%------------------------------------------------------------------------------ % File : ePrincess---1.0 % Problem : GEO192+1 : TPTP v8.1.0. Released v3.3.0. % Transfm : none % Format : tptp:raw % Command : ePrincess-casc -timeout=%d %s % Computer : n024.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 03:48:28 EDT 2022 % Result : Theorem 4.26s 1.64s % Output : Proof 6.02s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.09/0.12 % Problem : GEO192+1 : TPTP v8.1.0. Released v3.3.0. % 0.09/0.12 % Command : ePrincess-casc -timeout=%d %s % 0.12/0.33 % Computer : n024.cluster.edu % 0.12/0.33 % Model : x86_64 x86_64 % 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.33 % Memory : 8042.1875MB % 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.33 % CPULimit : 300 % 0.12/0.33 % WCLimit : 600 % 0.12/0.33 % DateTime : Sat Jun 18 05:27:17 EDT 2022 % 0.12/0.34 % CPUTime : % 0.20/0.52 ____ _ % 0.20/0.52 ___ / __ \_____(_)___ ________ __________ % 0.20/0.52 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/ % 0.20/0.52 / __/ ____/ / / / / / / /__/ __(__ |__ ) % 0.20/0.52 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/ % 0.20/0.52 % 0.20/0.52 A Theorem Prover for First-Order Logic % 0.20/0.52 (ePrincess v.1.0) % 0.20/0.52 % 0.20/0.52 (c) Philipp Rümmer, 2009-2015 % 0.20/0.52 (c) Peter Backeman, 2014-2015 % 0.20/0.52 (contributions by Angelo Brillout, Peter Baumgartner) % 0.20/0.52 Free software under GNU Lesser General Public License (LGPL). % 0.20/0.52 Bug reports to peter@backeman.se % 0.20/0.52 % 0.20/0.52 For more information, visit http://user.uu.se/~petba168/breu/ % 0.20/0.52 % 0.20/0.52 Loading /export/starexec/sandbox2/benchmark/theBenchmark.p ... % 0.20/0.57 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 1.62/0.89 Prover 0: Preprocessing ... % 1.75/1.01 Prover 0: Warning: ignoring some quantifiers % 2.01/1.03 Prover 0: Constructing countermodel ... % 2.67/1.24 Prover 0: gave up % 2.67/1.24 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all % 2.67/1.26 Prover 1: Preprocessing ... % 2.97/1.35 Prover 1: Constructing countermodel ... % 3.20/1.41 Prover 1: gave up % 3.20/1.41 Prover 2: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 3.20/1.42 Prover 2: Preprocessing ... % 3.88/1.54 Prover 2: Warning: ignoring some quantifiers % 3.88/1.54 Prover 2: Constructing countermodel ... % 4.26/1.64 Prover 2: proved (233ms) % 4.26/1.64 % 4.26/1.64 No countermodel exists, formula is valid % 4.26/1.64 % SZS status Theorem for theBenchmark % 4.26/1.64 % 4.26/1.64 Generating proof ... Warning: ignoring some quantifiers % 5.64/1.93 found it (size 63) % 5.64/1.93 % 5.64/1.93 % SZS output start Proof for theBenchmark % 5.64/1.93 Assumed formulas after preprocessing and simplification: % 5.64/1.93 | (0) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ? [v5] : (intersection_point(v0, v1) = v3 & apart_point_and_line(v3, v2) = 0 & convergent_lines(v0, v1) = 0 & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : (v11 = 0 | v10 = 0 | ~ (apart_point_and_line(v7, v9) = v11) | ~ (apart_point_and_line(v7, v8) = v10) | ~ (distinct_points(v6, v7) = 0) | ? [v12] : ((v12 = 0 & apart_point_and_line(v6, v9) = 0) | (v12 = 0 & apart_point_and_line(v6, v8) = 0) | ( ~ (v12 = 0) & distinct_lines(v8, v9) = v12))) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : (v11 = 0 | v10 = 0 | ~ (apart_point_and_line(v7, v9) = v11) | ~ (apart_point_and_line(v6, v9) = v10) | ~ (distinct_lines(v8, v9) = 0) | ? [v12] : ((v12 = 0 & apart_point_and_line(v7, v8) = 0) | (v12 = 0 & apart_point_and_line(v6, v8) = 0) | ( ~ (v12 = 0) & distinct_points(v6, v7) = v12))) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : (v11 = 0 | v10 = 0 | ~ (apart_point_and_line(v7, v9) = v11) | ~ (apart_point_and_line(v6, v8) = v10) | ? [v12] : ((v12 = 0 & apart_point_and_line(v7, v8) = 0) | (v12 = 0 & apart_point_and_line(v6, v9) = 0) | ( ~ (v12 = 0) & distinct_lines(v8, v9) = v12) | ( ~ (v12 = 0) & distinct_points(v6, v7) = v12))) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : (v11 = 0 | v10 = 0 | ~ (apart_point_and_line(v7, v8) = v11) | ~ (apart_point_and_line(v6, v9) = v10) | ? [v12] : ((v12 = 0 & apart_point_and_line(v7, v9) = 0) | (v12 = 0 & apart_point_and_line(v6, v8) = 0) | ( ~ (v12 = 0) & distinct_lines(v8, v9) = v12) | ( ~ (v12 = 0) & distinct_points(v6, v7) = v12))) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : (v11 = 0 | v10 = 0 | ~ (apart_point_and_line(v7, v8) = v11) | ~ (apart_point_and_line(v6, v8) = v10) | ~ (distinct_lines(v8, v9) = 0) | ? [v12] : ((v12 = 0 & apart_point_and_line(v7, v9) = 0) | (v12 = 0 & apart_point_and_line(v6, v9) = 0) | ( ~ (v12 = 0) & distinct_points(v6, v7) = v12))) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : (v11 = 0 | v10 = 0 | ~ (apart_point_and_line(v6, v9) = v11) | ~ (apart_point_and_line(v6, v8) = v10) | ~ (distinct_points(v6, v7) = 0) | ? [v12] : ((v12 = 0 & apart_point_and_line(v7, v9) = 0) | (v12 = 0 & apart_point_and_line(v7, v8) = 0) | ( ~ (v12 = 0) & distinct_lines(v8, v9) = v12))) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | v9 = 0 | ~ (apart_point_and_line(v8, v7) = v10) | ~ (distinct_points(v6, v8) = v9) | ? [v11] : ( ~ (v11 = 0) & apart_point_and_line(v6, v7) = v11)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | v9 = 0 | ~ (apart_point_and_line(v6, v8) = v10) | ~ (distinct_lines(v7, v8) = v9) | ? [v11] : ( ~ (v11 = 0) & apart_point_and_line(v6, v7) = v11)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | v9 = 0 | ~ (convergent_lines(v7, v8) = v10) | ~ (convergent_lines(v6, v8) = v9) | ? [v11] : ( ~ (v11 = 0) & convergent_lines(v6, v7) = v11)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | v9 = 0 | ~ (convergent_lines(v6, v8) = v10) | ~ (distinct_lines(v7, v8) = v9) | ? [v11] : ( ~ (v11 = 0) & convergent_lines(v6, v7) = v11)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | v9 = 0 | ~ (distinct_lines(v7, v8) = v10) | ~ (distinct_lines(v6, v8) = v9) | ? [v11] : ( ~ (v11 = 0) & distinct_lines(v6, v7) = v11)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | v9 = 0 | ~ (distinct_points(v7, v8) = v10) | ~ (distinct_points(v6, v8) = v9) | ? [v11] : ( ~ (v11 = 0) & distinct_points(v6, v7) = v11)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (apart_point_and_line(v8, v7) = v9) | ~ (apart_point_and_line(v6, v7) = 0) | distinct_points(v6, v8) = 0) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (apart_point_and_line(v6, v8) = v9) | ~ (apart_point_and_line(v6, v7) = 0) | distinct_lines(v7, v8) = 0) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (apart_point_and_line(v6, v7) = 0) | ~ (distinct_lines(v7, v8) = v9) | apart_point_and_line(v6, v8) = 0) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (apart_point_and_line(v6, v7) = 0) | ~ (distinct_points(v6, v8) = v9) | apart_point_and_line(v8, v7) = 0) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (convergent_lines(v7, v8) = v9) | ~ (convergent_lines(v6, v7) = 0) | convergent_lines(v6, v8) = 0) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (convergent_lines(v6, v8) = v9) | ~ (convergent_lines(v6, v7) = 0) | convergent_lines(v7, v8) = 0) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (convergent_lines(v6, v8) = v9) | ~ (convergent_lines(v6, v7) = 0) | distinct_lines(v7, v8) = 0) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (convergent_lines(v6, v7) = 0) | ~ (distinct_lines(v7, v8) = v9) | convergent_lines(v6, v8) = 0) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (distinct_lines(v7, v8) = v9) | ~ (distinct_lines(v6, v7) = 0) | distinct_lines(v6, v8) = 0) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (distinct_lines(v6, v8) = v9) | ~ (distinct_lines(v6, v7) = 0) | distinct_lines(v7, v8) = 0) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (distinct_points(v7, v8) = v9) | ~ (distinct_points(v6, v7) = 0) | distinct_points(v6, v8) = 0) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | ~ (distinct_points(v6, v8) = v9) | ~ (distinct_points(v6, v7) = 0) | distinct_points(v7, v8) = 0) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v7 = v6 | ~ (intersection_point(v9, v8) = v7) | ~ (intersection_point(v9, v8) = v6)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v7 = v6 | ~ (line_connecting(v9, v8) = v7) | ~ (line_connecting(v9, v8) = v6)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v7 = v6 | ~ (apart_point_and_line(v9, v8) = v7) | ~ (apart_point_and_line(v9, v8) = v6)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v7 = v6 | ~ (convergent_lines(v9, v8) = v7) | ~ (convergent_lines(v9, v8) = v6)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v7 = v6 | ~ (distinct_lines(v9, v8) = v7) | ~ (distinct_lines(v9, v8) = v6)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v7 = v6 | ~ (distinct_points(v9, v8) = v7) | ~ (distinct_points(v9, v8) = v6)) & ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (distinct_lines(v8, v9) = 0) | ~ (distinct_points(v6, v7) = 0) | ? [v10] : ((v10 = 0 & apart_point_and_line(v7, v9) = 0) | (v10 = 0 & apart_point_and_line(v7, v8) = 0) | (v10 = 0 & apart_point_and_line(v6, v9) = 0) | (v10 = 0 & apart_point_and_line(v6, v8) = 0))) & ! [v6] : ! [v7] : ! [v8] : ( ~ (intersection_point(v6, v7) = v8) | ? [v9] : (( ~ (v9 = 0) & apart_point_and_line(v8, v7) = v9) | ( ~ (v9 = 0) & convergent_lines(v6, v7) = v9))) & ! [v6] : ! [v7] : ! [v8] : ( ~ (intersection_point(v6, v7) = v8) | ? [v9] : (( ~ (v9 = 0) & apart_point_and_line(v8, v6) = v9) | ( ~ (v9 = 0) & convergent_lines(v6, v7) = v9))) & ! [v6] : ! [v7] : ! [v8] : ( ~ (line_connecting(v6, v7) = v8) | ? [v9] : (( ~ (v9 = 0) & apart_point_and_line(v7, v8) = v9) | ( ~ (v9 = 0) & distinct_points(v6, v7) = v9))) & ! [v6] : ! [v7] : ! [v8] : ( ~ (line_connecting(v6, v7) = v8) | ? [v9] : (( ~ (v9 = 0) & apart_point_and_line(v6, v8) = v9) | ( ~ (v9 = 0) & distinct_points(v6, v7) = v9))) & ! [v6] : ! [v7] : ( ~ (convergent_lines(v6, v7) = 0) | ? [v8] : ? [v9] : ( ~ (v9 = 0) & intersection_point(v6, v7) = v8 & apart_point_and_line(v8, v7) = v9)) & ! [v6] : ! [v7] : ( ~ (convergent_lines(v6, v7) = 0) | ? [v8] : ? [v9] : ( ~ (v9 = 0) & intersection_point(v6, v7) = v8 & apart_point_and_line(v8, v6) = v9)) & ! [v6] : ! [v7] : ( ~ (distinct_points(v6, v7) = 0) | ? [v8] : ? [v9] : ( ~ (v9 = 0) & line_connecting(v6, v7) = v8 & apart_point_and_line(v7, v8) = v9)) & ! [v6] : ! [v7] : ( ~ (distinct_points(v6, v7) = 0) | ? [v8] : ? [v9] : ( ~ (v9 = 0) & line_connecting(v6, v7) = v8 & apart_point_and_line(v6, v8) = v9)) & ! [v6] : ~ (convergent_lines(v6, v6) = 0) & ! [v6] : ~ (distinct_lines(v6, v6) = 0) & ! [v6] : ~ (distinct_points(v6, v6) = 0) & ? [v6] : ? [v7] : ? [v8] : intersection_point(v7, v6) = v8 & ? [v6] : ? [v7] : ? [v8] : line_connecting(v7, v6) = v8 & ? [v6] : ? [v7] : ? [v8] : apart_point_and_line(v7, v6) = v8 & ? [v6] : ? [v7] : ? [v8] : convergent_lines(v7, v6) = v8 & ? [v6] : ? [v7] : ? [v8] : distinct_lines(v7, v6) = v8 & ? [v6] : ? [v7] : ? [v8] : distinct_points(v7, v6) = v8 & (( ~ (v5 = 0) & distinct_lines(v1, v2) = v5) | ( ~ (v4 = 0) & distinct_lines(v0, v2) = v4))) % 5.64/1.96 | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3, all_0_4_4, all_0_5_5 