%------------------------------------------------------------------------------ % File : ePrincess---1.0 % Problem : GEO200+1 : TPTP v8.1.0. Released v3.3.0. % Transfm : none % Format : tptp:raw % Command : ePrincess-casc -timeout=%d %s % Computer : n021.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 03:48:36 EDT 2022 % Result : Theorem 4.51s 1.78s % Output : Proof 8.61s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.09/0.11 % Problem : GEO200+1 : TPTP v8.1.0. Released v3.3.0. % 0.09/0.12 % Command : ePrincess-casc -timeout=%d %s % 0.12/0.33 % Computer : n021.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 : Fri Jun 17 15:26:45 EDT 2022 % 0.12/0.33 % CPUTime : % 0.18/0.57 ____ _ % 0.18/0.57 ___ / __ \_____(_)___ ________ __________ % 0.18/0.57 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/ % 0.18/0.57 / __/ ____/ / / / / / / /__/ __(__ |__ ) % 0.18/0.57 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/ % 0.18/0.57 % 0.18/0.57 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/sandbox/benchmark/theBenchmark.p ... % 0.69/0.62 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 1.73/0.93 Prover 0: Preprocessing ... % 1.95/1.06 Prover 0: Warning: ignoring some quantifiers % 1.95/1.08 Prover 0: Constructing countermodel ... % 2.60/1.29 Prover 0: gave up % 2.60/1.29 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all % 2.86/1.32 Prover 1: Preprocessing ... % 3.15/1.40 Prover 1: Constructing countermodel ... % 3.49/1.48 Prover 1: gave up % 3.49/1.48 Prover 2: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 3.57/1.50 Prover 2: Preprocessing ... % 4.13/1.62 Prover 2: Warning: ignoring some quantifiers % 4.13/1.63 Prover 2: Constructing countermodel ... % 4.51/1.78 Prover 2: proved (295ms) % 4.51/1.78 % 4.51/1.78 No countermodel exists, formula is valid % 4.51/1.78 % SZS status Theorem for theBenchmark % 4.51/1.78 % 4.51/1.78 Generating proof ... Warning: ignoring some quantifiers % 7.75/2.47 found it (size 152) % 7.75/2.47 % 7.75/2.47 % SZS output start Proof for theBenchmark % 7.75/2.47 Assumed formulas after preprocessing and simplification: % 7.75/2.47 | (0) ? [v0] : ? [v1] : ? [v2] : ? [v3] : (line_connecting(v1, v0) = v3 & line_connecting(v0, v1) = v2 & distinct_lines(v2, v3) = 0 & distinct_points(v0, v1) = 0 & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | v8 = 0 | ~ (apart_point_and_line(v5, v7) = v9) | ~ (apart_point_and_line(v5, v6) = v8) | ~ (distinct_points(v4, v5) = 0) | ? [v10] : ((v10 = 0 & apart_point_and_line(v4, v7) = 0) | (v10 = 0 & apart_point_and_line(v4, v6) = 0) | ( ~ (v10 = 0) & distinct_lines(v6, v7) = v10))) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | v8 = 0 | ~ (apart_point_and_line(v5, v7) = v9) | ~ (apart_point_and_line(v4, v7) = v8) | ~ (distinct_lines(v6, v7) = 0) | ? [v10] : ((v10 = 0 & apart_point_and_line(v5, v6) = 0) | (v10 = 0 & apart_point_and_line(v4, v6) = 0) | ( ~ (v10 = 0) & distinct_points(v4, v5) = v10))) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | v8 = 0 | ~ (apart_point_and_line(v5, v7) = v9) | ~ (apart_point_and_line(v4, v6) = v8) | ? [v10] : ((v10 = 0 & apart_point_and_line(v5, v6) = 0) | (v10 = 0 & apart_point_and_line(v4, v7) = 0) | ( ~ (v10 = 0) & distinct_lines(v6, v7) = v10) | ( ~ (v10 = 0) & distinct_points(v4, v5) = v10))) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | v8 = 0 | ~ (apart_point_and_line(v5, v6) = v9) | ~ (apart_point_and_line(v4, v7) = v8) | ? [v10] : ((v10 = 0 & apart_point_and_line(v5, v7) = 0) | (v10 = 0 & apart_point_and_line(v4, v6) = 0) | ( ~ (v10 = 0) & distinct_lines(v6, v7) = v10) | ( ~ (v10 = 0) & distinct_points(v4, v5) = v10))) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | v8 = 0 | ~ (apart_point_and_line(v5, v6) = v9) | ~ (apart_point_and_line(v4, v6) = v8) | ~ (distinct_lines(v6, v7) = 0) | ? [v10] : ((v10 = 0 & apart_point_and_line(v5, v7) = 0) | (v10 = 0 & apart_point_and_line(v4, v7) = 0) | ( ~ (v10 = 0) & distinct_points(v4, v5) = v10))) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | v8 = 0 | ~ (apart_point_and_line(v4, v7) = v9) | ~ (apart_point_and_line(v4, v6) = v8) | ~ (distinct_points(v4, v5) = 0) | ? [v10] : ((v10 = 0 & apart_point_and_line(v5, v7) = 0) | (v10 = 0 & apart_point_and_line(v5, v6) = 0) | ( ~ (v10 = 0) & distinct_lines(v6, v7) = v10))) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (apart_point_and_line(v6, v5) = v8) | ~ (distinct_points(v4, v6) = v7) | ? [v9] : ( ~ (v9 = 0) & apart_point_and_line(v4, v5) = v9)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (apart_point_and_line(v4, v6) = v8) | ~ (distinct_lines(v5, v6) = v7) | ? [v9] : ( ~ (v9 = 0) & apart_point_and_line(v4, v5) = v9)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (convergent_lines(v5, v6) = v8) | ~ (convergent_lines(v4, v6) = v7) | ? [v9] : ( ~ (v9 = 0) & convergent_lines(v4, v5) = v9)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (convergent_lines(v4, v6) = v8) | ~ (distinct_lines(v5, v6) = v7) | ? [v9] : ( ~ (v9 = 0) & convergent_lines(v4, v5) = v9)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (distinct_lines(v5, v6) = v8) | ~ (distinct_lines(v4, v6) = v7) | ? [v9] : ( ~ (v9 = 0) & distinct_lines(v4, v5) = v9)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : (v8 = 0 | v7 = 0 | ~ (distinct_points(v5, v6) = v8) | ~ (distinct_points(v4, v6) = v7) | ? [v9] : ( ~ (v9 = 0) & distinct_points(v4, v5) = v9)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (apart_point_and_line(v6, v5) = v7) | ~ (apart_point_and_line(v4, v5) = 0) | distinct_points(v4, v6) = 0) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (apart_point_and_line(v4, v6) = v7) | ~ (apart_point_and_line(v4, v5) = 0) | distinct_lines(v5, v6) = 0) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (apart_point_and_line(v4, v5) = 0) | ~ (distinct_lines(v5, v6) = v7) | apart_point_and_line(v4, v6) = 0) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (apart_point_and_line(v4, v5) = 0) | ~ (distinct_points(v4, v6) = v7) | apart_point_and_line(v6, v5) = 0) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (convergent_lines(v5, v6) = v7) | ~ (convergent_lines(v4, v5) = 0) | convergent_lines(v4, v6) = 0) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (convergent_lines(v4, v6) = v7) | ~ (convergent_lines(v4, v5) = 0) | convergent_lines(v5, v6) = 0) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (convergent_lines(v4, v6) = v7) | ~ (convergent_lines(v4, v5) = 0) | distinct_lines(v5, v6) = 0) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (convergent_lines(v4, v5) = 0) | ~ (distinct_lines(v5, v6) = v7) | convergent_lines(v4, v6) = 0) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (distinct_lines(v5, v6) = v7) | ~ (distinct_lines(v4, v5) = 0) | distinct_lines(v4, v6) = 0) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (distinct_lines(v4, v6) = v7) | ~ (distinct_lines(v4, v5) = 0) | distinct_lines(v5, v6) = 0) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (distinct_points(v5, v6) = v7) | ~ (distinct_points(v4, v5) = 0) | distinct_points(v4, v6) = 0) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = 0 | ~ (distinct_points(v4, v6) = v7) | ~ (distinct_points(v4, v5) = 0) | distinct_points(v5, v6) = 0) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (intersection_point(v7, v6) = v5) | ~ (intersection_point(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (line_connecting(v7, v6) = v5) | ~ (line_connecting(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (apart_point_and_line(v7, v6) = v5) | ~ (apart_point_and_line(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (convergent_lines(v7, v6) = v5) | ~ (convergent_lines(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (distinct_lines(v7, v6) = v5) | ~ (distinct_lines(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v5 = v4 | ~ (distinct_points(v7, v6) = v5) | ~ (distinct_points(v7, v6) = v4)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (distinct_lines(v6, v7) = 0) | ~ (distinct_points(v4, v5) = 0) | ? [v8] : ((v8 = 0 & apart_point_and_line(v5, v7) = 0) | (v8 = 0 & apart_point_and_line(v5, v6) = 0) | (v8 = 0 & apart_point_and_line(v4, v7) = 0) | (v8 = 0 & apart_point_and_line(v4, v6) = 0))) & ! [v4] : ! [v5] : ! [v6] : ( ~ (intersection_point(v4, v5) = v6) | ? [v7] : (( ~ (v7 = 0) & apart_point_and_line(v6, v5) = v7) | ( ~ (v7 = 0) & convergent_lines(v4, v5) = v7))) & ! [v4] : ! [v5] : ! [v6] : ( ~ (intersection_point(v4, v5) = v6) | ? [v7] : (( ~ (v7 = 0) & apart_point_and_line(v6, v4) = v7) | ( ~ (v7 = 0) & convergent_lines(v4, v5) = v7))) & ! [v4] : ! [v5] : ! [v6] : ( ~ (line_connecting(v4, v5) = v6) | ? [v7] : (( ~ (v7 = 0) & apart_point_and_line(v5, v6) = v7) | ( ~ (v7 = 0) & distinct_points(v4, v5) = v7))) & ! [v4] : ! [v5] : ! [v6] : ( ~ (line_connecting(v4, v5) = v6) | ? [v7] : (( ~ (v7 = 0) & apart_point_and_line(v4, v6) = v7) | ( ~ (v7 = 0) & distinct_points(v4, v5) = v7))) & ! [v4] : ! [v5] : ( ~ (convergent_lines(v4, v5) = 0) | ? [v6] : ? [v7] : ( ~ (v7 = 0) & intersection_point(v4, v5) = v6 & apart_point_and_line(v6, v5) = v7)) & ! [v4] : ! [v5] : ( ~ (convergent_lines(v4, v5) = 0) | ? [v6] : ? [v7] : ( ~ (v7 = 0) & intersection_point(v4, v5) = v6 & apart_point_and_line(v6, v4) = v7)) & ! [v4] : ! [v5] : ( ~ (distinct_points(v4, v5) = 0) | ? [v6] : ? [v7] : ( ~ (v7 = 0) & line_connecting(v4, v5) = v6 & apart_point_and_line(v5, v6) = v7)) & ! [v4] : ! [v5] : ( ~ (distinct_points(v4, v5) = 0) | ? [v6] : ? [v7] : ( ~ (v7 = 0) & line_connecting(v4, v5) = v6 & apart_point_and_line(v4, v6) = v7)) & ! [v4] : ~ (convergent_lines(v4, v4) = 0) & ! [v4] : ~ (distinct_lines(v4, v4) = 0) & ! [v4] : ~ (distinct_points(v4, v4) = 0) & ? [v4] : ? [v5] : ? [v6] : intersection_point(v5, v4) = v6 & ? [v4] : ? [v5] : ? [v6] : line_connecting(v5, v4) = v6 & ? [v4] : ? [v5] : ? [v6] : apart_point_and_line(v5, v4) = v6 & ? [v4] : ? [v5] : ? [v6] : convergent_lines(v5, v4) = v6 & ? [v4] : ? [v5] : ? [v6] : distinct_lines(v5, v4) = v6 & ? [v4] : ? [v5] : ? [v6] : distinct_points(v5, v4) = v6) % 8.17/2.51 | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3 yields: % 8.17/2.51 | (1) line_connecting(all_0_2_2, all_0_3_3) = all_0_0_0 & line_connecting(all_0_3_3, all_0_2_2) = all_0_1_1 & distinct_lines(all_0_1_1, all_0_0_0) = 0 & distinct_points(all_0_3_3, all_0_2_2) = 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 % 8.17/2.53 | % 8.17/2.53 | Applying alpha-rule on (1) yields: % 8.17/2.53 | (2) ! [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))) % 8.17/2.53 | (3) ! [v0] : ~ (distinct_points(v0, v0) = 0) % 8.17/2.53 | (4) ? [v0] : ? [v1] : ? [v2] : convergent_lines(v1, v0) = v2 % 8.17/2.53 | (5) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v3) = v5) | ~ (apart_point_and_line(v0, v3) = v4) | ~ (distinct_lines(v2, v3) = 0) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v2) = 0) | (v6 = 0 & apart_point_and_line(v0, v2) = 0) | ( ~ (v6 = 0) & distinct_points(v0, v1) = v6))) % 8.17/2.53 | (6) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (convergent_lines(v1, v2) = v4) | ~ (convergent_lines(v0, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & convergent_lines(v0, v1) = v5)) % 