%------------------------------------------------------------------------------ % File : ePrincess---1.0 % Problem : GEO214+2 : TPTP v8.1.0. Released v3.3.0. % Transfm : none % Format : tptp:raw % Command : ePrincess-casc -timeout=%d %s % Computer : n010.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 03:48:48 EDT 2022 % Result : Theorem 6.18s 2.13s % Output : Proof 8.58s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.08/0.13 % Problem : GEO214+2 : TPTP v8.1.0. Released v3.3.0. % 0.08/0.14 % Command : ePrincess-casc -timeout=%d %s % 0.14/0.35 % Computer : n010.cluster.edu % 0.14/0.35 % Model : x86_64 x86_64 % 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.35 % Memory : 8042.1875MB % 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.35 % CPULimit : 300 % 0.14/0.35 % WCLimit : 600 % 0.14/0.35 % DateTime : Fri Jun 17 18:10:38 EDT 2022 % 0.14/0.35 % CPUTime : % 0.61/0.66 ____ _ % 0.61/0.66 ___ / __ \_____(_)___ ________ __________ % 0.61/0.66 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/ % 0.61/0.66 / __/ ____/ / / / / / / /__/ __(__ |__ ) % 0.61/0.66 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/ % 0.61/0.66 % 0.61/0.66 A Theorem Prover for First-Order Logic % 0.61/0.66 (ePrincess v.1.0) % 0.61/0.66 % 0.61/0.66 (c) Philipp Rümmer, 2009-2015 % 0.61/0.66 (c) Peter Backeman, 2014-2015 % 0.61/0.66 (contributions by Angelo Brillout, Peter Baumgartner) % 0.61/0.66 Free software under GNU Lesser General Public License (LGPL). % 0.61/0.66 Bug reports to peter@backeman.se % 0.61/0.66 % 0.61/0.66 For more information, visit http://user.uu.se/~petba168/breu/ % 0.61/0.66 % 0.61/0.66 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ... % 0.77/0.71 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 1.78/1.02 Prover 0: Preprocessing ... % 2.18/1.18 Prover 0: Warning: ignoring some quantifiers % 2.18/1.20 Prover 0: Constructing countermodel ... % 3.77/1.62 Prover 0: gave up % 3.77/1.62 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all % 4.08/1.66 Prover 1: Preprocessing ... % 4.56/1.77 Prover 1: Constructing countermodel ... % 4.78/1.82 Prover 1: gave up % 4.78/1.82 Prover 2: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 4.78/1.85 Prover 2: Preprocessing ... % 5.57/2.01 Prover 2: Warning: ignoring some quantifiers % 5.57/2.02 Prover 2: Constructing countermodel ... % 6.18/2.13 Prover 2: proved (309ms) % 6.18/2.13 % 6.18/2.13 No countermodel exists, formula is valid % 6.18/2.13 % SZS status Theorem for theBenchmark % 6.18/2.13 % 6.18/2.13 Generating proof ... Warning: ignoring some quantifiers % 7.90/2.54 found it (size 23) % 7.90/2.54 % 7.90/2.54 % SZS output start Proof for theBenchmark % 7.90/2.54 Assumed formulas after preprocessing and simplification: % 7.90/2.54 | (0) ? [v0] : ? [v1] : ? [v2] : ( ~ (v2 = 0) & unorthogonal_lines(v1, v0) = v2 & unorthogonal_lines(v0, v1) = 0 & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (unorthogonal_lines(v5, v6) = v8) | ~ (apart_point_and_line(v3, v5) = v7) | ~ (distinct_lines(v4, v5) = 0) | ? [v9] : ((v9 = 0 & unorthogonal_lines(v4, v6) = 0) | (v9 = 0 & apart_point_and_line(v3, v4) = 0))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (unorthogonal_lines(v5, v6) = v8) | ~ (apart_point_and_line(v3, v4) = v7) | ? [v9] : ((v9 = 0 & unorthogonal_lines(v4, v6) = 0) | (v9 = 0 & apart_point_and_line(v3, v5) = 0) | ( ~ (v9 = 0) & distinct_lines(v4, v5) = v9))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (unorthogonal_lines(v4, v6) = v8) | ~ (apart_point_and_line(v3, v5) = v7) | ? [v9] : ((v9 = 0 & unorthogonal_lines(v5, v6) = 0) | (v9 = 0 & apart_point_and_line(v3, v4) = 0) | ( ~ (v9 = 0) & distinct_lines(v4, v5) = v9))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (unorthogonal_lines(v4, v6) = v8) | ~ (apart_point_and_line(v3, v4) = v7) | ~ (distinct_lines(v4, v5) = 0) | ? [v9] : ((v9 = 0 & unorthogonal_lines(v5, v6) = 0) | (v9 = 0 & apart_point_and_line(v3, v5) = 0))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (apart_point_and_line(v4, v6) = v8) | ~ (apart_point_and_line(v4, v5) = v7) | ~ (distinct_points(v3, v4) = 0) | ? [v9] : ((v9 = 0 & apart_point_and_line(v3, v6) = 0) | (v9 = 0 & apart_point_and_line(v3, v5) = 0) | ( ~ (v9 = 0) & distinct_lines(v5, v6) = v9))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (apart_point_and_line(v4, v6) = v8) | ~ (apart_point_and_line(v3, v6) = v7) | ~ (distinct_lines(v5, v6) = 0) | ? [v9] : ((v9 = 0 & apart_point_and_line(v4, v5) = 0) | (v9 = 0 & apart_point_and_line(v3, v5) = 0) | ( ~ (v9 = 0) & distinct_points(v3, v4) = v9))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (apart_point_and_line(v4, v6) = v8) | ~ (apart_point_and_line(v3, v5) = v7) | ? [v9] : ((v9 = 0 & apart_point_and_line(v4, v5) = 0) | (v9 = 0 & apart_point_and_line(v3, v6) = 0) | ( ~ (v9 = 0) & distinct_lines(v5, v6) = v9) | ( ~ (v9 = 0) & distinct_points(v3, v4) = v9))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (apart_point_and_line(v4, v5) = v8) | ~ (apart_point_and_line(v3, v6) = v7) | ? [v9] : ((v9 = 0 & apart_point_and_line(v4, v6) = 0) | (v9 = 0 & apart_point_and_line(v3, v5) = 0) | ( ~ (v9 = 0) & distinct_lines(v5, v6) = v9) | ( ~ (v9 = 0) & distinct_points(v3, v4) = v9))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (apart_point_and_line(v4, v5) = v8) | ~ (apart_point_and_line(v3, v5) = v7) | ~ (distinct_lines(v5, v6) = 0) | ? [v9] : ((v9 = 0 & apart_point_and_line(v4, v6) = 0) | (v9 = 0 & apart_point_and_line(v3, v6) = 0) | ( ~ (v9 = 0) & distinct_points(v3, v4) = v9))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (apart_point_and_line(v3, v6) = v8) | ~ (apart_point_and_line(v3, v5) = v7) | ~ (distinct_points(v3, v4) = 0) | ? [v9] : ((v9 = 0 & apart_point_and_line(v4, v6) = 0) | (v9 = 0 & apart_point_and_line(v4, v5) = 0) | ( ~ (v9 = 0) & distinct_lines(v5, v6) = v9))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | v6 = 