%------------------------------------------------------------------------------ % File : ePrincess---1.0 % Problem : GEO218+2 : TPTP v8.1.0. Released v3.3.0. % Transfm : none % Format : tptp:raw % Command : ePrincess-casc -timeout=%d %s % Computer : n003.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 03:48:51 EDT 2022 % Result : Theorem 5.31s 1.90s % Output : Proof 18.64s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.06/0.11 % Problem : GEO218+2 : TPTP v8.1.0. Released v3.3.0. % 0.06/0.12 % Command : ePrincess-casc -timeout=%d %s % 0.11/0.33 % Computer : n003.cluster.edu % 0.11/0.33 % Model : x86_64 x86_64 % 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.11/0.33 % Memory : 8042.1875MB % 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.11/0.33 % CPULimit : 300 % 0.11/0.33 % WCLimit : 600 % 0.11/0.33 % DateTime : Sat Jun 18 16:41:28 EDT 2022 % 0.11/0.33 % CPUTime : % 0.18/0.56 ____ _ % 0.18/0.56 ___ / __ \_____(_)___ ________ __________ % 0.18/0.56 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/ % 0.18/0.56 / __/ ____/ / / / / / / /__/ __(__ |__ ) % 0.18/0.56 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/ % 0.18/0.56 % 0.18/0.56 A Theorem Prover for First-Order Logic % 0.18/0.57 (ePrincess v.1.0) % 0.18/0.57 % 0.18/0.57 (c) Philipp Rümmer, 2009-2015 % 0.18/0.57 (c) Peter Backeman, 2014-2015 % 0.18/0.57 (contributions by Angelo Brillout, Peter Baumgartner) % 0.18/0.57 Free software under GNU Lesser General Public License (LGPL). % 0.18/0.57 Bug reports to peter@backeman.se % 0.18/0.57 % 0.18/0.57 For more information, visit http://user.uu.se/~petba168/breu/ % 0.18/0.57 % 0.18/0.57 Loading /export/starexec/sandbox2/benchmark/theBenchmark.p ... % 0.71/0.61 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 1.62/0.90 Prover 0: Preprocessing ... % 1.97/1.04 Prover 0: Warning: ignoring some quantifiers % 1.97/1.06 Prover 0: Constructing countermodel ... % 3.66/1.49 Prover 0: gave up % 3.66/1.49 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all % 3.76/1.52 Prover 1: Preprocessing ... % 4.25/1.60 Prover 1: Constructing countermodel ... % 4.25/1.66 Prover 1: gave up % 4.25/1.66 Prover 2: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 4.25/1.68 Prover 2: Preprocessing ... % 5.11/1.80 Prover 2: Warning: ignoring some quantifiers % 5.11/1.81 Prover 2: Constructing countermodel ... % 5.31/1.90 Prover 2: proved (240ms) % 5.31/1.90 % 5.31/1.90 No countermodel exists, formula is valid % 5.31/1.90 % SZS status Theorem for theBenchmark % 5.31/1.90 % 5.31/1.90 Generating proof ... Warning: ignoring some quantifiers % 17.91/4.86 found it (size 88) % 17.91/4.86 % 17.91/4.86 % SZS output start Proof for theBenchmark % 17.91/4.87 Assumed formulas after preprocessing and simplification: % 17.91/4.87 | (0) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ( ~ (v4 = 0) & ~ (v3 = 0) & unorthogonal_lines(v1, v2) = 0 & unorthogonal_lines(v0, v2) = v4 & convergent_lines(v0, v1) = v3 & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | v9 = 0 | ~ (apart_point_and_line(v6, v8) = v10) | ~ (apart_point_and_line(v6, v7) = v9) | ~ (distinct_points(v5, v6) = 0) | ? [v11] : ((v11 = 0 & apart_point_and_line(v5, v8) = 0) | (v11 = 0 & apart_point_and_line(v5, v7) = 0) | ( ~ (v11 = 0) & distinct_lines(v7, v8) = v11))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | v9 = 0 | ~ (apart_point_and_line(v6, v8) = v10) | ~ (apart_point_and_line(v5, v8) = v9) | ~ (distinct_lines(v7, v8) = 0) | ? [v11] : ((v11 = 0 & apart_point_and_line(v6, v7) = 0) | (v11 = 0 & apart_point_and_line(v5, v7) = 0) | ( ~ (v11 = 0) & distinct_points(v5, v6) = v11))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | v9 = 0 | ~ (apart_point_and_line(v6, v8) = v10) | ~ (apart_point_and_line(v5, v7) = v9) | ? [v11] : ((v11 = 0 & apart_point_and_line(v6, v7) = 0) | (v11 = 0 & apart_point_and_line(v5, v8) = 0) | ( ~ (v11 = 0) & distinct_lines(v7, v8) = v11) | ( ~ (v11 = 0) & distinct_points(v5, v6) = v11))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | v9 = 0 | ~ (apart_point_and_line(v6, v7) = v10) | ~ (apart_point_and_line(v5, v8) = v9) | ? [v11] : ((v11 = 0 & apart_point_and_line(v6, v8) = 0) | (v11 = 0 & apart_point_and_line(v5, v7) = 0) | ( ~ (v11 = 0) & distinct_lines(v7, v8) = v11) | ( ~ (v11 = 0) & distinct_points(v5, v6) = v11))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | v9 = 0 | ~ (apart_point_and_line(v6, v7) = v10) | ~ (apart_point_and_line(v5, v7) = v9) | ~ (distinct_lines(v7, v8) = 0) | ? [v11] : ((v11 = 0 & apart_point_and_line(v6, v8) = 0) | (v11 = 0 & apart_point_and_line(v5, v8) = 0) | ( ~ (v11 = 0) & distinct_points(v5, v6) = v11))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | v9 = 0 | ~ (apart_point_and_line(v5, v8) = v10) | ~ (apart_point_and_line(v5, v7) = v9) | ~ (distinct_points(v5, v6) = 0) | ? [v11] : ((v11 = 0 & apart_point_and_line(v6, v8) = 0) | (v11 = 0 & apart_point_and_line(v6, v7) = 0) | ( ~ (v11 = 0) & distinct_lines(v7, v8) = v11))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | v8 = 0 | ~ (unorthogonal_lines(v5, v7) = v9) | ~ (unorthogonal_lines(v5, v6) = v8) | ? [v10] : ( ~ (v10 = 0) & convergent_lines(v6, v7) = v10)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | v8 = 0 | ~ (apart_point_and_line(v7, v6) = v9) | ~ (distinct_points(v5, v7) = v8) | ? [v10] : ( ~ (v10 = 0) & apart_point_and_line(v5, v6) = v10)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | v8 = 0 | ~ (apart_point_and_line(v5, v7) = v9) | ~ (distinct_lines(v6, v7) = v8) | ? [v10] : ( ~ (v10 = 0) & apart_point_and_line(v5, v6) = v10)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | v8 = 0 | ~ (convergent_lines(v6, v7) = v9) | ~ (convergent_lines(v5, v7) = v8) | ? [v10] : ( ~ (v10 = 0) & convergent_lines(v5, v6) = v10)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | v8 = 0 | ~ (distinct_lines(v6, v7) = v9) | ~ (distinct_lines(v5, v7) = v8) | ? [v10] : ( ~ (v10 = 0) & distinct_lines(v5, v6) = v10)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | v8 = 0 | ~ (distinct_points(v6, v7) = v9) | ~ (distinct_points(v5, v7) = v8) | ? [v10] : ( ~ (v10 = 0) & distinct_points(v5, v6) = v10)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (intersection_point(v5, v6) = v8) | ~ (distinct_points(v7, v8) = v9) | ? [v10] : ? [v11] : (( ~ (v11 = 0) & ~ (v10 = 0) & apart_point_and_line(v7, v6) = v11 & apart_point_and_line(v7, v5) = v10) | ( ~ (v10 = 0) & convergent_lines(v5, v6) = v10))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (unorthogonal_lines(v5, v7) = v9) | ~ (unorthogonal_lines(v5, v6) = v8) | ? [v10] : ((v10 = 0 & v9 = 0 & convergent_lines(v5, v7) = 0) | (v10 = 0 & v8 = 0 & convergent_lines(v5, v6) = 0) | ( ~ (v10 = 0) & unorthogonal_lines(v6, v7) = v10) | ( ~ (v10 = 0) & convergent_lines(v6, v7) = v10))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (unorthogonal_lines(v5, v7) = v9) | ~ (convergent_lines(v5, v6) = v8) | ? [v10] : ((v10 = 0 & v9 = 0 & convergent_lines(v5, v7) = 0) | (v10 = 0 & v8 = 0 & unorthogonal_lines(v5, v6) = 0) | ( ~ (v10 = 0) & unorthogonal_lines(v6, v7) = v10) | ( ~ (v10 = 0) & convergent_lines(v6, v7) = v10))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (unorthogonal_lines(v5, v6) = v8) | ~ (convergent_lines(v5, v7) = v9) | ? [v10] : ((v10 = 0 & v9 = 0 & unorthogonal_lines(v5, v7) = 0) | (v10 = 0 & v8 = 0 & convergent_lines(v5, v6) = 0) | ( ~ (v10 = 0) & unorthogonal_lines(v6, v7) = v10) | ( ~ (v10 = 0) & convergent_lines(v6, v7) = v10))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (convergent_lines(v5, v7) = v9) | ~ (convergent_lines(v5, v6) = v8) | ? [v10] : ((v10 = 0 & v9 = 0 & unorthogonal_lines(v5, v7) = 0) | (v10 = 0 & v8 = 0 & unorthogonal_lines(v5, v6) = 0) | ( ~ (v10 = 0) & unorthogonal_lines(v6, v7) = v10) | ( ~ (v10 = 0) & convergent_lines(v6, v7) = v10))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (unorthogonal_lines(v5, v7) = v8) | ~ (convergent_lines(v6, v7) = 0) | unorthogonal_lines(v5, v6) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (unorthogonal_lines(v5, v6) = v8) | ~ (convergent_lines(v6, v7) = 0) | unorthogonal_lines(v5, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (apart_point_and_line(v7, v6) = v8) | ~ (apart_point_and_line(v5, v6) = 0) | distinct_points(v5, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (apart_point_and_line(v5, v7) = v8) | ~ (apart_point_and_line(v5, v6) = 0) | distinct_lines(v6, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (apart_point_and_line(v5, v6) = 0) | ~ (distinct_lines(v6, v7) = v8) | apart_point_and_line(v5, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (apart_point_and_line(v5, v6) = 0) | ~ (distinct_points(v5, v7) = v8) | apart_point_and_line(v7, v6) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (convergent_lines(v6, v7) = v8) | ~ (convergent_lines(v5, v6) = 0) | convergent_lines(v5, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (convergent_lines(v5, v7) = v8) | ~ (convergent_lines(v5, v6) = 0) | convergent_lines(v6, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (distinct_lines(v6, v7) = v8) | ~ (distinct_lines(v5, v6) = 0) | distinct_lines(v5, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (distinct_lines(v5, v7) = v8) | ~ (distinct_lines(v5, v6) = 0) | distinct_lines(v6, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (distinct_points(v6, v7) = v8) | ~ (distinct_points(v5, v6) = 0) | distinct_points(v5, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | ~ (distinct_points(v5, v7) = v8) | ~ (distinct_points(v5, v6) = 0) | distinct_points(v6, v7) = 0) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (unorthogonal_lines(v8, v7) = v6) | ~ (unorthogonal_lines(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (intersection_point(v8, v7) = v6) | ~ (intersection_point(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (line_connecting(v8, v7) = v6) | ~ (line_connecting(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (apart_point_and_line(v8, v7) = v6) | ~ (apart_point_and_line(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (convergent_lines(v8, v7) = v6) | ~ (convergent_lines(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (distinct_lines(v8, v7) = v6) | ~ (distinct_lines(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v6 = v5 | ~ (distinct_points(v8, v7) = v6) | ~ (distinct_points(v8, v7) = v5)) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (unorthogonal_lines(v6, v7) = 0) | ~ (unorthogonal_lines(v5, v7) = v8) | ? [v9] : ? [v10] : ((v10 = 0 & v9 = 0 & unorthogonal_lines(v5, v6) = 0 & convergent_lines(v5, v6) = 0) | (v9 = 0 & v8 = 0 & convergent_lines(v5, v7) = 0) | ( ~ (v9 = 0) & convergent_lines(v6, v7) = v9))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (unorthogonal_lines(v6, v7) = 0) | ~ (unorthogonal_lines(v5, v6) = v8) | ? [v9] : ? [v10] : ((v10 = 0 & v9 = 0 & unorthogonal_lines(v5, v7) = 0 & convergent_lines(v5, v7) = 0) | (v9 = 0 & v8 = 0 & convergent_lines(v5, v6) = 0) | ( ~ (v9 = 0) & convergent_lines(v6, v7) = v9))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (unorthogonal_lines(v6, v7) = 0) | ~ (convergent_lines(v5, v7) = v8) | ? [v9] : ? [v10] : ((v10 = 0 & v9 = 0 & unorthogonal_lines(v5, v6) = 0 & convergent_lines(v5, v6) = 0) | (v9 = 0 & v8 = 0 & unorthogonal_lines(v5, v7) = 0) | ( ~ (v9 = 0) & convergent_lines(v6, v7) = v9))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (unorthogonal_lines(v6, v7) = 0) | ~ (convergent_lines(v5, v6) = v8) | ? [v9] : ? [v10] : ((v10 = 0 & v9 = 0 & unorthogonal_lines(v5, v7) = 0 & convergent_lines(v5, v7) = 0) | (v9 = 0 & v8 = 0 & unorthogonal_lines(v5, v6) = 0) | ( ~ (v9 = 