%------------------------------------------------------------------------------ % File : ePrincess---1.0 % Problem : GEO214+1 : TPTP v8.1.0. Released v3.3.0. % Transfm : none % Format : tptp:raw % Command : ePrincess-casc -timeout=%d %s % Computer : n025.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 6.27s 2.14s % Output : Proof 9.77s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.12/0.12 % Problem : GEO214+1 : TPTP v8.1.0. Released v3.3.0. % 0.12/0.12 % Command : ePrincess-casc -timeout=%d %s % 0.13/0.33 % Computer : n025.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 % 0.13/0.34 % WCLimit : 600 % 0.13/0.34 % DateTime : Fri Jun 17 21:59:42 EDT 2022 % 0.13/0.34 % CPUTime : % 0.61/0.61 ____ _ % 0.61/0.61 ___ / __ \_____(_)___ ________ __________ % 0.61/0.61 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/ % 0.61/0.61 / __/ ____/ / / / / / / /__/ __(__ |__ ) % 0.61/0.61 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/ % 0.61/0.61 % 0.61/0.61 A Theorem Prover for First-Order Logic % 0.61/0.61 (ePrincess v.1.0) % 0.61/0.61 % 0.61/0.61 (c) Philipp Rümmer, 2009-2015 % 0.61/0.61 (c) Peter Backeman, 2014-2015 % 0.61/0.61 (contributions by Angelo Brillout, Peter Baumgartner) % 0.61/0.61 Free software under GNU Lesser General Public License (LGPL). % 0.61/0.61 Bug reports to peter@backeman.se % 0.61/0.61 % 0.61/0.61 For more information, visit http://user.uu.se/~petba168/breu/ % 0.61/0.61 % 0.61/0.61 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ... % 0.66/0.68 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 1.72/0.99 Prover 0: Preprocessing ... % 2.12/1.16 Prover 0: Warning: ignoring some quantifiers % 2.12/1.18 Prover 0: Constructing countermodel ... % 3.83/1.60 Prover 0: gave up % 3.83/1.60 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all % 3.83/1.63 Prover 1: Preprocessing ... % 4.49/1.74 Prover 1: Constructing countermodel ... % 4.49/1.79 Prover 1: gave up % 4.49/1.79 Prover 2: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 4.88/1.82 Prover 2: Preprocessing ... % 5.30/1.98 Prover 2: Warning: ignoring some quantifiers % 5.71/1.99 Prover 2: Constructing countermodel ... % 6.27/2.14 Prover 2: proved (344ms) % 6.27/2.14 % 6.27/2.14 No countermodel exists, formula is valid % 6.27/2.14 % SZS status Theorem for theBenchmark % 6.27/2.14 % 6.27/2.14 Generating proof ... Warning: ignoring some quantifiers % 9.31/2.82 found it (size 76) % 9.31/2.82 % 9.31/2.82 % SZS output start Proof for theBenchmark % 9.31/2.82 Assumed formulas after preprocessing and simplification: % 9.31/2.82 | (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 | ~ (convergent_lines(v3, v5) = v7) | ~ (distinct_lines(v4, 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] : ( ~ (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 | ~ (convergent_lines(v3, v5) = v6) | ~ (convergent_lines(v3, v4) = 0) | distinct_lines(v4, v5) = 0) & ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v6 = 0 | ~ (convergent_lines(v3, v4) = 0) | ~ (distinct_lines(v4, v5) = v6) | convergent_lines(v3, 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] : ( ~ (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] : ( ~ (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] : ! [v5] : ( ~ (intersection_point(v3, v4) = v5) | ? [v6] : (( ~ (v6 = 0) & apart_point_and_line(v5, v4) = v6) | ( ~ (v6 = 0) & convergent_lines(v3, v4) = v6))) & ! [v3] : ! [v4] : ! [v5] : ( ~ (intersection_point(v3, v4) = v5) | ? [v6] : (( ~ (v6 = 0) & apart_point_and_line(v5, v3) = v6) | ( ~ (v6 = 0) & convergent_lines(v3, v4) = v6))) & ! [v3] : ! [v4] : ! [v5] : ( ~ (line_connecting(v3, v4) = v5) | ? [v6] : (( ~ (v6 = 0) & apart_point_and_line(v4, v5) = v6) | ( ~ (v6 = 0) & distinct_points(v3, v4) = v6))) & ! [v3] : ! [v4] : ! [v5] : ( ~ (line_connecting(v3, v4) = v5) | ? [v6] : (( ~ (v6 = 0) & apart_point_and_line(v3, v5) = v6) | ( ~ (v6 = 0) & distinct_points(v3, v4) = v6))) & ! [v3] : ! [v4] : ( ~ (convergent_lines(v3, v4) = 0) | ? [v5] : ? [v6] : ( ~ (v6 = 0) & intersection_point(v3, v4) = v5 & apart_point_and_line(v5, v4) = v6)) & ! [v3] : ! [v4] : ( ~ (convergent_lines(v3, v4) = 0) | ? [v5] : ? [v6] : ( ~ (v6 = 0) & intersection_point(v3, v4) = v5 & apart_point_and_line(v5, v3) = v6)) & ! [v3] : ! [v4] : ( ~ (distinct_points(v3, v4) = 0) | ? [v5] : ? [v6] : ( ~ (v6 = 0) & line_connecting(v3, v4) = v5 & apart_point_and_line(v4, v5) = v6)) & ! [v3] : ! [v4] : ( ~ (distinct_points(v3, v4) = 0) | ? [v5] : ? [v6] : ( ~ (v6 = 0) & line_connecting(v3, v4) = v5 & apart_point_and_line(v3, v5) = v6)) & ! [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) % 9.31/2.88 | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2 yields: % 9.31/2.88 | (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 | ~ (convergent_lines(v0, v2) = v4) | ~ (distinct_lines(v1, 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] : ( ~ (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 | ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | distinct_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(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] : ( ~ (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] : ( ~ (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] : ! [v2] : ( ~ (intersection_point(v0, v1) = v2) | ? [v3] : (( ~ (v3 = 0) & apart_point_and_line(v2, v1) = v3) | ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3))) & ! [v0] : ! [v1] : ! [v2] : ( ~ (intersection_point(v0, v1) = v2) | ? [v3] : (( ~ (v3 = 0) & apart_point_and_line(v2, v0) = v3) | ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3))) & ! [v0] : ! [v1] : ! [v2] : ( ~ (line_connecting(v0, v1) = v2) | ? [v3] : (( ~ (v3 = 0) & apart_point_and_line(v1, v2) = v3) | ( ~ (v3 = 0) & distinct_points(v0, v1) = v3))) & ! [v0] : ! [v1] : ! [v2] : ( ~ (line_connecting(v0, v1) = v2) | ? [v3] : (( ~ (v3 = 0) & apart_point_and_line(v0, v2) = v3) | ( ~ (v3 = 0) & distinct_points(v0, v1) = v3))) & ! [v0] : ! [v1] : ( ~ (convergent_lines(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & intersection_point(v0, v1) = v2 & apart_point_and_line(v2, v1) = v3)) & ! [v0] : ! [v1] : ( ~ (convergent_lines(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & intersection_point(v0, v1) = v2 & apart_point_and_line(v2, v0) = v3)) & ! [v0] : ! [v1] : ( ~ (distinct_points(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & line_connecting(v0, v1) = v2 & apart_point_and_line(v1, v2) = v3)) & ! [v0] : ! [v1] : ( ~ (distinct_points(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & line_connecting(v0, v1) = v2 & apart_point_and_line(v0, v2) = v3)) & ! [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 % 9.77/2.90 | % 9.77/2.90 | Applying alpha-rule on (1) yields: % 9.77/2.90 | (2) ! [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))) % 9.77/2.91 | (3) ! [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) % 9.77/2.91 | (4) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v1, v2) = 0) % 9.77/2.91 | (5) ! [v0] : ~ (distinct_points(v0, v0) = 0) % 9.77/2.91 | (6) ! [v0] : ! [v1] : ! [v2] : ( ~ (line_connecting(v0, v1) = v2) | ? [v3] : (( ~ (v3 = 0) & apart_point_and_line(v0, v2) = v3) | ( ~ (v3 = 0) & distinct_points(v0, v1) = v3))) % 9.77/2.91 | (7) ! [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)) % 9.77/2.91 | (8) ! [v0] : ~ (distinct_lines(v0, v0) = 0) % 9.77/2.91 | (9) ! [v0] : ! [v1] : ! [v2] : ( ~ (intersection_point(v0, v1) = v2) | ? [v3] : (( ~ (v3 = 0) & apart_point_and_line(v2, v0) = v3) | ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3))) % 9.77/2.91 | (10) ! [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))) % 9.77/2.91 | (11) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (convergent_lines(v3, v2) = v1) | ~ (convergent_lines(v3, v2) = v0)) % 9.77/2.91 | (12) ? [v0] : ? [v1] : ? [v2] : unorthogonal_lines(v1, v0) = v2 % 9.77/2.91 | (13) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_points(v3, v2) = v1) | ~ (distinct_points(v3, v2) = v0)) % 9.77/2.91 | (14) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (parallel_through_point(v3, v2) = v1) | ~ (parallel_through_point(v3, v2) = v0)) % 9.77/2.91 | (15) ! [v0] : ! [v1] : ( ~ (convergent_lines(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & intersection_point(v0, v1) = v2 & apart_point_and_line(v2, v0) = v3)) % 9.77/2.91 | (16) ! [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))) % 9.77/2.91 | (17) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v0, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0) % 9.77/2.91 | (18) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0) % 9.77/2.91 | (19) ! [v0] : ! [v1] : ! [v2] : ( ~ (intersection_point(v0, v1) = v2) | ? [v3] : (( ~ (v3 = 0) & apart_point_and_line(v2, v1) = v3) | ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3))) % 9.77/2.91 | (20) ! [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))) % 9.77/2.91 | (21) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v1, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v0, v2) = 0) % 9.77/2.91 | (22) ! [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))) % 9.77/2.91 | (23) ! [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))) % 9.77/2.91 | (24) ? [v0] : ? [v1] : ? [v2] : parallel_through_point(v1, v0) = v2 % 9.77/2.91 | (25) ? [v0] : ? [v1] : ? [v2] : intersection_point(v1, v0) = v2 % 9.77/2.91 | (26) ? [v0] : ? [v1] : ? [v2] : apart_point_and_line(v1, v0) = v2 % 9.77/2.91 | (27) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (intersection_point(v3, v2) = v1) | ~ (intersection_point(v3, v2) = v0)) % 9.77/2.91 | (28) ! [v0] : ! [v1] : ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) | ? [v3] : ( ~ (v3 = 0) & convergent_lines(v2, v1) = v3)) % 9.77/2.91 | (29) ! [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))) % 9.77/2.91 | (30) ! [v0] : ~ (convergent_lines(v0, v0) = 0) % 9.77/2.91 | (31) ! [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))) % 9.77/2.91 | (32) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v0, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v1, v2) = 0) % 9.77/2.91 | (33) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v1, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v0, v2) = 0) % 9.77/2.91 | (34) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (orthogonal_through_point(v3, v2) = v1) | ~ (orthogonal_through_point(v3, v2) = v0)) % 9.77/2.91 | (35) ? [v0] : ? [v1] : ? [v2] : distinct_lines(v1, v0) = v2 % 9.77/2.91 | (36) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (convergent_lines(v0, v1) = v2) | unorthogonal_lines(v0, v1) = 0) % 9.77/2.91 | (37) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (unorthogonal_lines(v0, v1) = v2) | convergent_lines(v0, v1) = 0) % 9.77/2.91 | (38) ! [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)) % 9.77/2.91 | (39) ! [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) % 9.77/2.91 | (40) ! [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))) % 9.77/2.92 | (41) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (apart_point_and_line(v3, v2) = v1) | ~ (apart_point_and_line(v3, v2) = v0)) % 9.77/2.92 | (42) ! [v0] : ! [v1] : ( ~ (distinct_points(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & line_connecting(v0, v1) = v2 & apart_point_and_line(v0, v2) = v3)) % 9.77/2.92 | (43) ! [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))) % 9.77/2.92 | (44) ! [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)) % 9.77/2.92 | (45) ! [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))) % 