8.17/2.53 | (7) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (convergent_lines(v3, v2) = v1) | ~ (convergent_lines(v3, v2) = v0)) % 8.17/2.53 | (8) ! [v0] : ! [v1] : ( ~ (distinct_points(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & line_connecting(v0, v1) = v2 & apart_point_and_line(v1, v2) = v3)) % 8.17/2.53 | (9) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v1, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v0, v2) = 0) % 8.17/2.53 | (10) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v3) = v5) | ~ (apart_point_and_line(v1, v2) = v4) | ~ (distinct_points(v0, v1) = 0) | ? [v6] : ((v6 = 0 & apart_point_and_line(v0, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v2) = 0) | ( ~ (v6 = 0) & distinct_lines(v2, v3) = v6))) % 8.17/2.53 | (11) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_points(v3, v2) = v1) | ~ (distinct_points(v3, v2) = v0)) % 8.17/2.53 | (12) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v1) = 0) | ~ (distinct_lines(v1, v2) = v3) | convergent_lines(v0, v2) = 0) % 8.17/2.53 | (13) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (intersection_point(v3, v2) = v1) | ~ (intersection_point(v3, v2) = v0)) % 8.17/2.53 | (14) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v3) = v5) | ~ (apart_point_and_line(v0, v2) = v4) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v2) = 0) | (v6 = 0 & apart_point_and_line(v0, v3) = 0) | ( ~ (v6 = 0) & distinct_lines(v2, v3) = v6) | ( ~ (v6 = 0) & distinct_points(v0, v1) = v6))) % 8.17/2.53 | (15) line_connecting(all_0_2_2, all_0_3_3) = all_0_0_0 % 8.17/2.53 | (16) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v2) = v3) | ~ (apart_point_and_line(v0, v1) = 0) | distinct_lines(v1, v2) = 0) % 8.17/2.53 | (17) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v1, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v0, v2) = 0) % 8.17/2.53 | (18) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v1) = 0) | ~ (distinct_lines(v1, v2) = v3) | apart_point_and_line(v0, v2) = 0) % 8.17/2.53 | (19) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v1, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v0, v2) = 0) % 8.17/2.53 | (20) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v1, v2) = 0) % 8.17/2.54 | (21) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (line_connecting(v3, v2) = v1) | ~ (line_connecting(v3, v2) = v0)) % 8.17/2.54 | (22) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (apart_point_and_line(v3, v2) = v1) | ~ (apart_point_and_line(v3, v2) = v0)) % 8.17/2.54 | (23) ? [v0] : ? [v1] : ? [v2] : intersection_point(v1, v0) = v2 % 8.17/2.54 | (24) ! [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))) % 8.17/2.54 | (25) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (distinct_lines(v2, v3) = 0) | ~ (distinct_points(v0, v1) = 0) | ? [v4] : ((v4 = 0 & apart_point_and_line(v1, v3) = 0) | (v4 = 0 & apart_point_and_line(v1, v2) = 0) | (v4 = 0 & apart_point_and_line(v0, v3) = 0) | (v4 = 0 & apart_point_and_line(v0, v2) = 0))) % 8.17/2.54 | (26) ? [v0] : ? [v1] : ? [v2] : distinct_lines(v1, v0) = v2 % 8.17/2.54 | (27) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v0, v1) = 0) | ~ (distinct_points(v0, v2) = v3) | apart_point_and_line(v2, v1) = 0) % 8.17/2.54 | (28) ? [v0] : ? [v1] : ? [v2] : apart_point_and_line(v1, v0) = v2 % 8.17/2.54 | (29) ? [v0] : ? [v1] : ? [v2] : line_connecting(v1, v0) = v2 % 8.17/2.54 | (30) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v0, v3) = v5) | ~ (apart_point_and_line(v0, v2) = v4) | ~ (distinct_points(v0, v1) = 0) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v3) = 0) | (v6 = 0 & apart_point_and_line(v1, v2) = 0) | ( ~ (v6 = 0) & distinct_lines(v2, v3) = v6))) % 