ~ (v4 = 0) | ~ (v3 = 0) | v5 = 0))) & ! [v0] : ! [v1] : ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) | ~ (apart_point_and_line(v0, v2) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) | ~ (convergent_lines(v2, v1) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (parallel_through_point(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (point(v1) = v4 & line(v2) = v5 & line(v0) = v3 & ( ~ (v4 = 0) | ~ (v3 = 0) | v5 = 0))) & ! [v0] : ! [v1] : ! [v2] : ( ~ (intersection_point(v0, v1) = v2) | ~ (apart_point_and_line(v2, v1) = 0) | ? [v3] : ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (intersection_point(v0, v1) = v2) | ~ (apart_point_and_line(v2, v0) = 0) | ? [v3] : ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (intersection_point(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : (point(v2) = v6 & line(v1) = v4 & line(v0) = v3 & convergent_lines(v0, v1) = v5 & ( ~ (v5 = 0) | ~ (v4 = 0) | ~ (v3 = 0) | v6 = 0))) & ! [v0] : ! [v1] : ! [v2] : ( ~ (line_connecting(v0, v1) = v2) | ~ (apart_point_and_line(v1, v2) = 0) | ? [v3] : ( ~ (v3 = 0) & distinct_points(v0, v1) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (line_connecting(v0, v1) = v2) | ~ (apart_point_and_line(v0, v2) = 0) | ? [v3] : ( ~ (v3 = 0) & distinct_points(v0, v1) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (line_connecting(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : (point(v1) = v4 & point(v0) = v3 & line(v2) = v6 & distinct_points(v0, v1) = v5 & ( ~ (v5 = 0) | ~ (v4 = 0) | ~ (v3 = 0) | v6 = 0))) & ! [v0] : ! [v1] : ( ~ (orthogonal_lines(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & unorthogonal_lines(v0, v1) = v2)) & ! [v0] : ! [v1] : ( ~ (incident_point_and_line(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & apart_point_and_line(v0, v1) = v2)) & ! [v0] : ! [v1] : ( ~ (parallel_lines(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & convergent_lines(v0, v1) = v2)) & ! [v0] : ! [v1] : ( ~ (equal_lines(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & distinct_lines(v0, v1) = v2)) & ! [v0] : ! [v1] : ( ~ (equal_points(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & distinct_points(v0, v1) = v2)) & ! [v0] : ! [v1] : ( ~ (distinct_lines(v0, v1) = 0) | convergent_lines(v0, v1) = 0) & ! [v0] : ~ (convergent_lines(v0, v0) = 0) & ! [v0] : ~ (distinct_lines(v0, v0) = 0) & ! [v0] : ~ (distinct_points(v0, v0) = 0) % 15.92/4.33 | % 15.92/4.33 | Applying alpha-rule on (1) yields: % 15.92/4.33 | (2) ! [v0] : ! [v1] : ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) | ~ (convergent_lines(v2, v1) = 0)) % 15.92/4.33 | (3) ! [v0] : ! [v1] : ( ~ (equal_lines(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & distinct_lines(v0, v1) = v2)) % 15.92/4.33 | (4) ! [v0] : ! [v1] : ! [v2] : ( ~ (intersection_point(v0, v1) = v2) | ~ (apart_point_and_line(v2, v0) = 0) | ? [v3] : ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3)) % 15.92/4.33 | (5) ! [v0] : ~ (distinct_points(v0, v0) = 0) % 15.92/4.33 | (6) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (apart_point_and_line(v3, v2) = v1) | ~ (apart_point_and_line(v3, v2) = v0)) % 15.92/4.33 | (7) ~ (all_0_0_0 = 