9.77/2.92 | (46) ! [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))) % 9.77/2.92 | (47) ! [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))) % 9.77/2.92 | (48) ! [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))) % 9.77/2.92 | (49) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v1) = 0) | ~ (distinct_lines(v1, v2) = v3) | convergent_lines(v0, v2) = 0) % 9.77/2.92 | (50) ! [v0] : ! [v1] : ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) | ? [v3] : ( ~ (v3 = 0) & apart_point_and_line(v0, v2) = v3)) % 9.77/2.92 | (51) ! [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))) % 9.77/2.92 | (52) ! [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))) % 9.77/2.92 | (53) ! [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))) % 9.77/2.92 | (54) ! [v0] : ! [v1] : ( ~ (distinct_points(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & line_connecting(v0, v1) = v2 & apart_point_and_line(v1, v2) = v3)) % 9.77/2.92 | (55) unorthogonal_lines(all_0_1_1, all_0_2_2) = all_0_0_0 % 9.77/2.92 | (56) ! [v0] : ! [v1] : ( ~ (convergent_lines(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & intersection_point(v0, v1) = v2 & apart_point_and_line(v2, v1) = v3)) % 9.77/2.92 | (57) ! [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))) % 9.77/2.92 | (58) ! [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))) % 9.77/2.92 | (59) ! [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) % 9.77/2.92 | (60) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (line_connecting(v3, v2) = v1) | ~ (line_connecting(v3, v2) = v0)) % 9.77/2.92 | (61) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_lines(v3, v2) = v1) | ~ (distinct_lines(v3, v2) = v0)) % 9.77/2.92 | (62) ! [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))) % 9.77/2.92 | (63) ! [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))) % 9.77/2.92 | (64) ! [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))) % 9.77/2.92 | (65) ! [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))) % 9.77/2.92 | (66) ? [v0] : ? [v1] : ? [v2] : distinct_points(v1, v0) = v2 % 9.77/2.92 | (67) ! [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) % 9.77/2.92 | (68) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (convergent_lines(v0, v2) = v4) | ~ (distinct_lines(v1, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & convergent_lines(v0, v1) = v5)) % 9.77/2.92 | (69) ! [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))) % 9.77/2.92 | (70) ! [v0] : ! [v1] : ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) | ? [v3] : ( ~ (v3 = 0) & unorthogonal_lines(v2, v1) = v3)) % 9.77/2.92 | (71) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (unorthogonal_lines(v3, v2) = v1) | ~ (unorthogonal_lines(v3, v2) = v0)) % 9.77/2.92 | (72) ? [v0] : ? [v1] : ? [v2] : line_connecting(v1, v0) = v2 % 9.77/2.92 | (73) ! [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)) % 9.77/2.93 | (74) unorthogonal_lines(all_0_2_2, all_0_1_1) = 0 % 9.77/2.93 | (75) ! [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))) % 9.77/2.93 | (76) ! [v0] : ! [v1] : ! [v2] : ( ~ (line_connecting(v0, v1) = v2) | ? [v3] : (( ~ (v3 = 0) & apart_point_and_line(v1, v2) = v3) | ( ~ (v3 = 0) & distinct_points(v0, v1) = v3))) % 9.77/2.93 | (77) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v1, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v0, v2) = 0) % 9.77/2.93 | (78) ? [v0] : ? [v1] : ? [v2] : orthogonal_through_point(v1, v0) = v2 % 9.77/2.93 | (79) ! [v0] : ! [v1] : ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) | ? [v3] : ( ~ (v3 = 0) & apart_point_and_line(v0, v2) = v3)) % 9.77/2.93 | (80) ? [v0] : ? [v1] : ? [v2] : convergent_lines(v1, v0) = v2 % 9.77/2.93 | (81) ! [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))) % 9.77/2.93 | (82) ! [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))) % 9.77/2.93 | (83) ! [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))) % 9.77/2.93 | (84) ~ (all_0_0_0 = 0) % 9.77/2.93 | (85) ! [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)) % 9.77/2.93 | % 9.77/2.93 | Instantiating formula (37) 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: % 9.77/2.93 | (86) all_0_0_0 = 0 | convergent_lines(all_0_1_1, all_0_2_2) = 0 % 9.77/2.93 | % 9.77/2.93 | Instantiating formula (48) 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: % 9.77/2.93 | (87) ? [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)) % 9.77/2.93 | % 9.77/2.93 | Instantiating (87) with all_26_0_30, all_26_1_31 yields: % 9.77/2.93 | (88) (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) % 9.77/2.93 | % 9.77/2.93 +-Applying beta-rule and splitting (86), into two cases. % 9.77/2.93 |-Branch one: % 9.77/2.93 | (89) convergent_lines(all_0_1_1, all_0_2_2) = 0 % 9.77/2.93 | % 9.77/2.93 | Instantiating formula (56) with all_0_2_2, all_0_1_1 and discharging atoms convergent_lines(all_0_1_1, all_0_2_2) = 0, yields: % 9.77/2.93 | (90) ? [v0] : ? [v1] : ( ~ (v1 = 0) & intersection_point(all_0_1_1, all_0_2_2) = v0 & apart_point_and_line(v0, all_0_2_2) = v1) % 9.77/2.93 | % 9.77/2.93 | Instantiating formula (15) with all_0_2_2, all_0_1_1 and discharging atoms convergent_lines(all_0_1_1, all_0_2_2) = 0, yields: % 9.77/2.93 | (91) ? [v0] : ? [v1] : ( ~ (v1 = 0) & intersection_point(all_0_1_1, all_0_2_2) = v0 & apart_point_and_line(v0, all_0_1_1) = v1) % 9.77/2.93 | % 9.77/2.93 | Instantiating (91) with all_37_0_36, all_37_1_37 yields: % 9.77/2.93 | (92) ~ (all_37_0_36 = 0) & intersection_point(all_0_1_1, all_0_2_2) = all_37_1_37 & apart_point_and_line(all_37_1_37, all_0_1_1) = all_37_0_36 % 9.77/2.93 | % 9.77/2.93 | Applying alpha-rule on (92) yields: % 9.77/2.93 | (93) ~ (all_37_0_36 = 0) % 9.77/2.93 | (94) intersection_point(all_0_1_1, all_0_2_2) = all_37_1_37 % 9.77/2.93 | (95) apart_point_and_line(all_37_1_37, all_0_1_1) = all_37_0_36 % 9.77/2.93 | % 9.77/2.93 | Instantiating (90) with all_39_0_38, all_39_1_39 yields: % 9.77/2.93 | (96) ~ (all_39_0_38 = 0) & intersection_point(all_0_1_1, all_0_2_2) = all_39_1_39 & apart_point_and_line(all_39_1_39, all_0_2_2) = all_39_0_38 % 9.77/2.93 | % 9.77/2.93 | Applying alpha-rule on (96) yields: % 9.77/2.93 | (97) ~ (all_39_0_38 = 0) % 9.77/2.93 | (98) intersection_point(all_0_1_1, all_0_2_2) = all_39_1_39 % 9.77/2.93 | (99) apart_point_and_line(all_39_1_39, all_0_2_2) = all_39_0_38 % 9.77/2.93 | % 9.77/2.93 | Instantiating formula (27) with all_0_1_1, all_0_2_2, all_37_1_37, all_39_1_39 and discharging atoms intersection_point(all_0_1_1, all_0_2_2) = all_39_1_39, intersection_point(all_0_1_1, all_0_2_2) = all_37_1_37, yields: % 9.77/2.93 | (100) all_39_1_39 = all_37_1_37 % 9.77/2.93 | % 9.77/2.93 | From (100) and (99) follows: % 9.77/2.93 | (101) apart_point_and_line(all_37_1_37, all_0_2_2) = all_39_0_38 % 9.77/2.93 | % 9.77/2.93 | Instantiating formula (69) with all_0_0_0, all_37_0_36, all_0_2_2, all_0_1_1, all_0_1_1, all_37_1_37 and discharging atoms unorthogonal_lines(all_0_1_1, all_0_2_2) = all_0_0_0, apart_point_and_line(all_37_1_37, all_0_1_1) = all_37_0_36, yields: % 9.77/2.93 | (102) all_37_0_36 = 0 | all_0_0_0 = 0 | ? [v0] : ((v0 = 0 & unorthogonal_lines(all_0_1_1, all_0_2_2) = 0) | (v0 = 0 & apart_point_and_line(all_37_1_37, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0)) % 9.77/2.93 | % 9.77/2.93 | Instantiating formula (65) with all_37_0_36, all_37_0_36, all_0_1_1, all_0_1_1, all_37_1_37 and discharging atoms apart_point_and_line(all_37_1_37, all_0_1_1) = all_37_0_36, yields: % 9.77/2.93 | (103) all_37_0_36 = 0 | ? [v0] : ((v0 = 0 & convergent_lines(all_0_1_1, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0)) % 