8.17/2.54 | (31) line_connecting(all_0_3_3, all_0_2_2) = all_0_1_1 % 8.17/2.54 | (32) distinct_points(all_0_3_3, all_0_2_2) = 0 % 8.17/2.54 | (33) ! [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)) % 8.17/2.54 | (34) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v2) = v5) | ~ (apart_point_and_line(v0, v3) = v4) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v2) = 0) | ( ~ (v6 = 0) & distinct_lines(v2, v3) = v6) | ( ~ (v6 = 0) & distinct_points(v0, v1) = v6))) % 8.17/2.54 | (35) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (distinct_points(v1, v2) = v4) | ~ (distinct_points(v0, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & distinct_points(v0, v1) = v5)) % 8.17/2.54 | (36) ! [v0] : ! [v1] : ( ~ (convergent_lines(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & intersection_point(v0, v1) = v2 & apart_point_and_line(v2, v0) = v3)) % 8.17/2.54 | (37) ? [v0] : ? [v1] : ? [v2] : distinct_points(v1, v0) = v2 % 8.17/2.54 | (38) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (distinct_lines(v1, v2) = v4) | ~ (distinct_lines(v0, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & distinct_lines(v0, v1) = v5)) % 8.17/2.54 | (39) ! [v0] : ! [v1] : ( ~ (convergent_lines(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & intersection_point(v0, v1) = v2 & apart_point_and_line(v2, v1) = v3)) % 8.17/2.54 | (40) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | v4 = 0 | ~ (apart_point_and_line(v1, v2) = v5) | ~ (apart_point_and_line(v0, v2) = v4) | ~ (distinct_lines(v2, v3) = 0) | ? [v6] : ((v6 = 0 & apart_point_and_line(v1, v3) = 0) | (v6 = 0 & apart_point_and_line(v0, v3) = 0) | ( ~ (v6 = 0) & distinct_points(v0, v1) = v6))) % 8.17/2.54 | (41) ! [v0] : ~ (distinct_lines(v0, v0) = 0) % 8.17/2.54 | (42) ! [v0] : ~ (convergent_lines(v0, v0) = 0) % 8.17/2.54 | (43) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (apart_point_and_line(v2, v1) = v4) | ~ (distinct_points(v0, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & apart_point_and_line(v0, v1) = v5)) % 8.17/2.54 | (44) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (apart_point_and_line(v2, v1) = v3) | ~ (apart_point_and_line(v0, v1) = 0) | distinct_points(v0, v2) = 0) % 8.17/2.54 | (45) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | v3 = 0 | ~ (apart_point_and_line(v0, v2) = v4) | ~ (distinct_lines(v1, v2) = v3) | ? [v5] : ( ~ (v5 = 0) & apart_point_and_line(v0, v1) = v5)) % 8.17/2.55 | (46) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v0, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0) % 8.17/2.55 | (47) ! [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))) % 8.17/2.55 | (48) ! [v0] : ! [v1] : ( ~ (distinct_points(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & line_connecting(v0, v1) = v2 & apart_point_and_line(v0, v2) = v3)) % 8.17/2.55 | (49) ! [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))) % 8.17/2.55 | (50) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0) % 8.17/2.55 | (51) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v0, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v1, v2) = 0) % 8.17/2.55 | (52) distinct_lines(all_0_1_1, all_0_0_0) = 0 % 8.17/2.55 | (53) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_lines(v3, v2) = v1) | ~ (distinct_lines(v3, v2) = v0)) % 8.17/2.55 | % 8.17/2.55 | Instantiating formula (47) with all_0_0_0, all_0_3_3, all_0_2_2 and discharging atoms line_connecting(all_0_2_2, all_0_3_3) = all_0_0_0, yields: % 8.17/2.55 | (54) ? [v0] : (( ~ (v0 = 0) & apart_point_and_line(all_0_3_3, all_0_0_0) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_3_3) = v0)) % 8.17/2.55 | % 8.17/2.55 | Instantiating formula (2) with all_0_0_0, all_0_3_3, all_0_2_2 