0) % 15.92/4.33 | (8) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_points(v3, v2) = v1) | ~ (distinct_points(v3, v2) = v0)) % 15.92/4.33 | (9) ! [v0] : ! [v1] : ( ~ (parallel_lines(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & convergent_lines(v0, v1) = v2)) % 15.92/4.33 | (10) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (orthogonal_lines(v3, v2) = v1) | ~ (orthogonal_lines(v3, v2) = v0)) % 15.92/4.33 | (11) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v1) = 0) | ~ (distinct_lines(v1, v2) = v3) | apart_point_and_line(v0, v2) = 0) % 15.98/4.33 | (12) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (point(v2) = v1) | ~ (point(v2) = v0)) % 15.98/4.33 | (13) ! [v0] : ! [v1] : ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) | ~ (apart_point_and_line(v0, v2) = 0)) % 15.98/4.33 | (14) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (orthogonal_lines(v0, v1) = v2) | unorthogonal_lines(v0, v1) = 0) % 15.98/4.33 | (15) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (line(v2) = v1) | ~ (line(v2) = v0)) % 15.98/4.33 | (16) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (incident_point_and_line(v3, v2) = v1) | ~ (incident_point_and_line(v3, v2) = v0)) % 15.98/4.33 | (17) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (parallel_lines(v3, v2) = v1) | ~ (parallel_lines(v3, v2) = v0)) % 15.98/4.33 | (18) ! [v0] : ! [v1] : ! [v2] : ( ~ (line_connecting(v0, v1) = v2) | ~ (apart_point_and_line(v1, v2) = 0) | ? [v3] : ( ~ (v3 = 0) & distinct_points(v0, v1) = v3)) % 15.98/4.33 | (19) ! [v0] : ! [v1] : ! [v2] : ( ~ (orthogonal_through_point(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (point(v1) = v4 & line(v2) = v5 & line(v0) = v3 & ( ~ (v4 = 0) | ~ (v3 = 0) | v5 = 0))) % 15.98/4.33 | (20) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (intersection_point(v3, v2) = v1) | ~ (intersection_point(v3, v2) = v0)) % 15.98/4.33 | (21) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (parallel_through_point(v3, v2) = v1) | ~ (parallel_through_point(v3, v2) = v0)) % 15.98/4.33 | (22) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (distinct_lines(v2, v3) = 0) | ~ (distinct_points(v0, v1) = 0) | ? [v4] : ? [v5] : ? [v6] : ? [v7] : (apart_point_and_line(v1, v3) = v7 & apart_point_and_line(v1, v2) = v6 & apart_point_and_line(v0, v3) = v5 & apart_point_and_line(v0, v2) = v4 & (v7 = 0 | v6 = 0 | v5 = 0 | v4 = 0))) % 15.98/4.33 | (23) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (equal_lines(v0, v1) = v2) | distinct_lines(v0, v1) = 0) % 15.98/4.33 | (24) distinct_points(all_0_4_4, all_0_3_3) = 0 % 15.98/4.33 | (25) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (unorthogonal_lines(v0, v1) = v2) | convergent_lines(v0, v1) = 0) % 15.98/4.33 | (26) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equal_points(v3, v2) = v1) | ~ (equal_points(v3, v2) = v0)) % 15.98/4.33 | (27) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (unorthogonal_lines(v0, v2) = v4) | ~ (unorthogonal_lines(v0, v1) = v3) | ? [v5] : ? [v6] : ? [v7] : ? [v8] : (unorthogonal_lines(v1, v2) = v8 & convergent_lines(v1, v2) = v7 & convergent_lines(v0, v2) = v6 & convergent_lines(v0, v1) = v5 & ( ~ (v8 = 0) | ~ (v7 = 0) | (v6 = 0 & v4 = 0) | (v5 = 0 & v3 = 0)))) % 15.98/4.33 | (28) line_connecting(all_0_4_4, all_0_3_3) = all_0_1_1 % 15.98/4.33 | (29) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (unorthogonal_lines(v1, v3) = v5) | ~ (apart_point_and_line(v0, v1) = v4) | ~ (distinct_lines(v1, v2) = 0) | ? [v6] : ? [v7] : (unorthogonal_lines(v2, v3) = v7 & apart_point_and_line(v0, v2) = v6 & (v7 = 0 | v6 = 0))) % 15.98/4.33 | (30) incident_point_and_line(all_0_3_3, all_0_2_2) = 0 % 15.98/4.33 | (31) ! [v0] : ! [v1] : ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) | ~ (unorthogonal_lines(v2, v1) = 0)) % 15.98/4.33 | (32) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (equal_points(v0, v1) = v2) | distinct_points(v0, v1) = 0) % 15.98/4.34 | (33) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (line_connecting(v3, v2) = v1) | ~ (line_connecting(v3, v2) = v0)) % 15.98/4.34 | (34) ! [v0] : ! [v1] : ! [v2] : ( ~ (parallel_through_point(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (point(v1) = v4 & line(v2) = v5 & line(v0) = v3 & ( ~ (v4 = 0) | ~ (v3 = 0) | v5 = 0))) % 15.98/4.34 | (35) equal_lines(all_0_2_2, all_0_1_1) = all_0_0_0 % 15.98/4.34 | (36) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v0, v2) = v3) | ~ (unorthogonal_lines(v0, v1) = 0) | ? [v4] : ? [v5] : ? [v6] : ? [v7] : (unorthogonal_lines(v1, v2) = v7 & convergent_lines(v1, v2) = v6 & convergent_lines(v0, v2) = v5 & convergent_lines(v0, v1) = v4 & ( ~ (v4 = 0) | (v7 = 0 & v6 = 0) | (v5 = 0 & v3 = 0)))) % 15.98/4.34 | (37) ! [v0] : ! [v1] : ( ~ (distinct_lines(v0, v1) = 0) | convergent_lines(v0, v1) = 0) % 15.98/4.34 | (38) incident_point_and_line(all_0_4_4, all_0_2_2) = 0 % 15.98/4.34 | (39) ! [v0] : ! [v1] : ! [v2] : ( ~ (line_connecting(v0, v1) = v2) | ~ (apart_point_and_line(v0, v2) = 0) | ? [v3] : ( ~ (v3 = 0) & distinct_points(v0, v1) = v3)) % 15.98/4.34 | (40) ! [v0] : ~ (convergent_lines(v0, v0) = 0) % 15.98/4.34 | (41) ! [v0] : ! [v1] : ( ~ (equal_points(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & distinct_points(v0, v1) = v2)) % 15.98/4.34 | (42) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v0, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v1, v2) = 0) % 15.98/4.34 | (43) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (convergent_lines(v3, v2) = v1) | ~ (convergent_lines(v3, v2) = v0)) % 15.98/4.34 | (44) ! [v0] : ! [v1] : ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) | ~ (apart_point_and_line(v0, v2) = 0)) % 15.98/4.34 | (45) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (incident_point_and_line(v0, v1) = v2) | apart_point_and_line(v0, v1) = 0) % 15.98/4.34 | (46) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v0, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0) % 15.98/4.34 | (47) ! [v0] : ~ (distinct_lines(v0, v0) = 0) % 15.98/4.34 | (48) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v1) = 0) | ~ (distinct_points(v0, v2) = v3) | apart_point_and_line(v2, v1) = 0) % 15.98/4.34 | (49) ! [v0] : ! [v1] : ! [v2] : ( ~ (intersection_point(v0, v1) = v2) | ~ (apart_point_and_line(v2, v1) = 0) | ? [v3] : ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3)) % 