yields: % 5.64/1.96 | (1) intersection_point(all_0_5_5, all_0_4_4) = all_0_2_2 & apart_point_and_line(all_0_2_2, all_0_3_3) = 0 & convergent_lines(all_0_5_5, all_0_4_4) = 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) | ~ (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 | ~ (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] : (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 | ~ (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 | ~ (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] : ( ~ (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] : ( ~ (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] : 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 & (( ~ (all_0_0_0 = 0) & distinct_lines(all_0_4_4, all_0_3_3) = all_0_0_0) | ( ~ (all_0_1_1 = 0) & distinct_lines(all_0_5_5, all_0_3_3) = all_0_1_1)) % 5.64/1.98 | % 5.64/1.98 | Applying alpha-rule on (1) yields: % 5.64/1.98 | (2) ! [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))) % 5.64/1.98 | (3) ! [v0] : ! [v1] : ( ~ (distinct_points(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & line_connecting(v0, v1) = v2 & apart_point_and_line(v0, v2) = v3)) % 5.64/1.98 | (4) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_points(v3, v2) = v1) | ~ (distinct_points(v3, v2) = v0)) % 5.64/1.98 | (5) ? [v0] : ? [v1] : ? [v2] : distinct_points(v1, v0) = v2 % 5.64/1.98 | (6) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0) % 5.64/1.98 | (7) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_lines(v3, v2) = v1) | ~ (distinct_lines(v3, v2) = v0)) % 5.64/1.98 | (8) ! [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)) % 5.64/1.98 | (9) ! [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) % 5.64/1.98 | (10) ! [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) % 5.64/1.98 | (11) ! [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) % 5.64/1.98 | (12) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v1, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v0, v2) = 0) % 5.64/1.98 | (13) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (line_connecting(v3, v2) = v1) | ~ (line_connecting(v3, v2) = v0)) % 5.64/1.99 | (14) ! [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))) % 5.64/1.99 | (15) ! [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))) % 5.64/1.99 | (16) ! [v0] : ! [v1] : ( ~ (distinct_points(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & line_connecting(v0, v1) = v2 & apart_point_and_line(v1, v2) = v3)) % 5.64/1.99 | (17) ! [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))) % 5.64/1.99 | (18) ! [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))) % 5.64/1.99 | (19) ! [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))) % 5.64/1.99 | (20) convergent_lines(all_0_5_5, all_0_4_4) = 0 % 5.64/1.99 | (21) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v1, v2) = 0) % 5.64/1.99 | (22) ? [v0] : ? [v1] : ? [v2] : apart_point_and_line(v1, v0) = v2 % 5.64/1.99 | (23) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (intersection_point(v3, v2) = v1) | ~ (intersection_point(v3, v2) = v0)) % 5.64/1.99 | (24) ! [v0] : ! [v1] : ( ~ (convergent_lines(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & intersection_point(v0, v1) = v2 & apart_point_and_line(v2, v1) = v3)) % 5.64/1.99 | (25) ? [v0] : ? [v1] : ? [v2] : distinct_lines(v1, v0) = v2 % 5.64/1.99 | (26) ! [v0] : ~ (distinct_lines(v0, v0) = 0) % 5.64/1.99 | (27) ? [v0] : ? [v1] : ? [v2] : convergent_lines(v1, v0) = v2 % 5.64/1.99 | (28) apart_point_and_line(all_0_2_2, all_0_3_3) = 0 % 5.64/1.99 | (29) ! [v0] : ~ (distinct_points(v0, v0) = 0) % 5.64/1.99 | (30) ! [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))) % 