%------------------------------------------------------------------------------ % File : ePrincess---1.0 % Problem : GEO210+3 : TPTP v8.1.0. Released v4.0.0. % Transfm : none % Format : tptp:raw % Command : ePrincess-casc -timeout=%d %s % Computer : n016.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 03:48:45 EDT 2022 % Result : Theorem 13.48s 3.86s % Output : Proof 15.44s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.08/0.14 % Problem : GEO210+3 : TPTP v8.1.0. Released v4.0.0. % 0.08/0.14 % Command : ePrincess-casc -timeout=%d %s % 0.14/0.36 % Computer : n016.cluster.edu % 0.14/0.36 % Model : x86_64 x86_64 % 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.36 % Memory : 8042.1875MB % 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.36 % CPULimit : 300 % 0.14/0.36 % WCLimit : 600 % 0.14/0.36 % DateTime : Fri Jun 17 18:31:21 EDT 2022 % 0.14/0.36 % CPUTime : % 0.57/0.61 ____ _ % 0.57/0.61 ___ / __ \_____(_)___ ________ __________ % 0.57/0.61 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/ % 0.57/0.61 / __/ ____/ / / / / / / /__/ __(__ |__ ) % 0.57/0.61 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/ % 0.57/0.61 % 0.57/0.61 A Theorem Prover for First-Order Logic % 0.57/0.61 (ePrincess v.1.0) % 0.57/0.61 % 0.57/0.61 (c) Philipp Rümmer, 2009-2015 % 0.57/0.61 (c) Peter Backeman, 2014-2015 % 0.57/0.61 (contributions by Angelo Brillout, Peter Baumgartner) % 0.57/0.61 Free software under GNU Lesser General Public License (LGPL). % 0.57/0.61 Bug reports to peter@backeman.se % 0.57/0.61 % 0.57/0.61 For more information, visit http://user.uu.se/~petba168/breu/ % 0.57/0.61 % 0.57/0.61 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ... % 0.69/0.66 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 1.70/1.00 Prover 0: Preprocessing ... % 2.49/1.22 Prover 0: Warning: ignoring some quantifiers % 2.59/1.25 Prover 0: Constructing countermodel ... % 12.52/3.65 Prover 0: gave up % 12.52/3.65 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all % 12.52/3.69 Prover 1: Preprocessing ... % 13.48/3.81 Prover 1: Constructing countermodel ... % 13.48/3.86 Prover 1: proved (209ms) % 13.48/3.86 % 13.48/3.86 No countermodel exists, formula is valid % 13.48/3.86 % SZS status Theorem for theBenchmark % 13.48/3.86 % 13.48/3.86 Generating proof ... found it (size 43) % 14.94/4.13 % 14.94/4.13 % SZS output start Proof for theBenchmark % 14.94/4.13 Assumed formulas after preprocessing and simplification: % 14.94/4.13 | (0) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ( ~ (v4 = 0) & orthogonal_lines(v1, v2) = 0 & incident_point_and_line(v0, v1) = 0 & equal_lines(v1, v3) = v4 & orthogonal_through_point(v2, v0) = v3 & ! [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)) % 14.94/4.17 | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3, all_0_4_4 yields: % 14.94/4.17 | (1) ~ (all_0_0_0 = 0) & orthogonal_lines(all_0_3_3, all_0_2_2) = 0 & incident_point_and_line(all_0_4_4, all_0_3_3) = 0 & equal_lines(all_0_3_3, all_0_1_1) = all_0_0_0 & orthogonal_through_point(all_0_2_2, all_0_4_4) = all_0_1_1 & ! [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) % 14.94/4.18 | % 14.94/4.18 | Applying alpha-rule on (1) yields: % 14.94/4.18 | (2) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (equal_points(v0, v1) = v2) | distinct_points(v0, v1) = 0) % 14.94/4.18 | (3) ! [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))) % 14.94/4.18 | (4) ! [v0] : ~ (convergent_lines(v0, v0) = 0) % 14.94/4.18 | (5) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (parallel_lines(v0, v1) = v2) | convergent_lines(v0, v1) = 0) % 14.94/4.18 | (6) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (equal_lines(v0, v1) = v2) | distinct_lines(v0, v1) = 0) % 14.94/4.18 | (7) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (parallel_through_point(v3, v2) = v1) | ~ (parallel_through_point(v3, v2) = v0)) % 14.94/4.18 | (8) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (convergent_lines(v3, v2) = v1) | ~ (convergent_lines(v3, v2) = v0)) % 14.94/4.18 | (9) ! [v0] : ! [v1] : ! [v2] : ( ~ (intersection_point(v0, v1) = v2) | ~ (apart_point_and_line(v2, v0) = 0) | ? [v3] : ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3)) % 14.94/4.18 | (10) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (apart_point_and_line(v3, v2) = v1) | ~ (apart_point_and_line(v3, v2) = v0)) % 14.94/4.18 | (11) ! [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))) % 14.94/4.18 | (12) ! [v0] : ! [v1] : ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) | ~ (unorthogonal_lines(v2, v1) = 0)) % 14.94/4.18 | (13) ~ (all_0_0_0 = 0) % 14.94/4.18 | (14) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v0, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0) % 14.94/4.19 | (15) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (unorthogonal_lines(v0, v1) = v2) | convergent_lines(v0, v1) = 0) % 14.94/4.19 | (16) incident_point_and_line(all_0_4_4, all_0_3_3) = 0 % 14.94/4.19 | (17) ! [v0] : ! [v1] : ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) | ~ (apart_point_and_line(v0, v2) = 0)) % 14.94/4.19 | (18) ! [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))) % 14.94/4.19 | (19) ! [v0] : ! [v1] : ( ~ (equal_points(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & distinct_points(v0, v1) = v2)) % 14.94/4.19 | (20) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (orthogonal_through_point(v3, v2) = v1) | ~ (orthogonal_through_point(v3, v2) = v0)) % 14.94/4.19 | (21) ! [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) % 14.94/4.19 | (22) ! [v0] : ! [v1] : ( ~ (orthogonal_lines(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & unorthogonal_lines(v0, v1) = v2)) % 14.94/4.19 | (23) orthogonal_through_point(all_0_2_2, all_0_4_4) = all_0_1_1 % 14.94/4.19 | (24) orthogonal_lines(all_0_3_3, all_0_2_2) = 0 % 14.94/4.19 | (25) ! [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))) % 14.94/4.19 | (26) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_points(v3, v2) = v1) | ~ (distinct_points(v3, v2) = v0)) % 14.94/4.19 | (27) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (incident_point_and_line(v3, v2) = v1) | ~ (incident_point_and_line(v3, v2) = v0)) % 14.94/4.19 | (28) ! [v0] : ! [v1] : ! [v2] : ( ~ (line_connecting(v0, v1) = v2) | ~ (apart_point_and_line(v0, v2) = 0) | ? [v3] : ( ~ (v3 = 0) & distinct_points(v0, v1) = v3)) % 14.94/4.19 | (29) ! [v0] : ! [v1] : ( ~ (distinct_lines(v0, v1) = 0) | convergent_lines(v0, v1) = 0) % 14.94/4.19 | (30) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (incident_point_and_line(v0, v1) = v2) | apart_point_and_line(v0, v1) = 0) % 14.94/4.19 | (31) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (point(v2) = v1) | ~ (point(v2) = v0)) % 14.94/4.19 | (32) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (unorthogonal_lines(v3, v2) = v1) | ~ (unorthogonal_lines(v3, v2) = v0)) % 14.94/4.19 | (33) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_lines(v3, v2) = v1) | ~ (distinct_lines(v3, v2) = v0)) % 14.94/4.19 | (34) ! [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)))) % 14.94/4.19 | (35) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v1) = 0) | ~ (distinct_lines(v1, v2) = v3) | convergent_lines(v0, v2) = 0) % 14.94/4.19 | (36) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v0, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v1, v2) = 0) % 14.94/4.19 | (37) ! [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))) % 14.94/4.19 | (38) ! [v0] : ! [v1] : ( ~ (incident_point_and_line(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & apart_point_and_line(v0, v1) = v2)) % 14.94/4.19 | (39) ! [v0] : ! [v1] : ( ~ (equal_lines(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & distinct_lines(v0, v1) = v2)) % 14.94/4.19 | (40) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v1, v2) = 0) % 14.94/4.19 | (41) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (intersection_point(v3, v2) = v1) | ~ (intersection_point(v3, v2) = v0)) % 14.94/4.19 | (42) ! [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))) % 14.94/4.19 | (43) ! [v0] : ~ (distinct_lines(v0, v0) = 0) % 14.94/4.19 | (44) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (line(v2) = v1) | ~ (line(v2) = v0)) % 14.94/4.19 | (45) ! [v0] : ! [v1] : ! [v2] : ( ~ (intersection_point(v0, v1) = v2) | ~ (apart_point_and_line(v2, v1) = 0) | ? [v3] : ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3)) % 14.94/4.19 | (46) ! [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)) % 14.94/4.19 | (47) ! [v0] : ! [v1] : ( ~ (parallel_lines(v0, v1) = 0) | ? [v2] : ( ~ (v2 = 0) & convergent_lines(v0, v1) = v2)) % 14.94/4.20 | (48) ! [v0] : ! [v1] : ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) | ~ (convergent_lines(v2, v1) = 0)) % 14.94/4.20 | (49) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equal_points(v3, v2) = v1) | ~ (equal_points(v3, v2) = v0)) % 14.94/4.20 | (50) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equal_lines(v3, v2) = v1) | ~ (equal_lines(v3, v2) = v0)) % 14.94/4.20 | (51) ! [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) % 14.94/4.20 | (52) ! [v0] : ! [v1] : ! [v2] : ( ~ (line_connecting(v0, v1) = v2) | ~ (apart_point_and_line(v1, v2) = 0) | ? [v3] : ( ~ (v3 = 0) & distinct_points(v0, v1) = v3)) % 14.94/4.20 | (53) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (orthogonal_lines(v0, v1) = v2) | unorthogonal_lines(v0, v1) = 0) % 14.94/4.20 | (54) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (orthogonal_lines(v3, v2) = v1) | ~ (orthogonal_lines(v3, v2) = v0)) % 14.94/4.20 | (55) ! [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)))) % 14.94/4.20 | (56) ! [v0] : ~ (distinct_points(v0, v0) = 0) % 14.94/4.20 | (57) equal_lines(all_0_3_3, all_0_1_1) = all_0_0_0 % 14.94/4.20 | (58) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (parallel_lines(v3, v2) = v1) | ~ (parallel_lines(v3, v2) = v0)) % 14.94/4.20 | (59) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (line_connecting(v3, v2) = v1) | ~ (line_connecting(v3, v2) = v0)) % 14.94/4.20 | (60) ! [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))) % 14.94/4.20 | (61) ! [v0] : ! [v1] : ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) | ~ (apart_point_and_line(v0, v2) = 0)) % 14.94/4.20 | % 14.94/4.20 | Instantiating formula (22) with all_0_2_2, all_0_3_3 and discharging atoms orthogonal_lines(all_0_3_3, all_0_2_2) = 0, yields: % 14.94/4.20 | (62) ? [v0] : ( ~ (v0 = 0) & unorthogonal_lines(all_0_3_3, all_0_2_2) = v0) % 14.94/4.20 | % 14.94/4.20 | Instantiating formula (38) with all_0_3_3, all_0_4_4 and discharging atoms incident_point_and_line(all_0_4_4, all_0_3_3) = 0, yields: % 14.94/4.20 | (63) ? [v0] : ( ~ (v0 = 0) & apart_point_and_line(all_0_4_4, all_0_3_3) = v0) % 14.94/4.20 | % 14.94/4.20 | Instantiating formula (6) with all_0_0_0, all_0_1_1, all_0_3_3 and discharging atoms equal_lines(all_0_3_3, all_0_1_1) = all_0_0_0, yields: % 14.94/4.20 | (64) all_0_0_0 = 0 | distinct_lines(all_0_3_3, all_0_1_1) = 0 % 14.94/4.20 | % 14.94/4.20 | Instantiating (63) with all_10_0_8 yields: % 14.94/4.20 | (65) ~ (all_10_0_8 = 0) & apart_point_and_line(all_0_4_4, all_0_3_3) = all_10_0_8 % 14.94/4.20 | % 14.94/4.20 | Applying alpha-rule on (65) yields: % 14.94/4.20 | (66) ~ (all_10_0_8 = 0) % 14.94/4.20 | (67) apart_point_and_line(all_0_4_4, all_0_3_3) = all_10_0_8 % 14.94/4.20 | % 14.94/4.20 | Instantiating (62) with all_12_0_9 yields: % 14.94/4.20 | (68) ~ (all_12_0_9 = 0) & unorthogonal_lines(all_0_3_3, all_0_2_2) = all_12_0_9 % 14.94/4.20 | % 14.94/4.20 | Applying alpha-rule on (68) yields: % 14.94/4.20 | (69) ~ (all_12_0_9 = 0) % 14.94/4.20 | (70) unorthogonal_lines(all_0_3_3, all_0_2_2) = all_12_0_9 % 14.94/4.20 | % 14.94/4.20 +-Applying beta-rule and splitting (64), into two cases. % 14.94/4.20 |-Branch one: % 14.94/4.20 | (71) distinct_lines(all_0_3_3, all_0_1_1) = 0 % 14.94/4.20 | % 14.94/4.20 | Instantiating formula (46) with all_12_0_9, all_12_0_9, all_0_2_2, all_0_2_2, all_0_3_3 and discharging atoms unorthogonal_lines(all_0_3_3, all_0_2_2) = all_12_0_9, yields: % 14.94/4.20 | (72) all_12_0_9 = 0 | ? [v0] : ( ~ (v0 = 0) & convergent_lines(all_0_2_2, all_0_2_2) = v0) % 14.94/4.20 | % 14.94/4.20 | Instantiating formula (3) with all_12_0_9, all_10_0_8, all_0_2_2, all_0_1_1, all_0_3_3, all_0_4_4 