15.98/4.34 | (50) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v1) = v3) | ~ (distinct_lines(v1, v2) = 0) | ? [v4] : ? [v5] : (apart_point_and_line(v0, v2) = v4 & convergent_lines(v1, v2) = v5 & (v5 = 0 | v4 = 0))) % 15.98/4.34 | (51) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (orthogonal_through_point(v3, v2) = v1) | ~ (orthogonal_through_point(v3, v2) = v0)) % 15.98/4.34 | (52) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (unorthogonal_lines(v3, v2) = v1) | ~ (unorthogonal_lines(v3, v2) = v0)) % 15.98/4.34 | (53) ! [v0] : ! [v1] : ( ~ (orthogonal_lines(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & unorthogonal_lines(v0, v1) = v2)) % 15.98/4.34 | (54) ! [v0] : ! [v1] : ( ~ (incident_point_and_line(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & apart_point_and_line(v0, v1) = v2)) % 15.98/4.34 | (55) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equal_lines(v3, v2) = v1) | ~ (equal_lines(v3, v2) = v0)) % 15.98/4.34 | (56) ! [v0] : ! [v1] : ! [v2] : ( ~ (line_connecting(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : (point(v1) = v4 & point(v0) = v3 & line(v2) = v6 & distinct_points(v0, v1) = v5 & ( ~ (v5 = 0) | ~ (v4 = 0) | ~ (v3 = 0) | v6 = 0))) % 15.98/4.34 | (57) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (parallel_lines(v0, v1) = v2) | convergent_lines(v0, v1) = 0) % 15.98/4.34 | (58) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (unorthogonal_lines(v0, v2) = v4) | ~ (unorthogonal_lines(v0, v1) = v3) | ? [v5] : ( ~ (v5 = 0) & convergent_lines(v1, v2) = v5)) % 15.98/4.34 | (59) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v1, v2) = 0) % 15.98/4.34 | (60) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_lines(v3, v2) = v1) | ~ (distinct_lines(v3, v2) = v0)) % 15.98/4.34 | (61) ! [v0] : ! [v1] : ! [v2] : ( ~ (intersection_point(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : (point(v2) = v6 & line(v1) = v4 & line(v0) = v3 & convergent_lines(v0, v1) = v5 & ( ~ (v5 = 0) | ~ (v4 = 0) | ~ (v3 = 0) | v6 = 0))) % 15.98/4.34 | (62) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v1) = 0) | ~ (distinct_lines(v1, v2) = v3) | convergent_lines(v0, v2) = 0) % 15.98/4.34 | % 15.98/4.35 | Instantiating formula (54) with all_0_2_2, all_0_3_3 and discharging atoms incident_point_and_line(all_0_3_3, all_0_2_2) = 0, yields: % 15.98/4.35 | (63) ? [v0] : ( ~ (v0 = 0) & apart_point_and_line(all_0_3_3, all_0_2_2) = v0) % 15.98/4.35 | % 15.98/4.35 | Instantiating formula (54) with all_0_2_2, all_0_4_4 and discharging atoms incident_point_and_line(all_0_4_4, all_0_2_2) = 0, yields: % 15.98/4.35 | (64) ? [v0] : ( ~ (v0 = 0) & apart_point_and_line(all_0_4_4, all_0_2_2) = v0) % 15.98/4.35 | % 15.98/4.35 | Instantiating formula (23) with all_0_0_0, all_0_1_1, all_0_2_2 and discharging atoms equal_lines(all_0_2_2, all_0_1_1) = all_0_0_0, yields: % 15.98/4.35 | (65) all_0_0_0 = 0 | distinct_lines(all_0_2_2, all_0_1_1) = 0 % 15.98/4.35 | % 15.98/4.35 | Instantiating formula (56) with all_0_1_1, all_0_3_3, all_0_4_4 and discharging atoms line_connecting(all_0_4_4, all_0_3_3) = all_0_1_1, yields: % 15.98/4.35 | (66) ? [v0] : ? [v1] : ? [v2] : ? [v3] : (point(all_0_3_3) = v1 & point(all_0_4_4) = v0 & line(all_0_1_1) = v3 & distinct_points(all_0_4_4, all_0_3_3) = v2 & ( ~ (v2 = 0) | ~ (v1 = 0) | ~ (v0 = 0) | v3 = 0)) % 15.98/4.35 | % 15.98/4.35 | Instantiating (66) with all_16_0_5, all_16_1_6, all_16_2_7, all_16_3_8 yields: % 15.98/4.35 | (67) point(all_0_3_3) = all_16_2_7 & point(all_0_4_4) = all_16_3_8 & line(all_0_1_1) = all_16_0_5 & distinct_points(all_0_4_4, all_0_3_3) = all_16_1_6 & ( ~ (all_16_1_6 = 0) | ~ (all_16_2_7 = 0) | ~ (all_16_3_8 = 0) | all_16_0_5 = 0) % 15.98/4.35 | % 15.98/4.35 | Applying alpha-rule on (67) yields: % 15.98/4.35 | (68) point(all_0_4_4) = all_16_3_8 % 15.98/4.35 | (69) distinct_points(all_0_4_4, all_0_3_3) = all_16_1_6 % 15.98/4.35 | (70) ~ (all_16_1_6 = 0) | ~ (all_16_2_7 = 0) | ~ (all_16_3_8 = 0) | all_16_0_5 = 0 % 15.98/4.35 | (71) point(all_0_3_3) = all_16_2_7 % 15.98/4.35 | (72) line(all_0_1_1) = all_16_0_5 % 15.98/4.35 | % 15.98/4.35 | Instantiating (64) with all_18_0_9 yields: % 15.98/4.35 | (73) ~ (all_18_0_9 = 0) & apart_point_and_line(all_0_4_4, all_0_2_2) = all_18_0_9 % 15.98/4.35 | % 15.98/4.35 | Applying alpha-rule on (73) yields: % 15.98/4.35 | (74) ~ (all_18_0_9 = 0) % 15.98/4.35 | (75) apart_point_and_line(all_0_4_4, all_0_2_2) = all_18_0_9 % 15.98/4.35 | % 15.98/4.35 | Instantiating (63) with all_20_0_10 yields: % 15.98/4.35 | (76) ~ (all_20_0_10 = 0) & apart_point_and_line(all_0_3_3, all_0_2_2) = all_20_0_10 % 15.98/4.35 | % 15.98/4.35 | Applying alpha-rule on (76) yields: % 15.98/4.35 | (77) ~ (all_20_0_10 = 0) % 15.98/4.35 | (78) apart_point_and_line(all_0_3_3, all_0_2_2) = all_20_0_10 % 15.98/4.35 | % 15.98/4.35 +-Applying beta-rule and splitting (65), into two cases. % 15.98/4.35 |-Branch one: % 15.98/4.35 | (79) distinct_lines(all_0_2_2, all_0_1_1) = 0 % 15.98/4.35 | % 15.98/4.35 | Instantiating formula (8) with all_0_4_4, all_0_3_3, all_16_1_6, 0 and discharging atoms distinct_points(all_0_4_4, all_0_3_3) = all_16_1_6, distinct_points(all_0_4_4, all_0_3_3) = 0, yields: % 15.98/4.35 | (80) all_16_1_6 = 0 % 15.98/4.35 | % 15.98/4.35 | From (80) and (69) follows: % 15.98/4.35 | (24) distinct_points(all_0_4_4, all_0_3_3) = 0 % 15.98/4.35 | % 15.98/4.35 | Instantiating formula (22) with all_0_1_1, all_0_2_2, all_0_3_3, all_0_4_4 and discharging atoms distinct_lines(all_0_2_2, all_0_1_1) = 0, distinct_points(all_0_4_4, all_0_3_3) = 0, yields: % 15.98/4.35 | (82) ? [v0] : ? [v1] : ? [v2] : ? [v3] : (apart_point_and_line(all_0_3_3, all_0_1_1) = v3 & apart_point_and_line(all_0_3_3, all_0_2_2) = v2 & apart_point_and_line(all_0_4_4, all_0_1_1) = v1 & apart_point_and_line(all_0_4_4, all_0_2_2) = v0 & (v3 = 0 | v2 = 0 | v1 = 0 | v0 = 0)) % 15.98/4.35 | % 15.98/4.35 | Instantiating (82) with all_40_0_11, all_40_1_12, all_40_2_13, all_40_3_14 yields: % 15.98/4.35 | (83) apart_point_and_line(all_0_3_3, all_0_1_1) = all_40_0_11 & apart_point_and_line(all_0_3_3, all_0_2_2) = all_40_1_12 & apart_point_and_line(all_0_4_4, all_0_1_1) = all_40_2_13 & apart_point_and_line(all_0_4_4, all_0_2_2) = all_40_3_14 & (all_40_0_11 = 0 | all_40_1_12 = 0 | all_40_2_13 = 0 | all_40_3_14 = 0) % 15.98/4.35 | % 15.98/4.35 | Applying alpha-rule on (83) yields: % 15.98/4.35 | (84) all_40_0_11 = 0 | all_40_1_12 = 0 | all_40_2_13 = 0 | all_40_3_14 = 0 % 15.98/4.35 | (85) apart_point_and_line(all_0_3_3, all_0_1_1) = all_40_0_11 % 15.98/4.35 | (86) apart_point_and_line(all_0_3_3, all_0_2_2) = all_40_1_12 % 15.98/4.35 | (87) apart_point_and_line(all_0_4_4, all_0_1_1) = all_40_2_13 % 15.98/4.35 | (88) apart_point_and_line(all_0_4_4, all_0_2_2) = all_40_3_14 % 15.98/4.35 | % 15.98/4.35 | Instantiating formula (6) with all_0_3_3, all_0_2_2, all_40_1_12, all_20_0_10 and discharging atoms apart_point_and_line(all_0_3_3, all_0_2_2) = all_40_1_12, apart_point_and_line(all_0_3_3, all_0_2_2) = all_20_0_10, yields: % 15.98/4.35 | (89) all_40_1_12 = all_20_0_10 % 15.98/4.35 | % 15.98/4.35 | Instantiating formula (6) with all_0_4_4, all_0_2_2, all_40_3_14, all_18_0_9 and discharging atoms apart_point_and_line(all_0_4_4, all_0_2_2) = all_40_3_14, apart_point_and_line(all_0_4_4, all_0_2_2) = all_18_0_9, yields: % 15.98/4.35 | (90) all_40_3_14 = all_18_0_9 % 15.98/4.35 | % 15.98/4.35 | Instantiating formula (18) with all_0_1_1, all_0_3_3, all_0_4_4 and discharging atoms line_connecting(all_0_4_4, all_0_3_3) = all_0_1_1, yields: % 15.98/4.35 | (91) ~ (apart_point_and_line(all_0_3_3, all_0_1_1) = 0) | ? [v0] : ( ~ (v0 = 0) & distinct_points(all_0_4_4, all_0_3_3) = v0) % 15.98/4.35 | % 15.98/4.35 | Instantiating formula (39) with all_0_1_1, all_0_3_3, all_0_4_4 and discharging atoms line_connecting(all_0_4_4, all_0_3_3) = all_0_1_1, yields: % 15.98/4.35 | (92) ~ (apart_point_and_line(all_0_4_4, all_0_1_1) = 0) | ? [v0] : ( ~ (v0 = 0) & distinct_points(all_0_4_4, all_0_3_3) = v0) % 15.98/4.35 | % 15.98/4.35 +-Applying beta-rule and splitting (92), into two cases. % 15.98/4.35 |-Branch one: % 15.98/4.35 | (93) ~ (apart_point_and_line(all_0_4_4, all_0_1_1) = 0) % 15.98/4.35 | % 15.98/4.35 +-Applying beta-rule and splitting (91), into two cases. % 15.98/4.35 |-Branch one: % 15.98/4.35 | (94) ~ (apart_point_and_line(all_0_3_3, all_0_1_1) = 0) % 15.98/4.35 | % 15.98/4.35 | Using (85) and (94) yields: % 15.98/4.35 | (95) ~ (all_40_0_11 = 0) % 15.98/4.35 | % 15.98/4.35 | Using (87) and (93) yields: % 15.98/4.35 | (96) ~ (all_40_2_13 = 0) % 15.98/4.35 | % 15.98/4.35 +-Applying beta-rule and splitting (84), into two cases. % 15.98/4.35 |-Branch one: % 15.98/4.35 | (97) all_40_0_11 = 0 % 15.98/4.35 | % 15.98/4.35 | Equations (97) can reduce 95 to: % 15.98/4.35 | (98) $false % 15.98/4.35 | % 15.98/4.35 |-The branch is then unsatisfiable % 15.98/4.35 |-Branch two: % 15.98/4.35 | (95) ~ (all_40_0_11 = 0) % 15.98/4.35 | (100) all_40_1_12 = 0 | all_40_2_13 = 0 | all_40_3_14 = 0 % 15.98/4.35 | % 15.98/4.35 +-Applying beta-rule and splitting (100), into two cases. % 15.98/4.35 |-Branch one: % 15.98/4.35 | (101) all_40_1_12 = 0 % 15.98/4.35 | % 15.98/4.35 | Combining equations (101,89) yields a new equation: % 15.98/4.35 | (102) all_20_0_10 = 0 % 15.98/4.35 | % 15.98/4.35 | Equations (102) can reduce 77 to: % 15.98/4.35 | (98) $false % 15.98/4.35 | % 15.98/4.35 |-The branch is then unsatisfiable % 15.98/4.35 |-Branch two: % 15.98/4.35 | (104) ~ (all_40_1_12 = 0) % 