5.64/1.99 | (31) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v1, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v0, v2) = 0) % 5.64/1.99 | (32) ! [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))) % 5.64/1.99 | (33) ! [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)) % 5.64/1.99 | (34) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v0, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0) % 6.02/1.99 | (35) ! [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)) % 6.02/1.99 | (36) ? [v0] : ? [v1] : ? [v2] : line_connecting(v1, v0) = v2 % 6.02/1.99 | (37) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v1) = 0) | ~ (distinct_lines(v1, v2) = v3) | convergent_lines(v0, v2) = 0) % 6.02/1.99 | (38) ! [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)) % 6.02/1.99 | (39) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (convergent_lines(v3, v2) = v1) | ~ (convergent_lines(v3, v2) = v0)) % 6.02/1.99 | (40) ! [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))) % 6.02/2.00 | (41) ! [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))) % 6.02/2.00 | (42) ! [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))) % 6.02/2.00 | (43) ! [v0] : ! [v1] : ( ~ (convergent_lines(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & intersection_point(v0, v1) = v2 & apart_point_and_line(v2, v0) = v3)) % 6.02/2.00 | (44) ? [v0] : ? [v1] : ? [v2] : intersection_point(v1, v0) = v2 % 6.02/2.00 | (45) intersection_point(all_0_5_5, all_0_4_4) = all_0_2_2 % 6.02/2.00 | (46) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (apart_point_and_line(v3, v2) = v1) | ~ (apart_point_and_line(v3, v2) = v0)) % 6.02/2.00 | (47) ! [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)) % 6.02/2.00 | (48) ( ~ (all_0_0_0 = 0) & distinct_lines(all_0_4_4, all_0_3_3) = all_0_0_0) | ( ~ (all_0_1_1 = 0) & distinct_lines(all_0_5_5, all_0_3_3) = all_0_1_1) % 6.02/2.00 | (49) ! [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) % 6.02/2.00 | (50) ! [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)) % 6.02/2.00 | (51) ! [v0] : ~ (convergent_lines(v0, v0) = 0) % 6.02/2.00 | (52) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v0, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v1, v2) = 0) % 6.02/2.00 | (53) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v1, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v0, v2) = 0) % 6.02/2.00 | % 6.02/2.00 | Instantiating formula (15) with all_0_2_2, all_0_4_4, all_0_5_5 and discharging atoms intersection_point(all_0_5_5, all_0_4_4) = all_0_2_2, yields: % 6.02/2.00 | (54) ? [v0] : (( ~ (v0 = 0) & apart_point_and_line(all_0_2_2, all_0_4_4) = v0) | ( ~ (v0 = 0) & convergent_lines(all_0_5_5, all_0_4_4) = v0)) % 6.02/2.00 | % 6.02/2.00 | Instantiating formula (19) with all_0_2_2, all_0_4_4, all_0_5_5 and discharging atoms intersection_point(all_0_5_5, all_0_4_4) = all_0_2_2, yields: % 6.02/2.00 | (55) ? [v0] : (( ~ (v0 = 0) & apart_point_and_line(all_0_2_2, all_0_5_5) = v0) | ( ~ (v0 = 0) & convergent_lines(all_0_5_5, all_0_4_4) = v0)) % 6.02/2.00 | % 6.02/2.00 | Instantiating formula (24) with all_0_4_4, all_0_5_5 and discharging atoms convergent_lines(all_0_5_5, all_0_4_4) = 0, yields: % 6.02/2.00 | (56) ? [v0] : ? [v1] : ( ~ (v1 = 0) & intersection_point(all_0_5_5, all_0_4_4) = v0 & apart_point_and_line(v0, all_0_4_4) = v1) % 6.02/2.00 | % 6.02/2.00 | Instantiating formula (43) with all_0_4_4, all_0_5_5 and discharging atoms convergent_lines(all_0_5_5, all_0_4_4) = 0, yields: % 6.02/2.00 | (57) ? [v0] : ? [v1] : ( ~ (v1 = 0) & intersection_point(all_0_5_5, all_0_4_4) = v0 & apart_point_and_line(v0, all_0_5_5) = v1) % 6.02/2.00 | % 6.02/2.00 | Instantiating (57) with all_20_0_24, all_20_1_25 yields: % 6.02/2.00 | (58) ~ (all_20_0_24 = 0) & intersection_point(all_0_5_5, all_0_4_4) = all_20_1_25 & apart_point_and_line(all_20_1_25, all_0_5_5) = all_20_0_24 % 6.02/2.00 | % 6.02/2.00 | Applying alpha-rule on (58) yields: % 6.02/2.00 | (59) ~ (all_20_0_24 = 0) % 6.02/2.00 | (60) intersection_point(all_0_5_5, all_0_4_4) = all_20_1_25 % 6.02/2.00 | (61) apart_point_and_line(all_20_1_25, all_0_5_5) = all_20_0_24 % 6.02/2.00 | % 6.02/2.00 | Instantiating (56) with all_22_0_26, all_22_1_27 yields: % 6.02/2.00 | (62) ~ (all_22_0_26 = 0) & intersection_point(all_0_5_5, all_0_4_4) = all_22_1_27 & apart_point_and_line(all_22_1_27, all_0_4_4) = all_22_0_26 % 6.02/2.00 | % 6.02/2.00 | Applying alpha-rule on (62) yields: % 6.02/2.00 | (63) ~ (all_22_0_26 = 0) % 6.02/2.00 | (64) intersection_point(all_0_5_5, all_0_4_4) = all_22_1_27 % 6.02/2.00 | (65) apart_point_and_line(all_22_1_27, all_0_4_4) = all_22_0_26 % 6.02/2.00 | % 6.02/2.00 | Instantiating (55) with all_24_0_28 yields: % 6.02/2.00 | (66) ( ~ (all_24_0_28 = 0) & apart_point_and_line(all_0_2_2, all_0_5_5) = all_24_0_28) | ( ~ (all_24_0_28 = 0) & convergent_lines(all_0_5_5, all_0_4_4) = all_24_0_28) % 6.02/2.00 | % 6.02/2.00 | Instantiating (54) with all_25_0_29 yields: % 6.02/2.00 | (67) ( ~ (all_25_0_29 = 0) & apart_point_and_line(all_0_2_2, all_0_4_4) = all_25_0_29) | ( ~ (all_25_0_29 = 0) & convergent_lines(all_0_5_5, all_0_4_4) = all_25_0_29) % 6.02/2.00 | % 6.02/2.00 +-Applying beta-rule and splitting (66), into two cases. % 6.02/2.00 |-Branch one: % 6.02/2.01 | (68) ~ (all_24_0_28 = 0) & apart_point_and_line(all_0_2_2, all_0_5_5) = all_24_0_28 % 6.02/2.01 | % 6.02/2.01 | Applying alpha-rule on (68) yields: % 6.02/2.01 | (69) ~ (all_24_0_28 = 0) % 6.02/2.01 | (70) apart_point_and_line(all_0_2_2, all_0_5_5) = all_24_0_28 % 6.02/2.01 | % 6.02/2.01 +-Applying beta-rule and splitting (67), into two cases. % 6.02/2.01 |-Branch one: % 6.02/2.01 | (71) ~ (all_25_0_29 = 0) & apart_point_and_line(all_0_2_2, all_0_4_4) = all_25_0_29 % 6.02/2.01 | % 6.02/2.01 | Applying alpha-rule on (71) yields: % 6.02/2.01 | (72) ~ (all_25_0_29 = 0) % 6.02/2.01 | (73) apart_point_and_line(all_0_2_2, all_0_4_4) = all_25_0_29 % 6.02/2.01 | % 6.02/2.01 | Instantiating formula (23) with all_0_5_5, all_0_4_4, all_22_1_27, all_0_2_2 and discharging atoms intersection_point(all_0_5_5, all_0_4_4) = all_22_1_27, intersection_point(all_0_5_5, all_0_4_4) = all_0_2_2, yields: % 6.02/2.01 | (74) all_22_1_27 = all_0_2_2 % 6.02/2.01 | % 6.02/2.01 | Instantiating formula (23) with all_0_5_5, all_0_4_4, all_20_1_25, all_22_1_27 and discharging atoms intersection_point(all_0_5_5, all_0_4_4) = all_22_1_27, intersection_point(all_0_5_5, all_0_4_4) = all_20_1_25, yields: % 6.02/2.01 | (75) all_22_1_27 = all_20_1_25 % 6.02/2.01 | % 6.02/2.01 | Combining equations (74,75) yields a new equation: % 6.02/2.01 | (76) all_20_1_25 = all_0_2_2 % 6.02/2.01 | % 6.02/2.01 | Combining equations (76,75) yields a new equation: % 6.02/2.01 | (74) all_22_1_27 = all_0_2_2 % 6.02/2.01 | % 6.02/2.01 | From (74) and (65) follows: % 6.02/2.01 | (78) apart_point_and_line(all_0_2_2, all_0_4_4) = all_22_0_26 % 6.02/2.01 | % 6.02/2.01 | From (76) and (61) follows: % 6.02/2.01 | (79) apart_point_and_line(all_0_2_2, all_0_5_5) = all_20_0_24 % 6.02/2.01 | % 6.02/2.01 | Instantiating formula (46) with all_0_2_2, all_0_4_4, all_22_0_26, all_25_0_29 and discharging atoms apart_point_and_line(all_0_2_2, all_0_4_4) = all_25_0_29, apart_point_and_line(all_0_2_2, all_0_4_4) = all_22_0_26, yields: % 6.02/2.01 | (80) all_25_0_29 = all_22_0_26 % 6.02/2.01 | % 6.02/2.01 | Instantiating formula (46) with