and discharging atoms unorthogonal_lines(all_0_3_3, all_0_2_2) = all_12_0_9, apart_point_and_line(all_0_4_4, all_0_3_3) = all_10_0_8, distinct_lines(all_0_3_3, all_0_1_1) = 0, yields: % 14.94/4.20 | (73) all_12_0_9 = 0 | all_10_0_8 = 0 | ? [v0] : ? [v1] : (unorthogonal_lines(all_0_1_1, all_0_2_2) = v1 & apart_point_and_line(all_0_4_4, all_0_1_1) = v0 & (v1 = 0 | v0 = 0)) % 14.94/4.20 | % 14.94/4.20 | Instantiating formula (18) with all_10_0_8, all_0_1_1, all_0_3_3, all_0_4_4 and discharging atoms apart_point_and_line(all_0_4_4, all_0_3_3) = all_10_0_8, distinct_lines(all_0_3_3, all_0_1_1) = 0, yields: % 14.94/4.20 | (74) all_10_0_8 = 0 | ? [v0] : ? [v1] : (apart_point_and_line(all_0_4_4, all_0_1_1) = v0 & convergent_lines(all_0_3_3, all_0_1_1) = v1 & (v1 = 0 | v0 = 0)) % 14.94/4.20 | % 14.94/4.20 +-Applying beta-rule and splitting (72), into two cases. % 14.94/4.20 |-Branch one: % 14.94/4.20 | (75) all_12_0_9 = 0 % 14.94/4.20 | % 14.94/4.21 | Equations (75) can reduce 69 to: % 14.94/4.21 | (76) $false % 14.94/4.21 | % 14.94/4.21 |-The branch is then unsatisfiable % 14.94/4.21 |-Branch two: % 14.94/4.21 | (69) ~ (all_12_0_9 = 0) % 14.94/4.21 | (78) ? [v0] : ( ~ (v0 = 0) & convergent_lines(all_0_2_2, all_0_2_2) = v0) % 14.94/4.21 | % 14.94/4.21 +-Applying beta-rule and splitting (74), into two cases. % 14.94/4.21 |-Branch one: % 14.94/4.21 | (79) all_10_0_8 = 0 % 14.94/4.21 | % 14.94/4.21 | Equations (79) can reduce 66 to: % 14.94/4.21 | (76) $false % 14.94/4.21 | % 14.94/4.21 |-The branch is then unsatisfiable % 14.94/4.21 |-Branch two: % 14.94/4.21 | (66) ~ (all_10_0_8 = 0) % 14.94/4.21 | (82) ? [v0] : ? [v1] : (apart_point_and_line(all_0_4_4, all_0_1_1) = v0 & convergent_lines(all_0_3_3, all_0_1_1) = v1 & (v1 = 0 | v0 = 0)) % 14.94/4.21 | % 14.94/4.21 | Instantiating (82) with all_47_0_15, all_47_1_16 yields: % 14.94/4.21 | (83) apart_point_and_line(all_0_4_4, all_0_1_1) = all_47_1_16 & convergent_lines(all_0_3_3, all_0_1_1) = all_47_0_15 & (all_47_0_15 = 0 | all_47_1_16 = 0) % 15.44/4.21 | % 15.44/4.21 | Applying alpha-rule on (83) yields: % 15.44/4.21 | (84) apart_point_and_line(all_0_4_4, all_0_1_1) = all_47_1_16 % 15.44/4.21 | (85) convergent_lines(all_0_3_3, all_0_1_1) = all_47_0_15 % 15.44/4.21 | (86) all_47_0_15 = 0 | all_47_1_16 = 0 % 15.44/4.21 | % 15.44/4.21 +-Applying beta-rule and splitting (73), into two cases. % 15.44/4.21 |-Branch one: % 15.44/4.21 | (75) all_12_0_9 = 0 % 15.44/4.21 | % 15.44/4.21 | Equations (75) can reduce 69 to: % 15.44/4.21 | (76) $false % 15.44/4.21 | % 15.44/4.21 |-The branch is then unsatisfiable % 15.44/4.21 |-Branch two: % 15.44/4.21 | (69) ~ (all_12_0_9 = 0) % 15.44/4.21 | (90) all_10_0_8 = 0 | ? [v0] : ? [v1] : (unorthogonal_lines(all_0_1_1, all_0_2_2) = v1 & apart_point_and_line(all_0_4_4, all_0_1_1) = v0 & (v1 = 0 | v0 = 0)) % 15.44/4.21 | % 15.44/4.21 +-Applying beta-rule and splitting (90), into two cases. % 15.44/4.21 |-Branch one: % 15.44/4.21 | (79) all_10_0_8 = 0 % 15.44/4.21 | % 15.44/4.21 | Equations (79) can reduce 66 to: % 15.44/4.21 | (76) $false % 15.44/4.21 | % 15.44/4.21 |-The branch is then unsatisfiable % 15.44/4.21 |-Branch two: % 15.44/4.21 | (66) ~ (all_10_0_8 = 0) % 15.44/4.21 | (94) ? [v0] : ? [v1] : (unorthogonal_lines(all_0_1_1, all_0_2_2) = v1 & apart_point_and_line(all_0_4_4, all_0_1_1) = v0 & (v1 = 0 | v0 = 0)) % 15.44/4.21 | % 15.44/4.21 | Instantiating (94) with all_57_0_17, all_57_1_18 yields: % 15.44/4.21 | (95) unorthogonal_lines(all_0_1_1, all_0_2_2) = all_57_0_17 & apart_point_and_line(all_0_4_4, all_0_1_1) = all_57_1_18 & (all_57_0_17 = 0 | all_57_1_18 = 0) % 15.44/4.21 | % 15.44/4.21 | Applying alpha-rule on (95) yields: % 15.44/4.21 | (96) unorthogonal_lines(all_0_1_1, all_0_2_2) = all_57_0_17 % 15.44/4.21 | (97) apart_point_and_line(all_0_4_4, all_0_1_1) = all_57_1_18 % 15.44/4.21 | (98) all_57_0_17 = 0 | all_57_1_18 = 0 % 15.44/4.21 | % 15.44/4.21 | Instantiating formula (12) with all_0_1_1, all_0_2_2, all_0_4_4 and discharging atoms orthogonal_through_point(all_0_2_2, all_0_4_4) = all_0_1_1, yields: % 15.44/4.21 | (99) ~ (unorthogonal_lines(all_0_1_1, all_0_2_2) = 0) % 15.44/4.21 | % 15.44/4.21 | Instantiating formula (61) with all_0_1_1, all_0_2_2, all_0_4_4 and discharging atoms orthogonal_through_point(all_0_2_2, all_0_4_4) = all_0_1_1, yields: % 15.44/4.21 | (100) ~ (apart_point_and_line(all_0_4_4, all_0_1_1) = 0) % 15.44/4.21 | % 15.44/4.21 | Instantiating formula (10) with all_0_4_4, all_0_1_1, all_47_1_16, all_57_1_18 and discharging atoms apart_point_and_line(all_0_4_4, all_0_1_1) = all_57_1_18, apart_point_and_line(all_0_4_4, all_0_1_1) = all_47_1_16, yields: % 15.44/4.21 | (101) all_57_1_18 = all_47_1_16 % 15.44/4.21 | % 15.44/4.21 | From (101) and (97) follows: % 15.44/4.21 | (84) apart_point_and_line(all_0_4_4, all_0_1_1) = all_47_1_16 % 15.44/4.21 | % 15.44/4.21 | Using (96) and (99) yields: % 15.44/4.21 | (103) ~ (all_57_0_17 = 0) % 15.44/4.21 | % 15.44/4.21 | Using (84) and (100) yields: % 15.44/4.21 | (104) ~ (all_47_1_16 = 0) % 15.44/4.21 | % 15.44/4.21 +-Applying beta-rule and splitting (98), into two cases. % 15.44/4.21 |-Branch one: % 15.44/4.21 | (105) all_57_0_17 = 0 % 15.44/4.21 | % 15.44/4.21 | Equations (105) can reduce 103 to: % 15.44/4.21 | (76) $false % 15.44/4.21 | % 15.44/4.21 |-The branch is then unsatisfiable % 15.44/4.21 |-Branch two: % 15.44/4.21 | (103) ~ (all_57_0_17 = 0) % 15.44/4.21 | (108) all_57_1_18 = 0 % 15.44/4.21 | % 15.44/4.21 | Combining equations (108,101) yields a new equation: % 15.44/4.21 | (109) all_47_1_16 = 0 % 15.44/4.21 | % 15.44/4.21 | Equations (109) can reduce 104 to: % 15.44/4.21 | (76) $false % 15.44/4.21 | % 15.44/4.21 |-The branch is then unsatisfiable % 15.44/4.21 |-Branch two: % 15.44/4.21 | (111) ~ (distinct_lines(all_0_3_3, all_0_1_1) = 0) % 15.44/4.21 | (112) all_0_0_0 = 0 % 15.44/4.21 | % 15.44/4.21 | Equations (112) can reduce 13 to: % 15.44/4.21 | (76) $false % 15.44/4.21 | % 15.44/4.21 |-The branch is then unsatisfiable % 15.44/4.21 % SZS output end Proof for theBenchmark % 15.44/4.21 % 15.44/4.21 3591ms %------------------------------------------------------------------------------