15.98/4.35 | (105) all_40_2_13 = 0 | all_40_3_14 = 0 % 15.98/4.35 | % 15.98/4.35 +-Applying beta-rule and splitting (105), into two cases. % 15.98/4.35 |-Branch one: % 15.98/4.35 | (106) all_40_2_13 = 0 % 15.98/4.35 | % 15.98/4.35 | Equations (106) can reduce 96 to: % 15.98/4.35 | (98) $false % 15.98/4.35 | % 15.98/4.35 |-The branch is then unsatisfiable % 15.98/4.35 |-Branch two: % 15.98/4.35 | (96) ~ (all_40_2_13 = 0) % 15.98/4.35 | (109) all_40_3_14 = 0 % 15.98/4.35 | % 15.98/4.36 | Combining equations (90,109) yields a new equation: % 15.98/4.36 | (110) all_18_0_9 = 0 % 15.98/4.36 | % 15.98/4.36 | Simplifying 110 yields: % 15.98/4.36 | (111) all_18_0_9 = 0 % 15.98/4.36 | % 15.98/4.36 | Equations (111) can reduce 74 to: % 15.98/4.36 | (98) $false % 15.98/4.36 | % 15.98/4.36 |-The branch is then unsatisfiable % 15.98/4.36 |-Branch two: % 15.98/4.36 | (113) apart_point_and_line(all_0_3_3, all_0_1_1) = 0 % 15.98/4.36 | (114) ? [v0] : ( ~ (v0 = 0) & distinct_points(all_0_4_4, all_0_3_3) = v0) % 15.98/4.36 | % 15.98/4.36 | Instantiating (114) with all_61_0_15 yields: % 15.98/4.36 | (115) ~ (all_61_0_15 = 0) & distinct_points(all_0_4_4, all_0_3_3) = all_61_0_15 % 15.98/4.36 | % 15.98/4.36 | Applying alpha-rule on (115) yields: % 15.98/4.36 | (116) ~ (all_61_0_15 = 0) % 15.98/4.36 | (117) distinct_points(all_0_4_4, all_0_3_3) = all_61_0_15 % 15.98/4.36 | % 15.98/4.36 | Instantiating formula (8) with all_0_4_4, all_0_3_3, all_61_0_15, 0 and discharging atoms distinct_points(all_0_4_4, all_0_3_3) = all_61_0_15, distinct_points(all_0_4_4, all_0_3_3) = 0, yields: % 15.98/4.36 | (118) all_61_0_15 = 0 % 15.98/4.36 | % 15.98/4.36 | Equations (118) can reduce 116 to: % 15.98/4.36 | (98) $false % 15.98/4.36 | % 15.98/4.36 |-The branch is then unsatisfiable % 15.98/4.36 |-Branch two: % 15.98/4.36 | (120) apart_point_and_line(all_0_4_4, all_0_1_1) = 0 % 15.98/4.36 | (114) ? [v0] : ( ~ (v0 = 0) & distinct_points(all_0_4_4, all_0_3_3) = v0) % 15.98/4.36 | % 15.98/4.36 | Instantiating (114) with all_57_0_16 yields: % 15.98/4.36 | (122) ~ (all_57_0_16 = 0) & distinct_points(all_0_4_4, all_0_3_3) = all_57_0_16 % 15.98/4.36 | % 15.98/4.36 | Applying alpha-rule on (122) yields: % 15.98/4.36 | (123) ~ (all_57_0_16 = 0) % 15.98/4.36 | (124) distinct_points(all_0_4_4, all_0_3_3) = all_57_0_16 % 15.98/4.36 | % 15.98/4.36 | Instantiating formula (8) with all_0_4_4, all_0_3_3, all_57_0_16, 0 and discharging atoms distinct_points(all_0_4_4, all_0_3_3) = all_57_0_16, distinct_points(all_0_4_4, all_0_3_3) = 0, yields: % 15.98/4.36 | (125) all_57_0_16 = 0 % 15.98/4.36 | % 15.98/4.36 | Equations (125) can reduce 123 to: % 15.98/4.36 | (98) $false % 15.98/4.36 | % 15.98/4.36 |-The branch is then unsatisfiable % 15.98/4.36 |-Branch two: % 15.98/4.36 | (127) ~ (distinct_lines(all_0_2_2, all_0_1_1) = 0) % 15.98/4.36 | (128) all_0_0_0 = 0 % 15.98/4.36 | % 15.98/4.36 | Equations (128) can reduce 7 to: % 15.98/4.36 | (98) $false % 15.98/4.36 | % 15.98/4.36 |-The branch is then unsatisfiable % 15.98/4.36 % SZS output end Proof for theBenchmark % 15.98/4.36 % 15.98/4.36 3777ms %------------------------------------------------------------------------------