0 | ~ (apart_point_and_line(v5, v4) = v7) | ~ (distinct_points(v3, v5) = v6) | ? [v8] : ( ~ (v8 = 0) & apart_point_and_line(v3, v4) = v8)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | v6 = 0 | ~ (apart_point_and_line(v3, v5) = v7) | ~ (apart_point_and_line(v3, v4) = v6) | ? [v8] : ((v8 = 0 & convergent_lines(v4, v5) = 0) | ( ~ (v8 = 0) & distinct_lines(v4, v5) = v8))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | v6 = 0 | ~ (apart_point_and_line(v3, v5) = v7) | ~ (distinct_lines(v4, v5) = v6) | ? [v8] : ( ~ (v8 = 0) & apart_point_and_line(v3, v4) = v8)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | v6 = 0 | ~ (apart_point_and_line(v3, v5) = v6) | ~ (convergent_lines(v4, v5) = v7) | ? [v8] : ((v8 = 0 & apart_point_and_line(v3, v4) = 0) | ( ~ (v8 = 0) & distinct_lines(v4, v5) = v8))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | v6 = 0 | ~ (apart_point_and_line(v3, v4) = v6) | ~ (convergent_lines(v4, v5) = v7) | ? [v8] : ((v8 = 0 & apart_point_and_line(v3, v5) = 0) | ( ~ (v8 = 0) & distinct_lines(v4, v5) = v8))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | v6 = 0 | ~ (convergent_lines(v4, v5) = v7) | ~ (convergent_lines(v3, v5) = v6) | ? [v8] : ( ~ (v8 = 0) & convergent_lines(v3, v4) = v8)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | v6 = 0 | ~ (distinct_lines(v4, v5) = v7) | ~ (distinct_lines(v3, v5) = v6) | ? [v8] : ( ~ (v8 = 0) & distinct_lines(v3, v4) = v8)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | v6 = 0 | ~ (distinct_points(v4, v5) = v7) | ~ (distinct_points(v3, v5) = v6) | ? [v8] : ( ~ (v8 = 0) & distinct_points(v3, v4) = v8)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (intersection_point(v3, v4) = v6) | ~ (distinct_points(v5, v6) = v7) | ? [v8] : ? [v9] : (( ~ (v9 = 0) & ~ (v8 = 0) & apart_point_and_line(v5, v4) = v9 & apart_point_and_line(v5, v3) = v8) | ( ~ (v8 = 0) & convergent_lines(v3, v4) = v8))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (unorthogonal_lines(v4, v5) = v7) | ~ (unorthogonal_lines(v3, v5) = v6) | ? [v8] : ((v8 = 0 & v7 = 0 & convergent_lines(v4, v5) = 0) | (v8 = 0 & v6 = 0 & convergent_lines(v3, v5) = 0) | ( ~ (v8 = 0) & unorthogonal_lines(v3, v4) = v8) | ( ~ (v8 = 0) & convergent_lines(v3, v4) = v8))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (unorthogonal_lines(v4, v5) = v7) | ~ (convergent_lines(v3, v5) = v6) | ? [v8] : ((v8 = 0 & v7 = 0 & convergent_lines(v4, v5) = 0) | (v8 = 0 & v6 = 0 & unorthogonal_lines(v3, v5) = 0) | ( ~ (v8 = 0) & unorthogonal_lines(v3, v4) = v8) | ( ~ (v8 = 0) & convergent_lines(v3, v4) = v8))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (unorthogonal_lines(v3, v5) = v6) | ~ (convergent_lines(v4, v5) = v7) | ? [v8] : ((v8 = 0 & v7 = 0 & unorthogonal_lines(v4, v5) = 0) | (v8 = 0 & v6 = 0 & convergent_lines(v3, v5) = 0) | ( ~ (v8 = 0) & unorthogonal_lines(v3, v4) = v8) | ( ~ (v8 = 0) & convergent_lines(v3, v4) = v8))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (convergent_lines(v4, v5) = v7) | ~ (convergent_lines(v3, v5) = v6) | ? [v8] : ((v8 = 0 & v7 = 0 & unorthogonal_lines(v4, v5) = 0) | (v8 = 0 & v6 = 0 & unorthogonal_lines(v3, v5) = 0) | ( ~ (v8 = 0) & unorthogonal_lines(v3, v4) = v8) | ( ~ (v8 = 0) & convergent_lines(v3, v4) = v8))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (apart_point_and_line(v5, v4) = v6) | ~ (apart_point_and_line(v3, v4) = 0) | distinct_points(v3, v5) = 0) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (apart_point_and_line(v3, v5) = v6) | ~ (apart_point_and_line(v3, v4) = 0) | distinct_lines(v4, v5) = 0) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (apart_point_and_line(v3, v5) = v6) | ~ (distinct_lines(v4, v5) = 0) | ? [v7] : ((v7 = 0 & apart_point_and_line(v3, v4) = 0) | (v7 = 0 & convergent_lines(v4, v5) = 0))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (apart_point_and_line(v3, v4) = v6) | ~ (distinct_lines(v4, v5) = 0) | ? [v7] : ((v7 = 0 & apart_point_and_line(v3, v5) = 0) | (v7 = 0 & convergent_lines(v4, v5) = 0))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (apart_point_and_line(v3, v4) = 0) | ~ (distinct_lines(v4, v5) = v6) | apart_point_and_line(v3, v5) = 0) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (apart_point_and_line(v3, v4) = 0) | ~ (distinct_points(v3, v5) = v6) | apart_point_and_line(v5, v4) = 0) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (convergent_lines(v4, v5) = v6) | ~ (convergent_lines(v3, v4) = 0) | convergent_lines(v3, v5) = 0) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (convergent_lines(v3, v5) = v6) | ~ (convergent_lines(v3, v4) = 0) | convergent_lines(v4, v5) = 0) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (distinct_lines(v4, v5) = v6) | ~ (distinct_lines(v3, v4) = 0) | distinct_lines(v3, v5) = 0) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (distinct_lines(v3, v5) = v6) | ~ (distinct_lines(v3, v4) = 0) | distinct_lines(v4, v5) = 0) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (distinct_points(v4, v5) = v6) | ~ (distinct_points(v3, v4) = 0) | distinct_points(v3, v5) = 0) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (distinct_points(v3, v5) = v6) | ~ (distinct_points(v3, v4) = 0) | distinct_points(v4, v5) = 0) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (orthogonal_through_point(v6, v5) = v4) | ~ (orthogonal_through_point(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (unorthogonal_lines(v6, v5) = v4) | ~ (unorthogonal_lines(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (parallel_through_point(v6, v5) = v4) | ~ (parallel_through_point(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (intersection_point(v6, v5) = v4) | ~ (intersection_point(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (line_connecting(v6, v5) = v4) | ~ (line_connecting(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (apart_point_and_line(v6, v5) = v4) | ~ (apart_point_and_line(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (convergent_lines(v6, v5) = v4) | ~ (convergent_lines(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (distinct_lines(v6, v5) = v4) | ~ (distinct_lines(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v4 = v3 | ~ (distinct_points(v6, v5) = v4) | ~ (distinct_points(v6, v5) = v3)) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (unorthogonal_lines(v4, v5) = v6) | ~ (unorthogonal_lines(v3, v4) = 0) | ? [v7] : ? [v8] : ((v8 = 0 & v7 = 0 & unorthogonal_lines(v3, v5) = 0 & convergent_lines(v3, v5) = 0) | (v7 = 0 & v6 = 0 & convergent_lines(v4, v5) = 0) | ( ~ (v7 = 0) & convergent_lines(v3, v4) = v7))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (unorthogonal_lines(v4, v5) = v6) | ~ (convergent_lines(v3, v4) = 0) | ? [v7] : ? [v8] : ((v8 = 0 & v7 = 0 & unorthogonal_lines(v3, v5) = 0 & convergent_lines(v3, v5) = 0) | (v7 = 0 & v6 = 0 & convergent_lines(v4, v5) = 0) | ( ~ (v7 = 0) & unorthogonal_lines(v3, v4) = v7))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (unorthogonal_lines(v3, v5) = v6) | ~ (unorthogonal_lines(v3, v4) = 0) | ? [v7] : ? [v8] : ((v8 = 0 & v7 = 0 & unorthogonal_lines(v4, v5) = 0 & convergent_lines(v4, v5) = 0) | (v7 = 0 & v6 = 0 & convergent_lines(v3, v5) = 0) | ( ~ (v7 = 0) & convergent_lines(v3, v4) = v7))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (unorthogonal_lines(v3, v5) = v6) | ~ (convergent_lines(v3, v4) = 0) | ? [v7] : ? [v8] : ((v8 = 0 & v7 = 0 & unorthogonal_lines(v4, v5) = 0 & convergent_lines(v4, v5) = 0) | (v7 = 0 & v6 = 0 & convergent_lines(v3, v5) = 0) | ( ~ (v7 = 0) & unorthogonal_lines(v3, v4) = v7))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (unorthogonal_lines(v3, v4) = 0) | ~ (convergent_lines(v4, v5) = v6) | ? [v7] : ? [v8] : ((v8 = 0 & v7 = 0 & unorthogonal_lines(v3, v5) = 0 & convergent_lines(v3, v5) = 0) | (v7 = 0 & v6 = 0 & unorthogonal_lines(v4, v5) = 0) | ( ~ (v7 = 0) & convergent_lines(v3, v4) = v7))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (unorthogonal_lines(v3, v4) = 0) | ~ (convergent_lines(v3, v5) = v6) | ? [v7] : ? [v8] : ((v8 = 0 & v7 = 0 & unorthogonal_lines(v4, v5) = 0 & convergent_lines(v4, v5) = 0) | (v7 = 0 & v6 = 0 & unorthogonal_lines(v3, v5) = 0) | ( ~ (v7 = 0) & convergent_lines(v3, v4) = v7))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (line_connecting(v3, v4) = v6) | ~ (apart_point_and_line(v5, v6) = 0) | ? [v7] : ? [v8] : ((v8 = 0 & v7 = 0 & distinct_points(v5, v4) = 0 & distinct_points(v5, v3) = 0) | ( ~ (v7 = 0) & distinct_points(v3, v4) = v7))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (convergent_lines(v4, v5) = v6) | ~ (convergent_lines(v3, v4) = 0) | ? [v7] : ? [v8] : ((v8 = 0 & v7 = 0 & unorthogonal_lines(v3, v5) = 0 & convergent_lines(v3, v5) = 0) | (v7 = 0 & v6 = 0 & unorthogonal_lines(v4, v5) = 0) | ( ~ (v7 = 0) & unorthogonal_lines(v3, v4) = v7))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (convergent_lines(v3, v5) = v6) | ~ (convergent_lines(v3, v4) = 0) | ? [v7] : ? [v8] : ((v8 = 0 & v7 = 0 & unorthogonal_lines(v4, v5) = 0 & convergent_lines(v4, v5) = 0) | (v7 = 0 & v6 = 0 & unorthogonal_lines(v3, v5) = 0) | ( ~ (v7 = 0) & unorthogonal_lines(v3, v4) = v7))) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (distinct_lines(v5, v6) = 0) | ~ (distinct_points(v3, v4) = 0) | ? [v7] : ((v7 = 0 & apart_point_and_line(v4, v6) = 0) | (v7 = 0 & apart_point_and_line(v4, v5) = 0) | (v7 = 0 & apart_point_and_line(v3, v6) = 0) | (v7 = 0 & apart_point_and_line(v3, v5) = 0))) & ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (unorthogonal_lines(v3, v4) = v5) | convergent_lines(v3, v4) = 0) & ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (convergent_lines(v3, v4) = v5) | unorthogonal_lines(v3, v4) = 0) & ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (distinct_lines(v3, v4) = v5) | ? [v6] : ( ~ (v6 = 0) & convergent_lines(v3, v4) = v6)) & ! [v3] : ! [v4] : ! [v5] : ( ~ (orthogonal_through_point(v4, v3) = v5) | ? [v6] : ( ~ (v6 = 0) & unorthogonal_lines(v5, v4) = v6)) & ! [v3] : ! [v4] : ! [v5] : ( ~ (orthogonal_through_point(v4, v3) = v5) | ? [v6] : ( ~ (v6 = 0) & apart_point_and_line(v3, v5) = v6)) & ! [v3] : ! [v4] : ! [v5] : ( ~ (parallel_through_point(v4, v3) = v5) | ? [v6] : ( ~ (v6 = 0) & apart_point_and_line(v3, v5) = v6)) & ! [v3] : ! [v4] : ! [v5] : ( ~ (parallel_through_point(v4, v3) = v5) | ? [v6] : ( ~ (v6 = 0) & convergent_lines(v5, v4) = v6)) & ! [v3] : ! [v4] : ( ~ (convergent_lines(v3, v4) = 0) | distinct_lines(v3, v4) = 0) & ! [v3] : ~ (convergent_lines(v3, v3) = 0) & ! [v3] : ~ (distinct_lines(v3, v3) = 0) & ! [v3] : ~ (distinct_points(v3, v3) = 0) & ? [v3] : ? [v4] : ? [v5] : orthogonal_through_point(v4, v3) = v5 & ? [v3] : ? [v4] : ? [v5] : unorthogonal_lines(v4, v3) = v5 & ? [v3] : ? [v4] : ? [v5] : parallel_through_point(v4, v3) = v5 & ? [v3] : ? [v4] : ? [v5] : intersection_point(v4, v3) = v5 & ? [v3] : ? [v4] : ? [v5] : line_connecting(v4, v3) = v5 & ? [v3] : ? [v4] : ? [v5] : apart_point_and_line(v4, v3) = v5 & ? [v3] : ? [v4] : ? [v5] : convergent_lines(v4, v3) = v5 & ? [v3] : ? [v4] : ? [v5] : distinct_lines(v4, v3) = v5 & ? [v3] : ? [v4] : ? [v5] : distinct_points(v4, v3) = v5) % 8.13/2.59 | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2 yields: % 8.13/2.59 | (1) ~ (all_0_0_0 = 0) & unorthogonal_lines(all_0_1_1, all_0_2_2) = all_0_0_0 & unorthogonal_lines(all_0_2_2, all_0_1_1) = 0 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (unorthogonal_lines(v2, v3) = v5) | ~ (apart_point_and_line(v0, v2) = v4) | ~ (distinct_lines(v1, v2) = 0) | ? [v6] : ((v6 = 0 & unorthogonal_lines(v1, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v1) = 0))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (unorthogonal_lines(v2, v3) = v5) | ~ (apart_point_and_line(v0, v1) = v4) | ? [v6] : ((v6 = 0 & unorthogonal_lines(v1, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v2) = 