0) & convergent_lines(v6, v7) = v9))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (unorthogonal_lines(v5, v7) = v8) | ~ (convergent_lines(v6, v7) = 0) | ? [v9] : ? [v10] : ((v10 = 0 & v9 = 0 & unorthogonal_lines(v5, v6) = 0 & convergent_lines(v5, v6) = 0) | (v9 = 0 & v8 = 0 & convergent_lines(v5, v7) = 0) | ( ~ (v9 = 0) & unorthogonal_lines(v6, v7) = v9))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (unorthogonal_lines(v5, v6) = v8) | ~ (convergent_lines(v6, v7) = 0) | ? [v9] : ? [v10] : ((v10 = 0 & v9 = 0 & unorthogonal_lines(v5, v7) = 0 & convergent_lines(v5, v7) = 0) | (v9 = 0 & v8 = 0 & convergent_lines(v5, v6) = 0) | ( ~ (v9 = 0) & unorthogonal_lines(v6, v7) = v9))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (line_connecting(v5, v6) = v8) | ~ (apart_point_and_line(v7, v8) = 0) | ? [v9] : ? [v10] : ((v10 = 0 & v9 = 0 & distinct_points(v7, v6) = 0 & distinct_points(v7, v5) = 0) | ( ~ (v9 = 0) & distinct_points(v5, v6) = v9))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (convergent_lines(v6, v7) = 0) | ~ (convergent_lines(v5, v7) = v8) | ? [v9] : ? [v10] : ((v10 = 0 & v9 = 0 & unorthogonal_lines(v5, v6) = 0 & convergent_lines(v5, v6) = 0) | (v9 = 0 & v8 = 0 & unorthogonal_lines(v5, v7) = 0) | ( ~ (v9 = 0) & unorthogonal_lines(v6, v7) = v9))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (convergent_lines(v6, v7) = 0) | ~ (convergent_lines(v5, v6) = v8) | ? [v9] : ? [v10] : ((v10 = 0 & v9 = 0 & unorthogonal_lines(v5, v7) = 0 & convergent_lines(v5, v7) = 0) | (v9 = 0 & v8 = 0 & unorthogonal_lines(v5, v6) = 0) | ( ~ (v9 = 0) & unorthogonal_lines(v6, v7) = v9))) & ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (distinct_lines(v7, v8) = 0) | ~ (distinct_points(v5, v6) = 0) | ? [v9] : ((v9 = 0 & apart_point_and_line(v6, v8) = 0) | (v9 = 0 & apart_point_and_line(v6, v7) = 0) | (v9 = 0 & apart_point_and_line(v5, v8) = 0) | (v9 = 0 & apart_point_and_line(v5, v7) = 0))) & ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (unorthogonal_lines(v5, v6) = v7) | convergent_lines(v5, v6) = 0) & ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (convergent_lines(v5, v6) = v7) | unorthogonal_lines(v5, v6) = 0) & ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (distinct_lines(v5, v6) = v7) | ? [v8] : ( ~ (v8 = 0) & convergent_lines(v5, v6) = v8)) & ! [v5] : ! [v6] : ( ~ (convergent_lines(v5, v6) = 0) | distinct_lines(v5, v6) = 0) & ! [v5] : ~ (convergent_lines(v5, v5) = 0) & ! [v5] : ~ (distinct_lines(v5, v5) = 0) & ! [v5] : ~ (distinct_points(v5, v5) = 0) & ? [v5] : ? [v6] : ? [v7] : unorthogonal_lines(v6, v5) = v7 & ? [v5] : ? [v6] : ? [v7] : intersection_point(v6, v5) = v7 & ? [v5] : ? [v6] : ? [v7] : line_connecting(v6, v5) = v7 & ? [v5] : ? [v6] : ? [v7] : apart_point_and_line(v6, v5) = v7 & ? [v5] : ? [v6] : ? [v7] : convergent_lines(v6, v5) = v7 & ? [v5] : ? [v6] : ? [v7] : distinct_lines(v6, v5) = v7 & ? [v5] : ? [v6] : ? [v7] : distinct_points(v6, v5) = v7) % 18.26/4.93 | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3, all_0_4_4 yields: % 18.26/4.93 | (1) ~ (all_0_0_0 = 0) & ~ (all_0_1_1 = 0) & unorthogonal_lines(all_0_3_3, all_0_2_2) = 0 & unorthogonal_lines(all_0_4_4, all_0_2_2) = all_0_0_0 & convergent_lines(all_0_4_4, all_0_3_3) = all_0_1_1 & ! [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 | ~ (unorthogonal_lines(v0, v2) = v4) | ~ (unorthogonal_lines(v0, v1) = v3) | ? [v5] : ( ~ (v5 = 0) & convergent_lines(v1, v2) = v5)) & ! [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) | ~ (distinct_lines(v1, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & apart_point_and_line(v0, v1) = 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(v0, v2) = v4) | ~ (unorthogonal_lines(v0, v1) = v3) | ? [v5] : ((v5 = 0 & v4 = 0 & convergent_lines(v0, v2) = 0) | (v5 = 0 & v3 = 0 & convergent_lines(v0, v1) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v1, v2) = v5) | ( ~ (v5 = 0) & convergent_lines(v1, v2) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (unorthogonal_lines(v0, v2) = v4) | ~ (convergent_lines(v0, v1) = v3) | ? [v5] : ((v5 = 0 & v4 = 0 & convergent_lines(v0, v2) = 0) | (v5 = 0 & v3 = 0 & unorthogonal_lines(v0, v1) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v1, v2) = v5) | ( ~ (v5 = 0) & convergent_lines(v1, v2) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (unorthogonal_lines(v0, v1) = v3) | ~ (convergent_lines(v0, v2) = v4) | ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0) | (v5 = 0 & v3 = 0 & convergent_lines(v0, v1) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v1, v2) = v5) | ( ~ (v5 = 0) & convergent_lines(v1, v2) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (convergent_lines(v0, v2) = v4) | ~ (convergent_lines(v0, v1) = v3) | ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0) | (v5 = 0 & v3 = 0 & unorthogonal_lines(v0, v1) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v1, v2) = v5) | ( ~ (v5 = 0) & convergent_lines(v1, v2) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (unorthogonal_lines(v0, v2) = v3) | ~ (convergent_lines(v1, v2) = 0) | unorthogonal_lines(v0, v1) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (unorthogonal_lines(v0, v1) = v3) | ~ (convergent_lines(v1, v2) = 0) | unorthogonal_lines(v0, v2) = 0) & ! [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, 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 | ~ (unorthogonal_lines(v3, v2) = v1) | ~ (unorthogonal_lines(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) = 0) | ~ (unorthogonal_lines(v0, v2) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v1) = 0 & convergent_lines(v0, v1) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v0, v2) = 0) | ( ~ (v4 = 0) & convergent_lines(v1, v2) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v1, v2) = 0) | ~ (unorthogonal_lines(v0, v1) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v0, v1) = 