all_0_2_2, all_0_5_5, all_20_0_24, all_24_0_28 and discharging atoms apart_point_and_line(all_0_2_2, all_0_5_5) = all_24_0_28, apart_point_and_line(all_0_2_2, all_0_5_5) = all_20_0_24, yields: % 6.02/2.01 | (81) all_24_0_28 = all_20_0_24 % 6.02/2.01 | % 6.02/2.01 | Equations (80) can reduce 72 to: % 6.02/2.01 | (63) ~ (all_22_0_26 = 0) % 6.02/2.01 | % 6.02/2.01 | Equations (81) can reduce 69 to: % 6.02/2.01 | (59) ~ (all_20_0_24 = 0) % 6.02/2.01 | % 6.02/2.01 | From (80) and (73) follows: % 6.02/2.01 | (78) apart_point_and_line(all_0_2_2, all_0_4_4) = all_22_0_26 % 6.02/2.01 | % 6.02/2.01 | From (81) and (70) follows: % 6.02/2.01 | (79) apart_point_and_line(all_0_2_2, all_0_5_5) = all_20_0_24 % 6.02/2.01 | % 6.02/2.01 | Instantiating formula (9) with all_22_0_26, all_0_4_4, all_0_3_3, all_0_2_2 and discharging atoms apart_point_and_line(all_0_2_2, all_0_3_3) = 0, apart_point_and_line(all_0_2_2, all_0_4_4) = all_22_0_26, yields: % 6.02/2.01 | (86) all_22_0_26 = 0 | distinct_lines(all_0_3_3, all_0_4_4) = 0 % 6.02/2.01 | % 6.02/2.01 | Instantiating formula (9) with all_20_0_24, all_0_5_5, all_0_3_3, all_0_2_2 and discharging atoms apart_point_and_line(all_0_2_2, all_0_3_3) = 0, apart_point_and_line(all_0_2_2, all_0_5_5) = all_20_0_24, yields: % 6.02/2.01 | (87) all_20_0_24 = 0 | distinct_lines(all_0_3_3, all_0_5_5) = 0 % 6.02/2.01 | % 6.02/2.01 | Instantiating formula (2) with all_20_0_24, all_20_0_24, all_0_5_5, all_0_5_5, all_0_2_2, all_0_2_2 and discharging atoms apart_point_and_line(all_0_2_2, all_0_5_5) = all_20_0_24, yields: % 6.02/2.01 | (88) all_20_0_24 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_5_5) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_5_5, all_0_5_5) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_2_2) = v0)) % 6.02/2.01 | % 6.02/2.01 +-Applying beta-rule and splitting (48), into two cases. % 6.02/2.01 |-Branch one: % 6.02/2.01 | (89) ~ (all_0_0_0 = 0) & distinct_lines(all_0_4_4, all_0_3_3) = all_0_0_0 % 6.02/2.01 | % 6.02/2.01 | Applying alpha-rule on (89) yields: % 6.02/2.01 | (90) ~ (all_0_0_0 = 0) % 6.02/2.01 | (91) distinct_lines(all_0_4_4, all_0_3_3) = all_0_0_0 % 6.02/2.01 | % 6.02/2.01 +-Applying beta-rule and splitting (86), into two cases. % 6.02/2.01 |-Branch one: % 6.02/2.01 | (92) distinct_lines(all_0_3_3, all_0_4_4) = 0 % 6.02/2.01 | % 6.02/2.01 | Instantiating formula (12) with all_0_0_0, all_0_3_3, all_0_4_4, all_0_3_3 and discharging atoms distinct_lines(all_0_3_3, all_0_4_4) = 0, distinct_lines(all_0_4_4, all_0_3_3) = all_0_0_0, yields: % 6.02/2.01 | (93) all_0_0_0 = 0 | distinct_lines(all_0_3_3, all_0_3_3) = 0 % 6.02/2.01 | % 6.02/2.01 +-Applying beta-rule and splitting (93), into two cases. % 6.02/2.01 |-Branch one: % 6.02/2.01 | (94) distinct_lines(all_0_3_3, all_0_3_3) = 0 % 6.02/2.01 | % 6.02/2.01 | Instantiating formula (26) with all_0_3_3 and discharging atoms distinct_lines(all_0_3_3, all_0_3_3) = 0, yields: % 6.02/2.01 | (95) $false % 6.02/2.01 | % 6.02/2.02 |-The branch is then unsatisfiable % 6.02/2.02 |-Branch two: % 6.02/2.02 | (96) ~ (distinct_lines(all_0_3_3, all_0_3_3) = 0) % 6.02/2.02 | (97) all_0_0_0 = 0 % 6.02/2.02 | % 6.02/2.02 | Equations (97) can reduce 90 to: % 6.02/2.02 | (98) $false % 6.02/2.02 | % 6.02/2.02 |-The branch is then unsatisfiable % 6.02/2.02 |-Branch two: % 6.02/2.02 | (99) ~ (distinct_lines(all_0_3_3, all_0_4_4) = 0) % 6.02/2.02 | (100) all_22_0_26 = 0 % 6.02/2.02 | % 6.02/2.02 | Equations (100) can reduce 63 to: % 6.02/2.02 | (98) $false % 6.02/2.02 | % 6.02/2.02 |-The branch is then unsatisfiable % 6.02/2.02 |-Branch two: % 6.02/2.02 | (102) ~ (all_0_1_1 = 0) & distinct_lines(all_0_5_5, all_0_3_3) = all_0_1_1 % 6.02/2.02 | % 6.02/2.02 | Applying