0) | ( ~ (v6 = 0) & distinct_lines(v1, v2) = v6))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (unorthogonal_lines(v1, v3) = v5) | ~ (apart_point_and_line(v0, v2) = v4) | ? [v6] : ((v6 = 0 & unorthogonal_lines(v2, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v1) = 0) | ( ~ (v6 = 0) & distinct_lines(v1, v2) = v6))) & ! [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] : ((v6 = 0 & unorthogonal_lines(v2, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v2) = 0))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v3) = v5) | ~ (apart_point_and_line(v1, v2) = v4) | ~ (distinct_points(v0, v1) = 0) | ? [v6] : ((v6 = 0 & apart_point_and_line(v0, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v2) = 0) | ( ~ (v6 = 0) & distinct_lines(v2, v3) = v6))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v3) = v5) | ~ (apart_point_and_line(v0, v3) = v4) | ~ (distinct_lines(v2, v3) = 0) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v2) = 0) | (v6 = 0 & apart_point_and_line(v0, v2) = 0) | ( ~ (v6 = 0) & distinct_points(v0, v1) = v6))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v3) = v5) | ~ (apart_point_and_line(v0, v2) = v4) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v2) = 0) | (v6 = 0 & apart_point_and_line(v0, v3) = 0) | ( ~ (v6 = 0) & distinct_lines(v2, v3) = v6) | ( ~ (v6 = 0) & distinct_points(v0, v1) = v6))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v2) = v5) | ~ (apart_point_and_line(v0, v3) = v4) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v2) = 0) | ( ~ (v6 = 0) & distinct_lines(v2, v3) = v6) | ( ~ (v6 = 0) & distinct_points(v0, v1) = v6))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v2) = v5) | ~ (apart_point_and_line(v0, v2) = v4) | ~ (distinct_lines(v2, v3) = 0) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v3) = 0) | ( ~ (v6 = 0) & distinct_points(v0, v1) = v6))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v0, v3) = v5) | ~ (apart_point_and_line(v0, v2) = v4) | ~ (distinct_points(v0, v1) = 0) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v3) = 0) | (v6 = 0 & apart_point_and_line(v1, v2) = 0) | ( ~ (v6 = 0) & distinct_lines(v2, v3) = v6))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (apart_point_and_line(v2, v1) = v4) | ~ (distinct_points(v0, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & apart_point_and_line(v0, v1) = v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (apart_point_and_line(v0, v2) = v4) | ~ (apart_point_and_line(v0, v1) = v3) | ? [v5] : ((v5 = 0 & convergent_lines(v1, v2) = 0) | ( ~ (v5 = 0) & distinct_lines(v1, v2) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (apart_point_and_line(v0, v2) = v4) | ~ (distinct_lines(v1, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & apart_point_and_line(v0, v1) = v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (apart_point_and_line(v0, v2) = v3) | ~ (convergent_lines(v1, v2) = v4) | ? [v5] : ((v5 = 0 & apart_point_and_line(v0, v1) = 0) | ( ~ (v5 = 0) & distinct_lines(v1, v2) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (apart_point_and_line(v0, v1) = v3) | ~ (convergent_lines(v1, v2) = v4) | ? [v5] : ((v5 = 0 & apart_point_and_line(v0, v2) = 0) | ( ~ (v5 = 0) & distinct_lines(v1, v2) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (convergent_lines(v1, v2) = v4) | ~ (convergent_lines(v0, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & convergent_lines(v0, v1) = v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (distinct_lines(v1, v2) = v4) | ~ (distinct_lines(v0, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & distinct_lines(v0, v1) = v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (distinct_points(v1, v2) = v4) | ~ (distinct_points(v0, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & distinct_points(v0, v1) = v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | ~ (intersection_point(v0, v1) = v3) | ~ (distinct_points(v2, v3) = v4) | ? [v5] : ? [v6] : (( ~ (v6 = 0) & ~ (v5 = 0) & apart_point_and_line(v2, v1) = v6 & apart_point_and_line(v2, v0) = v5) | ( ~ (v5 = 0) & convergent_lines(v0, v1) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (unorthogonal_lines(v1, v2) = v4) | ~ (unorthogonal_lines(v0, v2) = v3) | ? [v5] : ((v5 = 0 & v4 = 0 & convergent_lines(v1, v2) = 0) | (v5 = 0 & v3 = 0 & convergent_lines(v0, v2) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v0, v1) = v5) | ( ~ (v5 = 0) & convergent_lines(v0, v1) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (unorthogonal_lines(v1, v2) = v4) | ~ (convergent_lines(v0, v2) = v3) | ? [v5] : ((v5 = 0 & v4 = 0 & convergent_lines(v1, v2) = 0) | (v5 = 0 & v3 = 0 & unorthogonal_lines(v0, v2) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v0, v1) = v5) | ( ~ (v5 = 0) & convergent_lines(v0, v1) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (unorthogonal_lines(v0, v2) = v3) | ~ (convergent_lines(v1, v2) = v4) | ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v1, v2) = 0) | (v5 = 0 & v3 = 0 & convergent_lines(v0, v2) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v0, v1) = v5) | ( ~ (v5 = 0) & convergent_lines(v0, v1) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (convergent_lines(v1, v2) = v4) | ~ (convergent_lines(v0, v2) = v3) | ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v1, v2) = 0) | (v5 = 0 & v3 = 0 & unorthogonal_lines(v0, v2) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v0, v1) = v5) | ( ~ (v5 = 0) & convergent_lines(v0, v1) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v2, v1) = v3) | ~ (apart_point_and_line(v0, v1) = 0) | distinct_points(v0, v2) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v2) = v3) | ~ (apart_point_and_line(v0, v1) = 0) | distinct_lines(v1, v2) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v2) = v3) | ~ (distinct_lines(v1, v2) = 0) | ? [v4] : ((v4 = 0 & apart_point_and_line(v0, v1) = 0) | (v4 = 0 & convergent_lines(v1, v2) = 0))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v1) = v3) | ~ (distinct_lines(v1, v2) = 0) | ? [v4] : ((v4 = 0 & apart_point_and_line(v0, v2) = 0) | (v4 = 0 & convergent_lines(v1, v2) = 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(v1, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v0, v2) = 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 | ~ (distinct_lines(v1, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_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(v1, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v0, 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_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(v1, v2) = v3) | ~ (unorthogonal_lines(v0, v1) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v1, v2) = 0) | ( ~ (v4 = 0) & convergent_lines(v0, v1) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v1, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v1, v2) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v0, v1) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v0, v2) = v3) | ~ (unorthogonal_lines(v0, v1) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v1, v2) = 0 & convergent_lines(v1, v2) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v0, v2) = 0) | ( ~ (v4 = 0) & convergent_lines(v0, v1) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v1, v2) = 0 & convergent_lines(v1, v2) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v0, v2) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v0, v1) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v0, v1) = 0) | ~ (convergent_lines(v1, v2) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v1, v2) = 0) | ( ~ (v4 = 0) & convergent_lines(v0, v1) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v0, v1) = 0) | ~ (convergent_lines(v0, v2) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v1, v2) = 0 & convergent_lines(v1, v2) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v0, v2) = 0) | ( ~ (v4 = 0) & convergent_lines(v0, v1) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (line_connecting(v0, v1) = v3) | ~ (apart_point_and_line(v2, v3) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & distinct_points(v2, v1) = 0 & distinct_points(v2, v0) = 0) | ( ~ (v4 = 0) & distinct_points(v0, v1) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (convergent_lines(v1, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v1, v2) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v0, v1) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v1, v2) = 0 & convergent_lines(v1, v2) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v0, v2) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v0, v1) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (distinct_lines(v2, v3) = 0) | ~ (distinct_points(v0, v1) = 0) | ? [v4] : ((v4 = 0 & apart_point_and_line(v1, v3) = 0) | (v4 = 0 & apart_point_and_line(v1, v2) = 0) | (v4 = 0 & apart_point_and_line(v0, v3) = 0) | (v4 = 0 & apart_point_and_line(v0, v2) = 0))) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (unorthogonal_lines(v0, v1) = v2) | convergent_lines(v0, v1) = 0) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (convergent_lines(v0, v1) = v2) | unorthogonal_lines(v0, v1) = 0) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (distinct_lines(v0, v1) = v2) | ? [v3] : ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) | ? [v3] : ( ~ (v3 = 0) & unorthogonal_lines(v2, v1) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) | ? [v3] : ( ~ (v3 = 0) & apart_point_and_line(v0, v2) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) | ? [v3] : ( ~ (v3 = 0) & apart_point_and_line(v0, v2) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) | ? [v3] : ( ~ (v3 = 0) & convergent_lines(v2, v1) = v3)) & ! [v0] : ! [v1] : ( ~ (convergent_lines(v0, v1) = 0) | distinct_lines(v0, v1) = 0) & ! [v0] : ~ (convergent_lines(v0, v0) = 0) & ! [v0] : ~ (distinct_lines(v0, v0) = 0) & ! [v0] : ~ (distinct_points(v0, v0) = 0) & ? [v0] : ? [v1] : ? [v2] : orthogonal_through_point(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : unorthogonal_lines(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : parallel_through_point(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : intersection_point(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : line_connecting(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : apart_point_and_line(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : convergent_lines(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : distinct_lines(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : distinct_points(v1, v0) = v2 % 8.36/2.62 | % 8.36/2.62 | Applying alpha-rule on (1) yields: % 8.36/2.62 | (2) ? [v0] : ? [v1] : ? [v2] : unorthogonal_lines(v1, v0) = v2 % 8.36/2.62 | (3) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v2) = v5) | ~ (apart_point_and_line(v0, v2) = v4) | ~ (distinct_lines(v2, v3) = 0) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v3) = 0) | ( ~ (v6 = 0) & distinct_points(v0, v1) = v6))) % 8.36/2.62 | (4) ? [v0] : ? [v1] : ? [v2] : orthogonal_through_point(v1, v0) = v2 % 8.36/2.62 | (5) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v3) = v5) | ~ (apart_point_and_line(v0, v3) = v4) | ~ (distinct_lines(v2, v3) = 0) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v2) = 0) | (v6 = 0 & apart_point_and_line(v0, v2) = 0) | ( ~ (v6 = 0) & distinct_points(v0, v1) = v6))) % 8.36/2.62 | (6) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (apart_point_and_line(v0, v1) = v3) | ~ (convergent_lines(v1, v2) = v4) | ? [v5] : ((v5 = 0 & apart_point_and_line(v0, v2) = 0) | ( ~ (v5 = 0) & distinct_lines(v1, v2) = v5))) % 8.36/2.62 | (7) ? [v0] : ? [v1] : ? [v2] : convergent_lines(v1, v0) = v2 % 8.36/2.62 | (8) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v1, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v0, v2) = 