0) | ( ~ (v4 = 0) & convergent_lines(v1, v2) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v1, v2) = 0) | ~ (convergent_lines(v0, v2) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v1) = 0 & convergent_lines(v0, v1) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v0, v2) = 0) | ( ~ (v4 = 0) & convergent_lines(v1, v2) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v1, v2) = 0) | ~ (convergent_lines(v0, v1) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v0, v1) = 0) | ( ~ (v4 = 0) & convergent_lines(v1, v2) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v0, v2) = v3) | ~ (convergent_lines(v1, v2) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v1) = 0 & convergent_lines(v0, v1) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v0, v2) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v1, v2) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v0, v1) = v3) | ~ (convergent_lines(v1, v2) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v0, v1) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v1, v2) = 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) = 0) | ~ (convergent_lines(v0, v2) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v1) = 0 & convergent_lines(v0, v1) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v0, v2) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v1, v2) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (convergent_lines(v1, v2) = 0) | ~ (convergent_lines(v0, v1) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v0, v1) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v1, v2) = 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] : ( ~ (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] : unorthogonal_lines(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 % 18.26/4.95 | % 18.26/4.95 | Applying alpha-rule on (1) yields: % 18.26/4.95 | (2) ! [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) % 18.26/4.95 | (3) ? [v0] : ? [v1] : ? [v2] : line_connecting(v1, v0) = v2 % 18.26/4.95 | (4) ! [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) % 18.26/4.95 | (5) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v1, v2) = 0) | ~ (convergent_lines(v0, v2) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v1) = 0 & convergent_lines(v0, v1) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v0, v2) = 0) | ( ~ (v4 = 0) & convergent_lines(v1, v2) = v4))) % 18.26/4.95 | (6) ! [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))) % 18.26/4.95 | (7) ! [v0] : ~ (distinct_points(v0, v0) = 0) % 18.26/4.95 | (8) ! [v0] : ! [v1] : ( ~ (convergent_lines(v0, v1) = 0) | distinct_lines(v0, v1) = 0) % 18.26/4.95 | (9) ! [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)) % 18.26/4.95 | (10) ? [v0] : ? [v1] : ? [v2] : apart_point_and_line(v1, v0) = v2 % 18.26/4.95 | (11) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (unorthogonal_lines(v3, v2) = v1) | ~ (unorthogonal_lines(v3, v2) = v0)) % 18.26/4.95 | (12) ~ (all_0_0_0 = 0) % 18.26/4.95 | (13) ! [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))) % 18.26/4.95 | (14) convergent_lines(all_0_4_4, all_0_3_3) = all_0_1_1 % 18.26/4.95 | (15) ? [v0] : ? [v1] : ? [v2] : intersection_point(v1, v0) = v2 % 18.26/4.95 | (16) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (unorthogonal_lines(v0, v1) = v3) | ~ (convergent_lines(v0, v2) = v4) | ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0) | (v5 = 0 & v3 = 0 & convergent_lines(v0, v1) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v1, v2) = v5) | ( ~ (v5 = 0) & convergent_lines(v1, v2) = v5))) % 18.26/4.95 | (17) ? [v0] : ? [v1] : ? [v2] : distinct_points(v1, v0) = v2 % 18.26/4.95 | (18) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (unorthogonal_lines(v0, v2) = v4) | ~ (convergent_lines(v0, v1) = v3) | ? [v5] : ((v5 = 0 & v4 = 0 & convergent_lines(v0, v2) = 0) | (v5 = 0 & v3 = 0 & unorthogonal_lines(v0, v1) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v1, v2) = v5) | ( ~ (v5 = 0) & convergent_lines(v1, v2) = v5))) % 18.26/4.95 | (19) unorthogonal_lines(all_0_4_4, all_0_2_2) = all_0_0_0 % 18.26/4.95 | (20) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v0, v1) = v3) | ~ (convergent_lines(v1, v2) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v0, v1) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v1, v2) = v4))) % 18.26/4.95 | (21) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (unorthogonal_lines(v0, v2) = v4) | ~ (unorthogonal_lines(v0, v1) = v3) | ? [v5] : ((v5 = 0 & v4 = 0 & convergent_lines(v0, v2) = 0) | (v5 = 0 & v3 = 0 & convergent_lines(v0, v1) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v1, v2) = v5) | ( ~ (v5 = 0) & convergent_lines(v1, v2) = v5))) % 18.26/4.95 | (22) ? [v0] : ? [v1] : ? [v2] : convergent_lines(v1, v0) = v2 % 18.26/4.95 | (23) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (unorthogonal_lines(v0, v1) = v3) | ~ (convergent_lines(v1, v2) = 0) | unorthogonal_lines(v0, v2) = 0) % 18.26/4.95 | (24) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (intersection_point(v3, v2) = v1) | ~ (intersection_point(v3, v2) = v0)) % 18.26/4.95 | (25) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v0, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v1, v2) = 0) % 18.26/4.95 | (26) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v1, v2) = 0) | ~ (convergent_lines(v0, v1) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v0, v1) = 0) | ( ~ (v4 = 0) & convergent_lines(v1, v2) = v4))) % 18.26/4.95 | (27) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (convergent_lines(v0, v1) = v2) | unorthogonal_lines(v0, v1) = 0) % 18.26/4.95 | (28) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (unorthogonal_lines(v0, v1) = v2) | convergent_lines(v0, v1) = 0) % 18.26/4.96 | (29) ~ (all_0_1_1 = 0) % 