9.77/2.93 | % 9.77/2.93 | Instantiating formula (63) with all_37_0_36, all_37_0_36, all_0_1_1, all_0_1_1, all_37_1_37, all_37_1_37 and discharging atoms apart_point_and_line(all_37_1_37, all_0_1_1) = all_37_0_36, yields: % 9.77/2.93 | (104) all_37_0_36 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_37_1_37, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_37_1_37, all_37_1_37) = v0)) % 9.77/2.93 | % 9.77/2.93 | Instantiating formula (52) with all_0_0_0, all_39_0_38, all_0_2_2, all_0_2_2, all_0_1_1, all_37_1_37 and discharging atoms unorthogonal_lines(all_0_1_1, all_0_2_2) = all_0_0_0, apart_point_and_line(all_37_1_37, all_0_2_2) = all_39_0_38, yields: % 9.77/2.93 | (105) all_39_0_38 = 0 | all_0_0_0 = 0 | ? [v0] : ((v0 = 0 & unorthogonal_lines(all_0_2_2, all_0_2_2) = 0) | (v0 = 0 & apart_point_and_line(all_37_1_37, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_2_2) = v0)) % 9.77/2.93 | % 9.77/2.93 | Instantiating formula (65) with all_39_0_38, all_37_0_36, all_0_2_2, all_0_1_1, all_37_1_37 and discharging atoms apart_point_and_line(all_37_1_37, all_0_1_1) = all_37_0_36, apart_point_and_line(all_37_1_37, all_0_2_2) = all_39_0_38, yields: % 9.77/2.93 | (106) all_39_0_38 = 0 | all_37_0_36 = 0 | ? [v0] : ((v0 = 0 & convergent_lines(all_0_1_1, all_0_2_2) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_2_2) = v0)) % 9.77/2.93 | % 9.77/2.93 | Instantiating formula (65) with all_37_0_36, all_39_0_38, all_0_1_1, all_0_2_2, all_37_1_37 and discharging atoms apart_point_and_line(all_37_1_37, all_0_1_1) = all_37_0_36, apart_point_and_line(all_37_1_37, all_0_2_2) = all_39_0_38, yields: % 9.77/2.94 | (107) all_39_0_38 = 0 | all_37_0_36 = 0 | ? [v0] : ((v0 = 0 & convergent_lines(all_0_2_2, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_2_2, all_0_1_1) = v0)) % 9.77/2.94 | % 9.77/2.94 | Instantiating formula (65) with all_39_0_38, all_39_0_38, all_0_2_2, all_0_2_2, all_37_1_37 and discharging atoms apart_point_and_line(all_37_1_37, all_0_2_2) = all_39_0_38, yields: % 9.77/2.94 | (108) all_39_0_38 = 0 | ? [v0] : ((v0 = 0 & convergent_lines(all_0_2_2, all_0_2_2) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_2_2, all_0_2_2) = v0)) % 9.77/2.94 | % 9.77/2.94 | Instantiating formula (63) with all_39_0_38, all_39_0_38, all_0_2_2, all_0_2_2, all_37_1_37, all_37_1_37 and discharging atoms apart_point_and_line(all_37_1_37, all_0_2_2) = all_39_0_38, yields: % 9.77/2.94 | (109) all_39_0_38 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_37_1_37, all_0_2_2) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_2_2, all_0_2_2) = v0) | ( ~ (v0 = 0) & distinct_points(all_37_1_37, all_37_1_37) = v0)) % 9.77/2.94 | % 9.77/2.94 +-Applying beta-rule and splitting (88), into two cases. % 9.77/2.94 |-Branch one: % 9.77/2.94 | (110) (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) % 9.77/2.94 | % 9.77/2.94 +-Applying beta-rule and splitting (110), into two cases. % 9.77/2.94 |-Branch one: % 9.77/2.94 | (111) 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 % 9.77/2.94 | % 9.77/2.94 | Applying alpha-rule on (111) yields: % 9.77/2.94 | (112) all_26_0_30 = 0 % 9.77/2.94 | (113) all_26_1_31 = 0 % 9.77/2.94 | (114) unorthogonal_lines(all_0_2_2, all_0_2_2) = 0 % 9.77/2.94 | (115) convergent_lines(all_0_2_2, all_0_2_2) = 0 % 9.77/2.94 | % 9.77/2.94 | Instantiating formula (30) with all_0_2_2 and discharging atoms convergent_lines(all_0_2_2, all_0_2_2) = 0, yields: % 9.77/2.94 | (116) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (117) all_26_1_31 = 0 & all_0_0_0 = 0 & convergent_lines(all_0_1_1, all_0_2_2) = 0 % 9.77/2.94 | % 9.77/2.94 | Applying alpha-rule on (117) yields: % 9.77/2.94 | (113) all_26_1_31 = 0 % 9.77/2.94 | (119) all_0_0_0 = 0 % 9.77/2.94 | (89) convergent_lines(all_0_1_1, all_0_2_2) = 0 % 9.77/2.94 | % 9.77/2.94 | Equations (119) can reduce 84 to: % 9.77/2.94 | (121) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (122) ~ (all_26_1_31 = 0) & convergent_lines(all_0_2_2, all_0_1_1) = all_26_1_31 % 9.77/2.94 | % 9.77/2.94 | Applying alpha-rule on (122) yields: % 9.77/2.94 | (123) ~ (all_26_1_31 = 0) % 9.77/2.94 | (124) convergent_lines(all_0_2_2, all_0_1_1) = all_26_1_31 % 9.77/2.94 | % 9.77/2.94 +-Applying beta-rule and splitting (104), into two cases. % 9.77/2.94 |-Branch one: % 9.77/2.94 | (125) all_37_0_36 = 0 % 9.77/2.94 | % 9.77/2.94 | Equations (125) can reduce 93 to: % 9.77/2.94 | (121) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (93) ~ (all_37_0_36 = 0) % 9.77/2.94 | (128) ? [v0] : ((v0 = 0 & apart_point_and_line(all_37_1_37, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_37_1_37, all_37_1_37) = v0)) % 9.77/2.94 | % 9.77/2.94 +-Applying beta-rule and splitting (109), into two cases. % 9.77/2.94 |-Branch one: % 9.77/2.94 | (129) all_39_0_38 = 0 % 9.77/2.94 | % 9.77/2.94 | Equations (129) can reduce 97 to: % 9.77/2.94 | (121) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (97) ~ (all_39_0_38 = 0) % 9.77/2.94 | (132) ? [v0] : ((v0 = 0 & apart_point_and_line(all_37_1_37, all_0_2_2) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_2_2, all_0_2_2) = v0) | ( ~ (v0 = 0) & distinct_points(all_37_1_37, all_37_1_37) = v0)) % 9.77/2.94 | % 9.77/2.94 +-Applying beta-rule and splitting (103), into two cases. % 9.77/2.94 |-Branch one: % 9.77/2.94 | (125) all_37_0_36 = 0 % 9.77/2.94 | % 9.77/2.94 | Equations (125) can reduce 93 to: % 9.77/2.94 | (121) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (93) ~ (all_37_0_36 = 0) % 9.77/2.94 | (136) ? [v0] : ((v0 = 0 & convergent_lines(all_0_1_1, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0)) % 9.77/2.94 | % 9.77/2.94 +-Applying beta-rule and splitting (108), into two cases. % 9.77/2.94 |-Branch one: % 9.77/2.94 | (129) all_39_0_38 = 0 % 9.77/2.94 | % 9.77/2.94 | Equations (129) can reduce 97 to: % 9.77/2.94 | (121) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (97) ~ (all_39_0_38 = 0) % 9.77/2.94 | (140) ? [v0] : ((v0 = 0 & convergent_lines(all_0_2_2, all_0_2_2) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_2_2, all_0_2_2) = v0)) % 9.77/2.94 | % 9.77/2.94 +-Applying beta-rule and splitting (105), into two cases. % 9.77/2.94 |-Branch one: % 9.77/2.94 | (129) all_39_0_38 = 0 % 9.77/2.94 | % 9.77/2.94 | Equations (129) can reduce 97 to: % 9.77/2.94 | (121) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (97) ~ (all_39_0_38 = 0) % 9.77/2.94 | (144) all_0_0_0 = 0 | ? [v0] : ((v0 = 0 & unorthogonal_lines(all_0_2_2, all_0_2_2) = 0) | (v0 = 0 & apart_point_and_line(all_37_1_37, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_2_2) = v0)) % 9.77/2.94 | % 9.77/2.94 +-Applying beta-rule and splitting (106), into two cases. % 9.77/2.94 |-Branch one: % 9.77/2.94 | (129) all_39_0_38 = 0 % 9.77/2.94 | % 9.77/2.94 | Equations (129) can reduce 97 to: % 9.77/2.94 | (121) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (97) ~ (all_39_0_38 = 0) % 9.77/2.94 | (148) all_37_0_36 = 0 | ? [v0] : ((v0 = 0 & convergent_lines(all_0_1_1, all_0_2_2) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_2_2) = v0)) % 9.77/2.94 | % 9.77/2.94 +-Applying beta-rule and splitting (102), into two cases. % 9.77/2.94 |-Branch one: % 9.77/2.94 | (125) all_37_0_36 = 0 % 9.77/2.94 | % 9.77/2.94 | Equations (125) can reduce 93 to: % 9.77/2.94 | (121) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (93) ~ (all_37_0_36 = 0) % 9.77/2.94 | (152) all_0_0_0 = 0 | ? [v0] : ((v0 = 0 & unorthogonal_lines(all_0_1_1, all_0_2_2) = 0) | (v0 = 0 & apart_point_and_line(all_37_1_37, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0)) % 9.77/2.94 | % 9.77/2.94 +-Applying beta-rule and splitting (107), into two cases. % 9.77/2.94 |-Branch one: % 9.77/2.94 | (129) all_39_0_38 = 0 % 9.77/2.94 | % 9.77/2.94 | Equations (129) can reduce 97 to: % 9.77/2.94 | (121) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (97) ~ (all_39_0_38 = 0) % 9.77/2.94 | (156) all_37_0_36 = 0 | ? [v0] : ((v0 = 0 & convergent_lines(all_0_2_2, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_2_2, all_0_1_1) = v0)) % 9.77/2.94 | % 9.77/2.94 +-Applying beta-rule and splitting (148), into two cases. % 9.77/2.94 |-Branch one: % 9.77/2.94 | (125) all_37_0_36 = 0 % 9.77/2.94 | % 9.77/2.94 | Equations (125) can reduce 93 to: % 9.77/2.94 | (121) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (93) ~ (all_37_0_36 = 0) % 9.77/2.94 | (160) ? [v0] : ((v0 = 0 & convergent_lines(all_0_1_1, all_0_2_2) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_2_2) = v0)) % 9.77/2.94 | % 9.77/2.94 +-Applying beta-rule and splitting (156), into two cases. % 9.77/2.94 |-Branch one: % 9.77/2.94 | (125) all_37_0_36 = 0 % 9.77/2.94 | % 9.77/2.94 | Equations (125) can reduce 93 to: % 9.77/2.94 | (121) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (93) ~ (all_37_0_36 = 0) % 9.77/2.94 | (164) ? [v0] : ((v0 = 0 & convergent_lines(all_0_2_2, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_2_2, all_0_1_1) = v0)) % 9.77/2.94 | % 9.77/2.94 | Instantiating (164) with all_112_0_75 yields: % 9.77/2.94 | (165) (all_112_0_75 = 0 & convergent_lines(all_0_2_2, all_0_1_1) = 0) | ( ~ (all_112_0_75 = 0) & distinct_lines(all_0_2_2, all_0_1_1) = all_112_0_75) % 9.77/2.94 | % 9.77/2.94 +-Applying beta-rule and splitting (165), into two cases. % 9.77/2.94 |-Branch one: % 9.77/2.94 | (166) all_112_0_75 = 0 & convergent_lines(all_0_2_2, all_0_1_1) = 0 % 9.77/2.94 | % 9.77/2.94 | Applying alpha-rule on (166) yields: % 9.77/2.94 | (167) all_112_0_75 = 0 % 9.77/2.94 | (168) convergent_lines(all_0_2_2, all_0_1_1) = 0 % 9.77/2.94 | % 9.77/2.94 | Instantiating formula (11) with all_0_2_2, all_0_1_1, 0, all_26_1_31 and discharging atoms convergent_lines(all_0_2_2, all_0_1_1) = all_26_1_31, convergent_lines(all_0_2_2, all_0_1_1) = 0, yields: % 9.77/2.94 | (113) all_26_1_31 = 0 % 9.77/2.94 | % 9.77/2.94 | Equations (113) can reduce 123 to: % 9.77/2.94 | (121) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (171) ~ (all_112_0_75 = 0) & distinct_lines(all_0_2_2, all_0_1_1) = all_112_0_75 % 9.77/2.94 | % 9.77/2.94 | Applying alpha-rule on (171) yields: % 9.77/2.94 | (172) ~ (all_112_0_75 = 0) % 9.77/2.94 | (173) distinct_lines(all_0_2_2, all_0_1_1) = all_112_0_75 % 9.77/2.94 | % 9.77/2.94 | Instantiating formula (49) with all_112_0_75, 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, distinct_lines(all_0_2_2, all_0_1_1) = all_112_0_75, yields: % 9.77/2.94 | (174) all_112_0_75 = 0 | convergent_lines(all_0_1_1, all_0_1_1) = 0 % 9.77/2.94 | % 9.77/2.94 +-Applying beta-rule and splitting (174), into two cases. % 9.77/2.94 |-Branch one: % 9.77/2.94 | (175) convergent_lines(all_0_1_1, all_0_1_1) = 0 % 9.77/2.94 | % 9.77/2.94 | Instantiating formula (30) with all_0_1_1 and discharging atoms convergent_lines(all_0_1_1, all_0_1_1) = 0, yields: % 9.77/2.94 | (116) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (177) ~ (convergent_lines(all_0_1_1, all_0_1_1) = 0) % 9.77/2.94 | (167) all_112_0_75 = 0 % 9.77/2.94 | % 9.77/2.94 | Equations (167) can reduce 172 to: % 9.77/2.94 | (121) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 |-Branch two: % 9.77/2.94 | (180) ~ (convergent_lines(all_0_1_1, all_0_2_2) = 0) % 9.77/2.94 | (119) all_0_0_0 = 0 % 9.77/2.94 | % 9.77/2.94 | Equations (119) can reduce 84 to: % 9.77/2.94 | (121) $false % 9.77/2.94 | % 9.77/2.94 |-The branch is then unsatisfiable % 9.77/2.94 % SZS output end Proof for theBenchmark % 9.77/2.95 % 9.77/2.95 2320ms %------------------------------------------------------------------------------