0) % 8.36/2.62 | (9) ! [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) % 8.36/2.62 | (10) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (convergent_lines(v1, v2) = v4) | ~ (convergent_lines(v0, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & convergent_lines(v0, v1) = v5)) % 8.36/2.62 | (11) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (intersection_point(v3, v2) = v1) | ~ (intersection_point(v3, v2) = v0)) % 8.36/2.62 | (12) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v1, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v1, v2) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v0, v1) = v4))) % 8.36/2.62 | (13) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v1, v2) = v3) | ~ (unorthogonal_lines(v0, v1) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v1, v2) = 0) | ( ~ (v4 = 0) & convergent_lines(v0, v1) = v4))) % 8.36/2.62 | (14) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (parallel_through_point(v3, v2) = v1) | ~ (parallel_through_point(v3, v2) = v0)) % 8.36/2.62 | (15) ! [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] : ((v6 = 0 & unorthogonal_lines(v2, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v2) = 0))) % 8.36/2.62 | (16) ~ (all_0_0_0 = 0) % 8.36/2.62 | (17) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v0, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0) % 8.36/2.62 | (18) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v3) = v5) | ~ (apart_point_and_line(v0, v2) = v4) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v2) = 0) | (v6 = 0 & apart_point_and_line(v0, v3) = 0) | ( ~ (v6 = 0) & distinct_lines(v2, v3) = v6) | ( ~ (v6 = 0) & distinct_points(v0, v1) = v6))) % 8.36/2.62 | (19) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v1, v2) = 0) % 8.36/2.62 | (20) ! [v0] : ! [v1] : ( ~ (convergent_lines(v0, v1) = 0) | distinct_lines(v0, v1) = 0) % 8.36/2.62 | (21) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (distinct_points(v1, v2) = v4) | ~ (distinct_points(v0, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & distinct_points(v0, v1) = v5)) % 8.36/2.62 | (22) ? [v0] : ? [v1] : ? [v2] : apart_point_and_line(v1, v0) = v2 % 8.36/2.62 | (23) ? [v0] : ? [v1] : ? [v2] : parallel_through_point(v1, v0) = v2 % 8.36/2.63 | (24) ! [v0] : ! [v1] : ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) | ? [v3] : ( ~ (v3 = 0) & convergent_lines(v2, v1) = v3)) % 8.36/2.63 | (25) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | ~ (intersection_point(v0, v1) = v3) | ~ (distinct_points(v2, v3) = v4) | ? [v5] : ? [v6] : (( ~ (v6 = 0) & ~ (v5 = 0) & apart_point_and_line(v2, v1) = v6 & apart_point_and_line(v2, v0) = v5) | ( ~ (v5 = 0) & convergent_lines(v0, v1) = v5))) % 8.36/2.63 | (26) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v2) = v5) | ~ (apart_point_and_line(v0, v3) = v4) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v2) = 0) | ( ~ (v6 = 0) & distinct_lines(v2, v3) = v6) | ( ~ (v6 = 0) & distinct_points(v0, v1) = v6))) % 8.36/2.63 | (27) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_points(v3, v2) = v1) | ~ (distinct_points(v3, v2) = v0)) % 8.36/2.63 | (28) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v1) = v3) | ~ (distinct_lines(v1, v2) = 0) | ? [v4] : ((v4 = 0 & apart_point_and_line(v0, v2) = 0) | (v4 = 0 & convergent_lines(v1, v2) = 0))) % 8.36/2.63 | (29) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (unorthogonal_lines(v2, v3) = v5) | ~ (apart_point_and_line(v0, v1) = v4) | ? [v6] : ((v6 = 0 & unorthogonal_lines(v1, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v2) = 0) | ( ~ (v6 = 0) & distinct_lines(v1, v2) = v6))) % 8.36/2.63 | (30) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v0, v1) = 0) | ~ (convergent_lines(v1, v2) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v1, v2) = 0) | ( ~ (v4 = 0) & convergent_lines(v0, v1) = v4))) % 8.36/2.63 | (31) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (convergent_lines(v1, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v1, v2) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v0, v1) = v4))) % 8.36/2.63 | (32) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (orthogonal_through_point(v3, v2) = v1) | ~ (orthogonal_through_point(v3, v2) = v0)) % 8.36/2.63 | (33) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (apart_point_and_line(v0, v2) = v4) | ~ (distinct_lines(v1, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & apart_point_and_line(v0, v1) = v5)) % 8.36/2.63 | (34) ? [v0] : ? [v1] : ? [v2] : distinct_lines(v1, v0) = v2 % 8.36/2.63 | (35) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v2) = v3) | ~ (apart_point_and_line(v0, v1) = 0) | distinct_lines(v1, v2) = 0) % 8.36/2.63 | (36) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (unorthogonal_lines(v3, v2) = v1) | ~ (unorthogonal_lines(v3, v2) = v0)) % 8.36/2.63 | (37) ! [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) % 8.36/2.63 | (38) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (apart_point_and_line(v3, v2) = v1) | ~ (apart_point_and_line(v3, v2) = v0)) % 8.36/2.63 | (39) ! [v0] : ! [v1] : ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) | ? [v3] : ( ~ (v3 = 0) & unorthogonal_lines(v2, v1) = v3)) % 8.36/2.63 | (40) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v2) = v3) | ~ (distinct_lines(v1, v2) = 0) | ? [v4] : ((v4 = 0 & apart_point_and_line(v0, v1) = 0) | (v4 = 0 & convergent_lines(v1, v2) = 0))) % 8.36/2.63 | (41) unorthogonal_lines(all_0_2_2, all_0_1_1) = 0 % 8.36/2.63 | (42) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v0, v1) = 0) | ~ (convergent_lines(v0, v2) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v1, v2) = 0 & convergent_lines(v1, v2) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v0, v2) = 0) | ( ~ (v4 = 0) & convergent_lines(v0, v1) = v4))) % 8.36/2.63 | (43) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v1, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v0, v2) = 0) % 8.36/2.63 | (44) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (line_connecting(v0, v1) = v3) | ~ (apart_point_and_line(v2, v3) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & distinct_points(v2, v1) = 0 & distinct_points(v2, v0) = 0) | ( ~ (v4 = 0) & distinct_points(v0, v1) = v4))) % 8.36/2.63 | (45) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v1, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v0, v2) = 0) % 8.36/2.63 | (46) ! [v0] : ~ (distinct_points(v0, v0) = 0) % 8.36/2.63 | (47) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (convergent_lines(v3, v2) = v1) | ~ (convergent_lines(v3, v2) = v0)) % 8.36/2.63 | (48) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (unorthogonal_lines(v0, v2) = v3) | ~ (convergent_lines(v1, v2) = v4) | ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v1, v2) = 0) | (v5 = 0 & v3 = 0 & convergent_lines(v0, v2) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v0, v1) = v5) | ( ~ (v5 = 0) & convergent_lines(v0, v1) = v5))) % 8.36/2.63 | (49) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_lines(v3, v2) = v1) | ~ (distinct_lines(v3, v2) = v0)) % 8.36/2.63 | (50) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v2, v1) = v3) | ~ (apart_point_and_line(v0, v1) = 0) | distinct_points(v0, v2) = 0) % 8.36/2.63 | (51) ? [v0] : ? [v1] : ? [v2] : distinct_points(v1, v0) = v2 % 8.36/2.63 | (52) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (apart_point_and_line(v2, v1) = v4) | ~ (distinct_points(v0, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & apart_point_and_line(v0, v1) = v5)) % 8.36/2.63 | (53) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (unorthogonal_lines(v1, v2) = v4) | ~ (unorthogonal_lines(v0, v2) = v3) | ? [v5] : ((v5 = 0 & v4 = 0 & convergent_lines(v1, v2) = 0) | (v5 = 0 & v3 = 0 & convergent_lines(v0, v2) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v0, v1) = v5) | ( ~ (v5 = 0) & convergent_lines(v0, v1) = v5))) % 8.36/2.63 | (54) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v0, v3) = v5) | ~ (apart_point_and_line(v0, v2) = v4) | ~ (distinct_points(v0, v1) = 0) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v3) = 0) | (v6 = 0 & apart_point_and_line(v1, v2) = 0) | ( ~ (v6 = 0) & distinct_lines(v2, v3) = v6))) % 8.36/2.63 | (55) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (unorthogonal_lines(v1, v3) = v5) | ~ (apart_point_and_line(v0, v2) = v4) | ? [v6] : ((v6 = 0 & unorthogonal_lines(v2, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v1) = 0) | ( ~ (v6 = 0) & distinct_lines(v1, v2) = v6))) % 8.36/2.63 | (56) ! [v0] : ~ (convergent_lines(v0, v0) = 0) % 8.36/2.63 | (57) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (convergent_lines(v0, v1) = v2) | unorthogonal_lines(v0, v1) = 0) % 8.36/2.63 | (58) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (unorthogonal_lines(v0, v1) = v2) | convergent_lines(v0, v1) = 0) % 8.36/2.64 | (59) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v1, v2) = 0 & convergent_lines(v1, v2) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v0, v2) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v0, v1) = v4))) % 8.36/2.64 | (60) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v0, v2) = v3) | ~ (unorthogonal_lines(v0, v1) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v1, v2) = 0 & convergent_lines(v1, v2) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v0, v2) = 0) | ( ~ (v4 = 0) & convergent_lines(v0, v1) = v4))) % 8.36/2.64 | (61) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (distinct_lines(v0, v1) = v2) | ? [v3] : ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3)) % 8.36/2.64 | (62) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v0, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v1, v2) = 0) % 8.36/2.64 | (63) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v3) = v5) | ~ (apart_point_and_line(v1, v2) = v4) | ~ (distinct_points(v0, v1) = 0) | ? [v6] : ((v6 = 0 & apart_point_and_line(v0, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v2) = 0) | ( ~ (v6 = 0) & distinct_lines(v2, v3) = v6))) % 8.36/2.64 | (64) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (unorthogonal_lines(v1, v2) = v4) | ~ (convergent_lines(v0, v2) = v3) | ? [v5] : ((v5 = 0 & v4 = 0 & convergent_lines(v1, v2) = 0) | (v5 = 0 & v3 = 0 & unorthogonal_lines(v0, v2) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v0, v1) = v5) | ( ~ (v5 = 0) & convergent_lines(v0, v1) = v5))) % 8.36/2.64 | (65) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v1, v2) = 0 & convergent_lines(v1, v2) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v0, v2) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v0, v1) = v4))) % 8.36/2.64 | (66) unorthogonal_lines(all_0_1_1, all_0_2_2) = all_0_0_0 % 8.36/2.64 | (67) ? [v0] : ? [v1] : ? [v2] : line_connecting(v1, v0) = v2 % 8.36/2.64 | (68) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (distinct_lines(v1, v2) = v4) | ~ (distinct_lines(v0, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & distinct_lines(v0, v1) = v5)) % 8.36/2.64 | (69) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (distinct_lines(v2, v3) = 0) | ~ (distinct_points(v0, v1) = 0) | ? [v4] : ((v4 = 0 & apart_point_and_line(v1, v3) = 0) | (v4 = 0 & apart_point_and_line(v1, v2) = 0) | (v4 = 0 & apart_point_and_line(v0, v3) = 0) | (v4 = 0 & apart_point_and_line(v0, v2) = 0))) % 8.36/2.64 | (70) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (apart_point_and_line(v0, v2) = v3) | ~ (convergent_lines(v1, v2) = v4) | ? [v5] : ((v5 = 0 & apart_point_and_line(v0, v1) = 0) | ( ~ (v5 = 0) & distinct_lines(v1, v2) = v5))) % 8.36/2.64 | (71) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (convergent_lines(v1, v2) = v4) | ~ (convergent_lines(v0, v2) = v3) | ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v1, v2) = 