alpha-rule on (102) yields: % 6.02/2.02 | (103) ~ (all_0_1_1 = 0) % 6.02/2.02 | (104) distinct_lines(all_0_5_5, all_0_3_3) = all_0_1_1 % 6.02/2.02 | % 6.02/2.02 +-Applying beta-rule and splitting (88), into two cases. % 6.02/2.02 |-Branch one: % 6.02/2.02 | (105) all_20_0_24 = 0 % 6.02/2.02 | % 6.02/2.02 | Equations (105) can reduce 59 to: % 6.02/2.02 | (98) $false % 6.02/2.02 | % 6.02/2.02 |-The branch is then unsatisfiable % 6.02/2.02 |-Branch two: % 6.02/2.02 | (59) ~ (all_20_0_24 = 0) % 6.02/2.02 | (108) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_5_5) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_5_5, all_0_5_5) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_2_2) = v0)) % 6.02/2.02 | % 6.02/2.02 +-Applying beta-rule and splitting (87), into two cases. % 6.02/2.02 |-Branch one: % 6.02/2.02 | (109) distinct_lines(all_0_3_3, all_0_5_5) = 0 % 6.02/2.02 | % 6.02/2.02 | Instantiating formula (12) with all_0_1_1, all_0_3_3, all_0_5_5, all_0_3_3 and discharging atoms distinct_lines(all_0_3_3, all_0_5_5) = 0, distinct_lines(all_0_5_5, all_0_3_3) = all_0_1_1, yields: % 6.02/2.02 | (110) all_0_1_1 = 0 | distinct_lines(all_0_3_3, all_0_3_3) = 0 % 6.02/2.02 | % 6.02/2.02 +-Applying beta-rule and splitting (110), into two cases. % 6.02/2.02 |-Branch one: % 6.02/2.02 | (94) distinct_lines(all_0_3_3, all_0_3_3) = 0 % 6.02/2.02 | % 6.02/2.02 | Instantiating formula (26) with all_0_3_3 and discharging atoms distinct_lines(all_0_3_3, all_0_3_3) = 0, yields: % 6.02/2.02 | (95) $false % 6.02/2.02 | % 6.02/2.02 |-The branch is then unsatisfiable % 6.02/2.02 |-Branch two: % 6.02/2.02 | (96) ~ (distinct_lines(all_0_3_3, all_0_3_3) = 0) % 6.02/2.02 | (114) all_0_1_1 = 0 % 6.02/2.02 | % 6.02/2.02 | Equations (114) can reduce 103 to: % 6.02/2.02 | (98) $false % 6.02/2.02 | % 6.02/2.02 |-The branch is then unsatisfiable % 6.02/2.02 |-Branch two: % 6.02/2.02 | (116) ~ (distinct_lines(all_0_3_3, all_0_5_5) = 0) % 6.02/2.02 | (105) all_20_0_24 = 0 % 6.02/2.02 | % 6.02/2.02 | Equations (105) can reduce 59 to: % 6.02/2.02 | (98) $false % 6.02/2.02 | % 6.02/2.02 |-The branch is then unsatisfiable % 6.02/2.02 |-Branch two: % 6.02/2.02 | (119) ~ (all_25_0_29 = 0) & convergent_lines(all_0_5_5, all_0_4_4) = all_25_0_29 % 6.02/2.02 | % 6.02/2.02 | Applying alpha-rule on (119) yields: % 6.02/2.02 | (72) ~ (all_25_0_29 = 0) % 6.02/2.02 | (121) convergent_lines(all_0_5_5, all_0_4_4) = all_25_0_29 % 6.02/2.02 | % 6.02/2.02 | Instantiating formula (39) with all_0_5_5, all_0_4_4, all_25_0_29, 0 and discharging atoms convergent_lines(all_0_5_5, all_0_4_4) = all_25_0_29, convergent_lines(all_0_5_5, all_0_4_4) = 0, yields: % 6.02/2.02 | (122) all_25_0_29 = 0 % 6.02/2.02 | % 6.02/2.02 | Equations (122) can reduce 72 to: % 6.02/2.02 | (98) $false % 6.02/2.02 | % 6.02/2.02 |-The branch is then unsatisfiable % 6.02/2.02 |-Branch two: % 6.02/2.02 | (124) ~ (all_24_0_28 = 0) & convergent_lines(all_0_5_5, all_0_4_4) = all_24_0_28 % 6.02/2.02 | % 6.02/2.02 | Applying alpha-rule on (124) yields: % 6.02/2.02 | (69) ~ (all_24_0_28 = 0) % 6.02/2.02 | (126) convergent_lines(all_0_5_5, all_0_4_4) = all_24_0_28 % 6.02/2.02 | % 6.02/2.02 | Instantiating formula (39) with all_0_5_5, all_0_4_4, all_24_0_28, 0 and discharging atoms convergent_lines(all_0_5_5, all_0_4_4) = all_24_0_28, convergent_lines(all_0_5_5, all_0_4_4) = 0, yields: % 6.02/2.02 | (127) all_24_0_28 = 0 % 6.02/2.02 | % 6.02/2.02 | Equations (127) can reduce 69 to: % 6.02/2.02 | (98) $false % 6.02/2.02 | % 6.02/2.02 |-The branch is then unsatisfiable % 6.02/2.02 % SZS output end Proof for theBenchmark % 6.02/2.02 % 6.02/2.02 1492ms %------------------------------------------------------------------------------