18.26/4.96 | (30) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (apart_point_and_line(v3, v2) = v1) | ~ (apart_point_and_line(v3, v2) = v0)) % 18.26/4.96 | (31) ! [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)) % 18.26/4.96 | (32) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v1, v2) = 0) % 18.26/4.96 | (33) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (line_connecting(v3, v2) = v1) | ~ (line_connecting(v3, v2) = v0)) % 18.26/4.96 | (34) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (convergent_lines(v3, v2) = v1) | ~ (convergent_lines(v3, v2) = v0)) % 18.26/4.96 | (35) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (unorthogonal_lines(v0, v2) = v4) | ~ (unorthogonal_lines(v0, v1) = v3) | ? [v5] : ( ~ (v5 = 0) & convergent_lines(v1, v2) = v5)) % 18.26/4.96 | (36) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_points(v3, v2) = v1) | ~ (distinct_points(v3, v2) = v0)) % 18.26/4.96 | (37) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v0, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0) % 18.26/4.96 | (38) ! [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)) % 18.26/4.96 | (39) ! [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))) % 18.26/4.96 | (40) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v1, v2) = 0) | ~ (unorthogonal_lines(v0, v1) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v0, v1) = 0) | ( ~ (v4 = 0) & convergent_lines(v1, v2) = v4))) % 18.26/4.96 | (41) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (distinct_lines(v0, v1) = v2) | ? [v3] : ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3)) % 18.26/4.96 | (42) ! [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))) % 18.26/4.96 | (43) ? [v0] : ? [v1] : ? [v2] : distinct_lines(v1, v0) = v2 % 18.26/4.96 | (44) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v1, v2) = 0) | ~ (unorthogonal_lines(v0, v2) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v1) = 0 & convergent_lines(v0, v1) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v0, v2) = 0) | ( ~ (v4 = 0) & convergent_lines(v1, v2) = v4))) % 18.26/4.96 | (45) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (convergent_lines(v0, v2) = v4) | ~ (convergent_lines(v0, v1) = v3) | ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0) | (v5 = 0 & v3 = 0 & unorthogonal_lines(v0, v1) = 0) | ( ~ (v5 = 0) & unorthogonal_lines(v1, v2) = v5) | ( ~ (v5 = 0) & convergent_lines(v1, v2) = v5))) % 18.26/4.96 | (46) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (unorthogonal_lines(v0, v2) = v3) | ~ (convergent_lines(v1, v2) = 0) | unorthogonal_lines(v0, v1) = 0) % 18.26/4.96 | (47) ! [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))) % 18.26/4.96 | (48) unorthogonal_lines(all_0_3_3, all_0_2_2) = 0 % 18.26/4.96 | (49) ! [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) % 18.26/4.96 | (50) ! [v0] : ~ (distinct_lines(v0, v0) = 0) % 18.26/4.96 | (51) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (convergent_lines(v1, v2) = 0) | ~ (convergent_lines(v0, v1) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v2) = 0 & convergent_lines(v0, v2) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v0, v1) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v1, v2) = v4))) % 18.26/4.96 | (52) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v1, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v0, v2) = 0) % 18.26/4.96 | (53) ! [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))) % 18.26/4.96 | (54) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v1, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v0, v2) = 0) % 18.26/4.96 | (55) ! [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))) % 18.26/4.97 | (56) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_lines(v3, v2) = v1) | ~ (distinct_lines(v3, v2) = v0)) % 18.26/4.97 | (57) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v1, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v0, v2) = 0) % 18.26/4.97 | (58) ! [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)) % 18.26/4.97 | (59) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (convergent_lines(v1, v2) = 0) | ~ (convergent_lines(v0, v2) = v3) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v1) = 0 & convergent_lines(v0, v1) = 0) | (v4 = 0 & v3 = 0 & unorthogonal_lines(v0, v2) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v1, v2) = v4))) % 18.26/4.97 | (60) ? [v0] : ? [v1] : ? [v2] : unorthogonal_lines(v1, v0) = v2 % 18.26/4.97 | (61) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (unorthogonal_lines(v0, v2) = v3) | ~ (convergent_lines(v1, v2) = 0) | ? [v4] : ? [v5] : ((v5 = 0 & v4 = 0 & unorthogonal_lines(v0, v1) = 0 & convergent_lines(v0, v1) = 0) | (v4 = 0 & v3 = 0 & convergent_lines(v0, v2) = 0) | ( ~ (v4 = 0) & unorthogonal_lines(v1, v2) = v4))) % 18.26/4.97 | (62) ! [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) % 18.26/4.97 | (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))) % 18.26/4.97 | (64) ! [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)) % 18.26/4.97 | (65) ! [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))) % 18.26/4.97 | (66) ! [v0] : ~ (convergent_lines(v0, v0) = 0) % 18.26/4.97 | % 18.26/4.97 | Instantiating formula (44) with all_0_0_0, all_0_2_2, all_0_3_3, all_0_4_4 and discharging atoms unorthogonal_lines(all_0_3_3, all_0_2_2) = 0, unorthogonal_lines(all_0_4_4, all_0_2_2) = all_0_0_0, yields: % 18.26/4.97 | (67) ? [v0] : ? [v1] : ((v1 = 0 & v0 = 0 & unorthogonal_lines(all_0_4_4, all_0_3_3) = 0 & convergent_lines(all_0_4_4, all_0_3_3) = 0) | (v0 = 0 & all_0_0_0 = 0 & convergent_lines(all_0_4_4, all_0_2_2) = 0) | ( ~ (v0 = 0) & convergent_lines(all_0_3_3, all_0_2_2) = v0)) % 18.26/4.97 | % 18.26/4.97 | Instantiating formula (35) with all_0_0_0, all_0_0_0, all_0_2_2, all_0_2_2, all_0_4_4 