0) | (v5 = 0 & v3 = 0 & unorthogonal_lines(v0, v2) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v0, v1) = v5) | ( ~ (v5 = 0) & convergent_lines(v0, v1) = v5))) % 8.36/2.64 | (72) ! [v0] : ! [v1] : ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) | ? [v3] : ( ~ (v3 = 0) & apart_point_and_line(v0, v2) = v3)) % 8.36/2.64 | (73) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (line_connecting(v3, v2) = v1) | ~ (line_connecting(v3, v2) = v0)) % 8.36/2.64 | (74) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (apart_point_and_line(v0, v2) = v4) | ~ (apart_point_and_line(v0, v1) = v3) | ? [v5] : ((v5 = 0 & convergent_lines(v1, v2) = 0) | ( ~ (v5 = 0) & distinct_lines(v1, v2) = v5))) % 8.36/2.64 | (75) ! [v0] : ! [v1] : ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) | ? [v3] : ( ~ (v3 = 0) & apart_point_and_line(v0, v2) = v3)) % 8.36/2.64 | (76) ? [v0] : ? [v1] : ? [v2] : intersection_point(v1, v0) = v2 % 8.36/2.64 | (77) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (unorthogonal_lines(v2, v3) = v5) | ~ (apart_point_and_line(v0, v2) = v4) | ~ (distinct_lines(v1, v2) = 0) | ? [v6] : ((v6 = 0 & unorthogonal_lines(v1, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v1) = 0))) % 8.36/2.64 | (78) ! [v0] : ~ (distinct_lines(v0, v0) = 0) % 8.36/2.64 | % 8.36/2.64 | Instantiating formula (58) with all_0_0_0, all_0_2_2, all_0_1_1 and discharging atoms unorthogonal_lines(all_0_1_1, all_0_2_2) = all_0_0_0, yields: % 8.36/2.64 | (79) all_0_0_0 = 0 | convergent_lines(all_0_1_1, all_0_2_2) = 0 % 8.36/2.64 | % 8.58/2.64 | Instantiating formula (13) with all_0_0_0, all_0_2_2, all_0_1_1, all_0_2_2 and discharging atoms unorthogonal_lines(all_0_1_1, all_0_2_2) = all_0_0_0, unorthogonal_lines(all_0_2_2, all_0_1_1) = 0, yields: % 8.58/2.64 | (80) ? [v0] : ? [v1] : ((v1 = 0 & v0 = 0 & unorthogonal_lines(all_0_2_2, all_0_2_2) = 0 & convergent_lines(all_0_2_2, all_0_2_2) = 0) | (v0 = 0 & all_0_0_0 = 0 & convergent_lines(all_0_1_1, all_0_2_2) = 0) | ( ~ (v0 = 0) & convergent_lines(all_0_2_2, all_0_1_1) = v0)) % 8.58/2.64 | % 8.58/2.64 | Instantiating (80) with all_26_0_30, all_26_1_31 yields: % 8.58/2.64 | (81) (all_26_0_30 = 0 & all_26_1_31 = 0 & unorthogonal_lines(all_0_2_2, all_0_2_2) = 0 & convergent_lines(all_0_2_2, all_0_2_2) = 0) | (all_26_1_31 = 0 & all_0_0_0 = 0 & convergent_lines(all_0_1_1, all_0_2_2) = 0) | ( ~ (all_26_1_31 = 0) & convergent_lines(all_0_2_2, all_0_1_1) = all_26_1_31) % 8.58/2.64 | % 8.58/2.64 +-Applying beta-rule and splitting (79), into two cases. % 8.58/2.64 |-Branch one: % 8.58/2.64 | (82) convergent_lines(all_0_1_1, all_0_2_2) = 0 % 8.58/2.64 | % 8.58/2.64 +-Applying beta-rule and splitting (81), into two cases. % 8.58/2.64 |-Branch one: % 8.58/2.64 | (83) (all_26_0_30 = 0 & all_26_1_31 = 0 & unorthogonal_lines(all_0_2_2, all_0_2_2) = 0 & convergent_lines(all_0_2_2, all_0_2_2) = 0) | (all_26_1_31 = 0 & all_0_0_0 = 0 & convergent_lines(all_0_1_1, all_0_2_2) = 0) % 8.58/2.64 | % 8.58/2.64 +-Applying beta-rule and splitting (83), into two cases. % 8.58/2.64 |-Branch one: % 8.58/2.64 | (84) all_26_0_30 = 0 & all_26_1_31 = 0 & unorthogonal_lines(all_0_2_2, all_0_2_2) = 0 & convergent_lines(all_0_2_2, all_0_2_2) = 0 % 8.58/2.64 | % 8.58/2.64 | Applying alpha-rule on (84) yields: % 8.58/2.64 | (85) all_26_0_30 = 0 % 8.58/2.64 | (86) all_26_1_31 = 0 % 8.58/2.64 | (87) unorthogonal_lines(all_0_2_2, all_0_2_2) = 0 % 8.58/2.64 | (88) convergent_lines(all_0_2_2, all_0_2_2) = 0 % 8.58/2.64 | % 8.58/2.64 | Instantiating formula (56) with all_0_2_2 and discharging atoms convergent_lines(all_0_2_2, all_0_2_2) = 0, yields: % 8.58/2.64 | (89) $false % 8.58/2.64 | % 8.58/2.65 |-The branch is then unsatisfiable % 8.58/2.65 |-Branch two: % 8.58/2.65 | (90) all_26_1_31 = 0 & all_0_0_0 = 0 & convergent_lines(all_0_1_1, all_0_2_2) = 0 % 8.58/2.65 | % 8.58/2.65 | Applying alpha-rule on (90) yields: % 8.58/2.65 | (86) all_26_1_31 = 0 % 8.58/2.65 | (92) all_0_0_0 = 0 % 8.58/2.65 | (82) convergent_lines(all_0_1_1, all_0_2_2) = 0 % 8.58/2.65 | % 8.58/2.65 | Equations (92) can reduce 16 to: % 8.58/2.65 | (94) $false % 8.58/2.65 | % 8.58/2.65 |-The branch is then unsatisfiable % 8.58/2.65 |-Branch two: % 8.58/2.65 | (95) ~ (all_26_1_31 = 0) & convergent_lines(all_0_2_2, all_0_1_1) = all_26_1_31 % 8.58/2.65 | % 8.58/2.65 | Applying alpha-rule on (95) yields: % 8.58/2.65 | (96) ~ (all_26_1_31 = 0) % 8.58/2.65 | (97) convergent_lines(all_0_2_2, all_0_1_1) = all_26_1_31 % 8.58/2.65 | % 8.58/2.65 | Instantiating formula (43) with all_26_1_31, all_0_1_1, all_0_2_2, all_0_1_1 and discharging atoms convergent_lines(all_0_1_1, all_0_2_2) = 0, convergent_lines(all_0_2_2, all_0_1_1) = all_26_1_31, yields: % 8.58/2.65 | (98) all_26_1_31 = 0 | convergent_lines(all_0_1_1, all_0_1_1) = 0 % 8.58/2.65 | % 8.58/2.65 +-Applying beta-rule and splitting (98), into two cases. % 8.58/2.65 |-Branch one: % 8.58/2.65 | (99) convergent_lines(all_0_1_1, all_0_1_1) = 0 % 8.58/2.65 | % 8.58/2.65 | Instantiating formula (56) with all_0_1_1 and discharging atoms convergent_lines(all_0_1_1, all_0_1_1) = 0, yields: % 8.58/2.65 | (89) $false % 8.58/2.65 | % 8.58/2.65 |-The branch is then unsatisfiable % 8.58/2.65 |-Branch two: % 8.58/2.65 | (101) ~ (convergent_lines(all_0_1_1, all_0_1_1) = 0) % 8.58/2.65 | (86) all_26_1_31 = 0 % 8.58/2.65 | % 8.58/2.65 | Equations (86) can reduce 96 to: % 8.58/2.65 | (94) $false % 8.58/2.65 | % 8.58/2.65 |-The branch is then unsatisfiable % 8.58/2.65 |-Branch two: % 8.58/2.65 | (104) ~ (convergent_lines(all_0_1_1, all_0_2_2) = 0) % 8.58/2.65 | (92) all_0_0_0 = 0 % 8.58/2.65 | % 8.58/2.65 | Equations (92) can reduce 16 to: % 8.58/2.65 | (94) $false % 8.58/2.65 | % 8.58/2.65 |-The branch is then unsatisfiable % 8.58/2.65 % SZS output end Proof for theBenchmark % 8.58/2.65 % 8.58/2.65 1976ms %------------------------------------------------------------------------------