and discharging atoms unorthogonal_lines(all_0_4_4, all_0_2_2) = all_0_0_0, yields: % 18.26/4.97 | (68) all_0_0_0 = 0 | ? [v0] : ( ~ (v0 = 0) & convergent_lines(all_0_2_2, all_0_2_2) = v0) % 18.26/4.97 | % 18.26/4.97 | Instantiating formula (28) with all_0_0_0, all_0_2_2, all_0_4_4 and discharging atoms unorthogonal_lines(all_0_4_4, all_0_2_2) = all_0_0_0, yields: % 18.26/4.97 | (69) all_0_0_0 = 0 | convergent_lines(all_0_4_4, all_0_2_2) = 0 % 18.26/4.97 | % 18.26/4.97 | Instantiating formula (31) with all_0_1_1, all_0_1_1, all_0_3_3, all_0_4_4, all_0_4_4 and discharging atoms convergent_lines(all_0_4_4, all_0_3_3) = all_0_1_1, yields: % 18.26/4.97 | (70) all_0_1_1 = 0 | ? [v0] : ( ~ (v0 = 0) & convergent_lines(all_0_4_4, all_0_4_4) = v0) % 18.26/4.97 | % 18.26/4.97 | Instantiating formula (26) with all_0_1_1, all_0_2_2, all_0_3_3, all_0_4_4 and discharging atoms unorthogonal_lines(all_0_3_3, all_0_2_2) = 0, convergent_lines(all_0_4_4, all_0_3_3) = all_0_1_1, yields: % 18.26/4.97 | (71) ? [v0] : ? [v1] : ((v1 = 0 & v0 = 0 & unorthogonal_lines(all_0_4_4, all_0_2_2) = 0 & convergent_lines(all_0_4_4, all_0_2_2) = 0) | (v0 = 0 & all_0_1_1 = 0 & unorthogonal_lines(all_0_4_4, all_0_3_3) = 0) | ( ~ (v0 = 0) & convergent_lines(all_0_3_3, all_0_2_2) = v0)) % 18.26/4.97 | % 18.26/4.97 | Instantiating formula (18) with all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3, all_0_4_4 and discharging atoms unorthogonal_lines(all_0_4_4, all_0_2_2) = all_0_0_0, convergent_lines(all_0_4_4, all_0_3_3) = all_0_1_1, yields: % 18.26/4.97 | (72) ? [v0] : ((v0 = 0 & all_0_0_0 = 0 & convergent_lines(all_0_4_4, all_0_2_2) = 0) | (v0 = 0 & all_0_1_1 = 0 & unorthogonal_lines(all_0_4_4, all_0_3_3) = 0) | ( ~ (v0 = 0) & unorthogonal_lines(all_0_3_3, all_0_2_2) = v0) | ( ~ (v0 = 0) & convergent_lines(all_0_3_3, all_0_2_2) = v0)) % 18.26/4.97 | % 18.26/4.97 | Instantiating (67) with all_25_0_29, all_25_1_30 yields: % 18.26/4.97 | (73) (all_25_0_29 = 0 & all_25_1_30 = 0 & unorthogonal_lines(all_0_4_4, all_0_3_3) = 0 & convergent_lines(all_0_4_4, all_0_3_3) = 0) | (all_25_1_30 = 0 & all_0_0_0 = 0 & convergent_lines(all_0_4_4, all_0_2_2) = 0) | ( ~ (all_25_1_30 = 0) & convergent_lines(all_0_3_3, all_0_2_2) = all_25_1_30) % 18.26/4.97 | % 18.26/4.97 | Instantiating (72) with all_28_0_34 yields: % 18.26/4.97 | (74) (all_28_0_34 = 0 & all_0_0_0 = 0 & convergent_lines(all_0_4_4, all_0_2_2) = 0) | (all_28_0_34 = 0 & all_0_1_1 = 0 & unorthogonal_lines(all_0_4_4, all_0_3_3) = 0) | ( ~ (all_28_0_34 = 0) & unorthogonal_lines(all_0_3_3, all_0_2_2) = all_28_0_34) | ( ~ (all_28_0_34 = 0) & convergent_lines(all_0_3_3, all_0_2_2) = all_28_0_34) % 18.26/4.97 | % 18.26/4.97 | Instantiating (71) with all_29_0_35, all_29_1_36 yields: % 18.26/4.97 | (75) (all_29_0_35 = 0 & all_29_1_36 = 0 & unorthogonal_lines(all_0_4_4, all_0_2_2) = 0 & convergent_lines(all_0_4_4, all_0_2_2) = 0) | (all_29_1_36 = 0 & all_0_1_1 = 0 & unorthogonal_lines(all_0_4_4, all_0_3_3) = 0) | ( ~ (all_29_1_36 = 0) & convergent_lines(all_0_3_3, all_0_2_2) = all_29_1_36) % 18.26/4.97 | % 18.26/4.98 +-Applying beta-rule and splitting (69), into two cases. % 18.26/4.98 |-Branch one: % 18.26/4.98 | (76) convergent_lines(all_0_4_4, all_0_2_2) = 0 % 18.26/4.98 | % 18.26/4.98 +-Applying beta-rule and splitting (75), into two cases. % 18.26/4.98 |-Branch one: % 18.26/4.98 | (77) (all_29_0_35 = 0 & all_29_1_36 = 0 & unorthogonal_lines(all_0_4_4, all_0_2_2) = 0 & convergent_lines(all_0_4_4, all_0_2_2) = 0) | (all_29_1_36 = 0 & all_0_1_1 = 0 & unorthogonal_lines(all_0_4_4, all_0_3_3) = 0) % 18.26/4.98 | % 18.26/4.98 +-Applying beta-rule and splitting (77), into two cases. % 18.26/4.98 |-Branch one: % 18.26/4.98 | (78) all_29_0_35 = 0 & all_29_1_36 = 0 & unorthogonal_lines(all_0_4_4, all_0_2_2) = 0 & convergent_lines(all_0_4_4, all_0_2_2) = 0 % 18.26/4.98 | % 18.26/4.98 | Applying alpha-rule on (78) yields: % 18.26/4.98 | (79) all_29_0_35 = 0 % 18.26/4.98 | (80) all_29_1_36 = 0 % 18.26/4.98 | (81) unorthogonal_lines(all_0_4_4, all_0_2_2) = 0 % 18.26/4.98 | (76) convergent_lines(all_0_4_4, all_0_2_2) = 0 % 18.26/4.98 | % 18.26/4.98 +-Applying beta-rule and splitting (68), into two cases. % 18.26/4.98 |-Branch one: % 18.26/4.98 | (83) all_0_0_0 = 0 % 18.26/4.98 | % 18.26/4.98 | Equations (83) can reduce 12 to: % 18.26/4.98 | (84) $false % 18.26/4.98 | % 18.26/4.98 |-The branch is then unsatisfiable % 18.26/4.98 |-Branch two: % 18.26/4.98 | (12) ~ (all_0_0_0 = 0) % 18.26/4.98 | (86) ? [v0] : ( ~ (v0 = 0) & convergent_lines(all_0_2_2, all_0_2_2) = v0) % 18.26/4.98 | % 18.26/4.98 | Instantiating formula (11) with all_0_4_4, all_0_2_2, 0, all_0_0_0 and discharging atoms unorthogonal_lines(all_0_4_4, all_0_2_2) = all_0_0_0, unorthogonal_lines(all_0_4_4, all_0_2_2) = 0, yields: % 18.26/4.98 | (83) all_0_0_0 = 0 % 18.26/4.98 | % 18.26/4.98 | Equations (83) can reduce 12 to: % 18.26/4.98 | (84) $false % 18.26/4.98 | % 18.26/4.98 |-The branch is then unsatisfiable % 18.26/4.98 |-Branch two: % 18.26/4.98 | (89) all_29_1_36 = 0 & all_0_1_1 = 0 & unorthogonal_lines(all_0_4_4, all_0_3_3) = 0 % 18.26/4.98 | % 18.26/4.98 | Applying alpha-rule on (89) yields: % 18.26/4.98 | (80) all_29_1_36 = 0 % 18.26/4.98 | (91) all_0_1_1 = 0 % 18.26/4.98 | (92) unorthogonal_lines(all_0_4_4, all_0_3_3) = 0 % 18.26/4.98 | % 18.26/4.98 | Equations (91) can reduce 29 to: % 18.26/4.98 | (84) $false % 18.26/4.98 | % 18.26/4.98 |-The branch is then unsatisfiable % 18.26/4.98 |-Branch two: % 18.26/4.98 | (94) ~ (all_29_1_36 = 0) & convergent_lines(all_0_3_3, all_0_2_2) = all_29_1_36 % 18.26/4.98 | % 18.26/4.98 | Applying alpha-rule on (94) yields: % 18.26/4.98 | (95) ~ (all_29_1_36 = 0) % 18.26/4.98 | (96) convergent_lines(all_0_3_3, all_0_2_2) = all_29_1_36 % 18.26/4.98 | % 18.26/4.98 +-Applying beta-rule and splitting (73), into two cases. % 18.26/4.98 |-Branch one: % 18.26/4.98 | (97) (all_25_0_29 = 0 & all_25_1_30 = 0 & unorthogonal_lines(all_0_4_4, all_0_3_3) = 0 & convergent_lines(all_0_4_4, all_0_3_3) = 0) | (all_25_1_30 = 0 & all_0_0_0 = 0 & convergent_lines(all_0_4_4, all_0_2_2) = 0) % 18.26/4.98 | % 18.26/4.98 +-Applying beta-rule and splitting (97), into two cases. % 18.26/4.98 |-Branch one: % 18.26/4.98 | (98) all_25_0_29 = 0 & all_25_1_30 = 0 & unorthogonal_lines(all_0_4_4, all_0_3_3) = 0 & convergent_lines(all_0_4_4, all_0_3_3) = 0 % 18.26/4.98 | % 18.26/4.98 | Applying alpha-rule on (98) yields: % 18.26/4.98 | (99) all_25_0_29 = 0 % 18.26/4.98 | (100) all_25_1_30 = 0 % 18.26/4.98 | (92) unorthogonal_lines(all_0_4_4, all_0_3_3) = 0 % 18.26/4.98 | (102) convergent_lines(all_0_4_4, all_0_3_3) = 0 % 18.26/4.98 | % 18.26/4.98 +-Applying beta-rule and splitting (70), into two cases. % 18.26/4.98 |-Branch one: % 18.26/4.98 | (91) all_0_1_1 = 0 % 18.26/4.98 | % 18.26/4.98 | Equations (91) can reduce 29 to: % 18.26/4.98 | (84) $false % 18.26/4.98 | % 18.26/4.98 |-The branch is then unsatisfiable % 18.26/4.98 |-Branch two: % 18.26/4.98 | (29) ~ (all_0_1_1 = 0) % 18.26/4.98 | (106) ? [v0] : ( ~ (v0 = 0) & convergent_lines(all_0_4_4, all_0_4_4) = v0) % 18.26/4.98 | % 18.26/4.98 | Instantiating formula (34) with all_0_4_4, all_0_3_3, 0, all_0_1_1 and discharging atoms convergent_lines(all_0_4_4, all_0_3_3) = all_0_1_1, convergent_lines(all_0_4_4, all_0_3_3) = 0, yields: % 18.26/4.98 | (91) all_0_1_1 = 0 % 18.26/4.98 | % 18.26/4.98 | Equations (91) can reduce 29 to: % 18.26/4.98 | (84) $false % 18.26/4.98 | % 18.26/4.98 |-The branch is then unsatisfiable % 18.26/4.98 |-Branch two: % 18.26/4.98 | (109) all_25_1_30 = 0 & all_0_0_0 = 0 & convergent_lines(all_0_4_4, all_0_2_2) = 0 % 18.26/4.98 | % 18.26/4.98 | Applying alpha-rule on (109) yields: % 18.26/4.98 | (100) all_25_1_30 = 0 % 18.26/4.98 | (83) all_0_0_0 = 0 % 18.26/4.98 | (76) convergent_lines(all_0_4_4, all_0_2_2) = 0 % 18.26/4.98 | % 18.26/4.98 | Equations (83) can reduce 12 to: % 18.26/4.98 | (84) $false % 18.26/4.98 | % 18.26/4.98 |-The branch is then unsatisfiable % 18.26/4.98 |-Branch two: % 18.26/4.98 | (114) ~ (all_25_1_30 = 0) & convergent_lines(all_0_3_3, all_0_2_2) = all_25_1_30 % 18.26/4.98 | % 18.26/4.98 | Applying alpha-rule on (114) yields: % 18.26/4.98 | (115) ~ (all_25_1_30 = 0) % 18.26/4.98 | (116) convergent_lines(all_0_3_3, all_0_2_2) = all_25_1_30 % 18.26/4.98 | % 18.26/4.98 +-Applying beta-rule and splitting (70), into two cases. % 18.26/4.98 |-Branch one: % 18.26/4.98 | (91) all_0_1_1 = 0 % 18.26/4.98 | % 18.26/4.98 | Equations (91) can reduce 29 to: % 18.26/4.98 | (84) $false % 18.26/4.98 | % 18.26/4.98 |-The branch is then unsatisfiable % 18.26/4.98 |-Branch two: % 18.26/4.98 | (29) ~ (all_0_1_1 = 0) % 18.26/4.98 | (106) ? [v0] : ( ~ (v0 = 0) & convergent_lines(all_0_4_4, all_0_4_4) = v0) % 18.26/4.98 | % 18.64/4.98 +-Applying beta-rule and splitting (74), into two cases. % 18.64/4.98 |-Branch one: % 18.64/4.98 | (121) (all_28_0_34 = 0 & all_0_0_0 = 0 & convergent_lines(all_0_4_4, all_0_2_2) = 0) | (all_28_0_34 = 0 & all_0_1_1 = 0 & unorthogonal_lines(all_0_4_4, all_0_3_3) = 0) | ( ~ (all_28_0_34 = 0) & unorthogonal_lines(all_0_3_3, all_0_2_2) = all_28_0_34) % 18.64/4.98 | % 18.64/4.98 +-Applying beta-rule and splitting (121), into two cases. % 18.64/4.98 |-Branch one: % 18.64/4.98 | (122) (all_28_0_34 = 0 & all_0_0_0 = 0 & convergent_lines(all_0_4_4, all_0_2_2) = 0) | (all_28_0_34 = 0 & all_0_1_1 = 0 & unorthogonal_lines(all_0_4_4, all_0_3_3) = 0) % 18.64/4.98 | % 18.64/4.98 +-Applying beta-rule and splitting (122), into two cases. % 18.64/4.98 |-Branch one: % 18.64/4.98 | (123) all_28_0_34 = 0 & all_0_0_0 = 0 & convergent_lines(all_0_4_4, all_0_2_2) = 0 % 18.64/4.98 | % 18.64/4.98 | Applying alpha-rule on (123) yields: % 18.64/4.98 | (124) all_28_0_34 = 0 % 18.64/4.98 | (83) all_0_0_0 = 0 % 18.64/4.98 | (76) convergent_lines(all_0_4_4, all_0_2_2) = 0 % 18.64/4.98 | % 18.64/4.98 | Equations (83) can reduce 12 to: % 18.64/4.98 | (84) $false % 18.64/4.98 | % 18.64/4.98 |-The branch is then unsatisfiable % 18.64/4.98 |-Branch two: % 18.64/4.98 | (128) all_28_0_34 = 0 & all_0_1_1 = 0 & unorthogonal_lines(all_0_4_4, all_0_3_3) = 0 % 18.64/4.98 | % 18.64/4.98 | Applying alpha-rule on (128) yields: % 18.64/4.98 | (124) all_28_0_34 = 0 % 18.64/4.98 | (91) all_0_1_1 = 0 % 18.64/4.98 | (92) unorthogonal_lines(all_0_4_4, all_0_3_3) = 0 % 18.64/4.98 | % 18.64/4.98 | Equations (91) can reduce 29 to: % 18.64/4.98 | (84) $false % 18.64/4.98 | % 18.64/4.98 |-The branch is then unsatisfiable % 18.64/4.98 |-Branch two: % 18.64/4.99 | (133) ~ (all_28_0_34 = 0) & unorthogonal_lines(all_0_3_3, all_0_2_2) = all_28_0_34 % 18.64/4.99 | % 18.64/4.99 | Applying alpha-rule on (133) yields: % 18.64/4.99 | (134) ~ (all_28_0_34 = 0) % 18.64/4.99 | (135) unorthogonal_lines(all_0_3_3, all_0_2_2) = all_28_0_34 % 18.64/4.99 | % 18.64/4.99 | Instantiating formula (11) with all_0_3_3, all_0_2_2, all_28_0_34, 0 and discharging atoms unorthogonal_lines(all_0_3_3, all_0_2_2) = all_28_0_34, unorthogonal_lines(all_0_3_3, all_0_2_2) = 0, yields: % 18.64/4.99 | (124) all_28_0_34 = 0 % 18.64/4.99 | % 18.64/4.99 | Equations (124) can reduce 134 to: % 18.64/4.99 | (84) $false % 18.64/4.99 | % 18.64/4.99 |-The branch is then unsatisfiable % 18.64/4.99 |-Branch two: % 18.64/4.99 | (138) ~ (all_28_0_34 = 0) & convergent_lines(all_0_3_3, all_0_2_2) = all_28_0_34 % 18.64/4.99 | % 18.64/4.99 | Applying alpha-rule on (138) yields: % 18.64/4.99 | (134) ~ (all_28_0_34 = 0) % 18.64/4.99 | (140) convergent_lines(all_0_3_3, all_0_2_2) = all_28_0_34 % 18.64/4.99 | % 18.64/4.99 +-Applying beta-rule and splitting (68), into two cases. % 18.64/4.99 |-Branch one: % 18.64/4.99 | (83) all_0_0_0 = 0 % 18.64/4.99 | % 18.64/4.99 | Equations (83) can reduce 12 to: % 18.64/4.99 | (84) $false % 18.64/4.99 | % 18.64/4.99 |-The branch is then unsatisfiable % 18.64/4.99 |-Branch two: % 18.64/4.99 | (12) ~ (all_0_0_0 = 0) % 18.64/4.99 | (86) ? [v0] : ( ~ (v0 = 0) & convergent_lines(all_0_2_2, all_0_2_2) = v0) % 18.64/4.99 | % 18.64/4.99 | Instantiating (86) with all_57_0_45 yields: % 18.64/4.99 | (145) ~ (all_57_0_45 = 0) & convergent_lines(all_0_2_2, all_0_2_2) = all_57_0_45 % 18.64/4.99 | % 18.64/4.99 | Applying alpha-rule on (145) yields: % 18.64/4.99 | (146) ~ (all_57_0_45 = 0) % 18.64/4.99 | (147) convergent_lines(all_0_2_2, all_0_2_2) = all_57_0_45 % 18.64/4.99 | % 18.64/4.99 | Instantiating formula (34) with all_0_3_3, all_0_2_2, all_28_0_34, all_29_1_36 and discharging atoms convergent_lines(all_0_3_3, all_0_2_2) = all_29_1_36, convergent_lines(all_0_3_3, all_0_2_2) = all_28_0_34, yields: % 18.64/4.99 | (148) all_29_1_36 = all_28_0_34 % 18.64/4.99 | % 18.64/4.99 | Instantiating formula (34) with all_0_3_3, all_0_2_2, all_25_1_30, all_29_1_36 and discharging atoms convergent_lines(all_0_3_3, all_0_2_2) = all_29_1_36, convergent_lines(all_0_3_3, all_0_2_2) = all_25_1_30, yields: % 18.64/4.99 | (149) all_29_1_36 = all_25_1_30 % 18.64/4.99 | % 18.64/4.99 | Combining equations (148,149) yields a new equation: % 18.64/4.99 | (150) all_28_0_34 = all_25_1_30 % 18.64/4.99 | % 18.64/4.99 | Simplifying 150 yields: % 18.64/4.99 | (151) all_28_0_34 = all_25_1_30 % 18.64/4.99 | % 18.64/4.99 | Equations (151) can reduce 134 to: % 18.64/4.99 | (115) ~ (all_25_1_30 = 0) % 18.64/4.99 | % 18.64/4.99 | From (151) and (140) follows: % 18.64/4.99 | (116) convergent_lines(all_0_3_3, all_0_2_2) = all_25_1_30 % 18.64/4.99 | % 18.64/4.99 | Instantiating formula (31) with all_25_1_30, all_57_0_45, all_0_2_2, all_0_3_3, all_0_2_2 and discharging atoms convergent_lines(all_0_2_2, all_0_2_2) = all_57_0_45, convergent_lines(all_0_3_3, all_0_2_2) = all_25_1_30, yields: % 18.64/4.99 | (154) all_57_0_45 = 0 | all_25_1_30 = 0 | ? [v0] : ( ~ (v0 = 0) & convergent_lines(all_0_2_2, all_0_3_3) = v0) % 18.64/4.99 | % 18.64/4.99 | Instantiating formula (31) with all_25_1_30, all_25_1_30, all_0_2_2, all_0_3_3, all_0_3_3 and discharging atoms convergent_lines(all_0_3_3, all_0_2_2) = all_25_1_30, yields: % 18.64/4.99 | (155) all_25_1_30 = 0 | ? [v0] : ( ~ (v0 = 0) & convergent_lines(all_0_3_3, all_0_3_3) = v0) % 18.64/4.99 | % 18.64/4.99 | Instantiating formula (32) with all_0_1_1, all_0_3_3, all_0_2_2, all_0_4_4 and discharging atoms convergent_lines(all_0_4_4, all_0_2_2) = 0, convergent_lines(all_0_4_4, all_0_3_3) = all_0_1_1, yields: % 18.64/4.99 | (156) all_0_1_1 = 0 | convergent_lines(all_0_2_2, all_0_3_3) = 0 % 18.64/4.99 | % 18.64/4.99 +-Applying beta-rule and splitting (155), into two cases. % 18.64/4.99 |-Branch one: % 18.64/4.99 | (100) all_25_1_30 = 0 % 18.64/4.99 | % 18.64/4.99 | Equations (100) can reduce 115 to: % 18.64/4.99 | (84) $false % 18.64/4.99 | % 18.64/4.99 |-The branch is then unsatisfiable % 18.64/4.99 |-Branch two: % 18.64/4.99 | (115) ~ (all_25_1_30 = 0) % 18.64/4.99 | (160) ? [v0] : ( ~ (v0 = 0) & convergent_lines(all_0_3_3, all_0_3_3) = v0) % 18.64/4.99 | % 18.64/4.99 +-Applying beta-rule and splitting (154), into two cases. % 18.64/4.99 |-Branch one: % 18.64/4.99 | (161) all_57_0_45 = 0 % 18.64/4.99 | % 18.64/4.99 | Equations (161) can reduce 146 to: % 18.64/4.99 | (84) $false % 18.64/4.99 | % 18.64/4.99 |-The branch is then unsatisfiable % 18.64/4.99 |-Branch two: % 18.64/4.99 | (146) ~ (all_57_0_45 = 0) % 18.64/4.99 | (164) all_25_1_30 = 0 | ? [v0] : ( ~ (v0 = 0) & convergent_lines(all_0_2_2, all_0_3_3) = v0) % 18.64/4.99 | % 18.64/4.99 +-Applying beta-rule and splitting (156), into two cases. % 18.64/4.99 |-Branch one: % 18.64/4.99 | (165) convergent_lines(all_0_2_2, all_0_3_3) = 0 % 18.64/4.99 | % 18.64/4.99 +-Applying beta-rule and splitting (164), into two cases. % 18.64/4.99 |-Branch one: % 18.64/4.99 | (100) all_25_1_30 = 0 % 18.64/4.99 | % 18.64/4.99 | Equations (100) can reduce 115 to: % 18.64/4.99 | (84) $false % 18.64/4.99 | % 18.64/4.99 |-The branch is then unsatisfiable % 18.64/4.99 |-Branch two: % 18.64/4.99 | (115) ~ (all_25_1_30 = 0) % 18.64/4.99 | (169) ? [v0] : ( ~ (v0 = 0) & convergent_lines(all_0_2_2, all_0_3_3) = v0) % 18.64/4.99 | % 18.64/4.99 | Instantiating (169) with all_189_0_229 yields: % 18.64/4.99 | (170) ~ (all_189_0_229 = 0) & convergent_lines(all_0_2_2, all_0_3_3) = all_189_0_229 % 18.64/4.99 | % 18.64/4.99 | Applying alpha-rule on (170) yields: % 18.64/4.99 | (171) ~ (all_189_0_229 = 0) % 18.64/4.99 | (172) convergent_lines(all_0_2_2, all_0_3_3) = all_189_0_229 % 18.64/4.99 | % 18.64/4.99 | Instantiating formula (34) with all_0_2_2, all_0_3_3, 0, all_189_0_229 and discharging atoms convergent_lines(all_0_2_2, all_0_3_3) = all_189_0_229, convergent_lines(all_0_2_2, all_0_3_3) = 0, yields: % 18.64/4.99 | (173) all_189_0_229 = 0 % 18.64/4.99 | % 18.64/4.99 | Equations (173) can reduce 171 to: % 18.64/4.99 | (84) $false % 18.64/4.99 | % 18.64/4.99 |-The branch is then unsatisfiable % 18.64/4.99 |-Branch two: % 18.64/4.99 | (175) ~ (convergent_lines(all_0_2_2, all_0_3_3) = 0) % 18.64/4.99 | (91) all_0_1_1 = 0 % 18.64/4.99 | % 18.64/4.99 | Equations (91) can reduce 29 to: % 18.64/4.99 | (84) $false % 18.64/4.99 | % 18.64/4.99 |-The branch is then unsatisfiable % 18.64/4.99 |-Branch two: % 18.64/4.99 | (178) ~ (convergent_lines(all_0_4_4, all_0_2_2) = 0) % 18.64/5.00 | (83) all_0_0_0 = 0 % 18.64/5.00 | % 18.64/5.00 | Equations (83) can reduce 12 to: % 18.64/5.00 | (84) $false % 18.64/5.00 | % 18.64/5.00 |-The branch is then unsatisfiable % 18.64/5.00 % SZS output end Proof for theBenchmark % 18.64/5.00 % 18.64/5.00 4419ms %------------------------------------------------------------------------------