and discharging atoms line_connecting(all_0_2_2, all_0_3_3) = all_0_0_0, yields: % 8.17/2.55 | (55) ? [v0] : (( ~ (v0 = 0) & apart_point_and_line(all_0_2_2, all_0_0_0) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_3_3) = v0)) % 8.17/2.55 | % 8.17/2.55 | Instantiating formula (47) with all_0_1_1, all_0_2_2, all_0_3_3 and discharging atoms line_connecting(all_0_3_3, all_0_2_2) = all_0_1_1, yields: % 8.17/2.55 | (56) ? [v0] : (( ~ (v0 = 0) & apart_point_and_line(all_0_2_2, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_2_2) = v0)) % 8.17/2.55 | % 8.17/2.55 | Instantiating formula (2) with all_0_1_1, all_0_2_2, all_0_3_3 and discharging atoms line_connecting(all_0_3_3, all_0_2_2) = all_0_1_1, yields: % 8.17/2.55 | (57) ? [v0] : (( ~ (v0 = 0) & apart_point_and_line(all_0_3_3, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_2_2) = v0)) % 8.17/2.55 | % 8.17/2.55 | Instantiating formula (25) with all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3 and discharging atoms distinct_lines(all_0_1_1, all_0_0_0) = 0, distinct_points(all_0_3_3, all_0_2_2) = 0, yields: % 8.17/2.55 | (58) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | (v0 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_0_0) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0)) % 8.17/2.55 | % 8.17/2.55 | Instantiating formula (8) with all_0_2_2, all_0_3_3 and discharging atoms distinct_points(all_0_3_3, all_0_2_2) = 0, yields: % 8.17/2.55 | (59) ? [v0] : ? [v1] : ( ~ (v1 = 0) & line_connecting(all_0_3_3, all_0_2_2) = v0 & apart_point_and_line(all_0_2_2, v0) = v1) % 8.17/2.55 | % 8.17/2.55 | Instantiating formula (48) with all_0_2_2, all_0_3_3 and discharging atoms distinct_points(all_0_3_3, all_0_2_2) = 0, yields: % 8.17/2.55 | (60) ? [v0] : ? [v1] : ( ~ (v1 = 0) & line_connecting(all_0_3_3, all_0_2_2) = v0 & apart_point_and_line(all_0_3_3, v0) = v1) % 8.17/2.55 | % 8.17/2.55 | Instantiating (60) with all_20_0_22, all_20_1_23 yields: % 8.17/2.55 | (61) ~ (all_20_0_22 = 0) & line_connecting(all_0_3_3, all_0_2_2) = all_20_1_23 & apart_point_and_line(all_0_3_3, all_20_1_23) = all_20_0_22 % 8.17/2.55 | % 8.17/2.55 | Applying alpha-rule on (61) yields: % 8.17/2.55 | (62) ~ (all_20_0_22 = 0) % 8.17/2.55 | (63) line_connecting(all_0_3_3, all_0_2_2) = all_20_1_23 % 8.17/2.55 | (64) apart_point_and_line(all_0_3_3, all_20_1_23) = all_20_0_22 % 8.17/2.55 | % 8.17/2.55 | Instantiating (57) with all_22_0_24 yields: % 8.17/2.55 | (65) ( ~ (all_22_0_24 = 0) & apart_point_and_line(all_0_3_3, all_0_1_1) = all_22_0_24) | ( ~ (all_22_0_24 = 0) & distinct_points(all_0_3_3, all_0_2_2) = all_22_0_24) % 8.17/2.55 | % 8.17/2.55 | Instantiating (55) with all_23_0_25 yields: % 8.17/2.55 | (66) ( ~ (all_23_0_25 = 0) & apart_point_and_line(all_0_2_2, all_0_0_0) = all_23_0_25) | ( ~ (all_23_0_25 = 0) & distinct_points(all_0_2_2, all_0_3_3) = all_23_0_25) % 8.17/2.55 | % 8.17/2.55 | Instantiating (54) with all_24_0_26 yields: % 8.17/2.55 | (67) ( ~ (all_24_0_26 = 0) & apart_point_and_line(all_0_3_3, all_0_0_0) = all_24_0_26) | ( ~ (all_24_0_26 = 0) & distinct_points(all_0_2_2, all_0_3_3) = all_24_0_26) % 8.17/2.55 | % 8.17/2.55 | Instantiating (56) with all_25_0_27 yields: % 8.17/2.55 | (68) ( ~ (all_25_0_27 = 0) & apart_point_and_line(all_0_2_2, all_0_1_1) = all_25_0_27) | ( ~ (all_25_0_27 = 0) & distinct_points(all_0_3_3, all_0_2_2) = all_25_0_27) % 8.17/2.55 | % 8.17/2.55 | Instantiating (59) with all_26_0_28, all_26_1_29 yields: % 8.17/2.55 | (69) ~ (all_26_0_28 = 0) & line_connecting(all_0_3_3, all_0_2_2) = all_26_1_29 & apart_point_and_line(all_0_2_2, all_26_1_29) = all_26_0_28 % 8.17/2.56 | % 8.17/2.56 | Applying alpha-rule on (69) yields: % 8.17/2.56 | (70) ~ (all_26_0_28 = 0) % 8.17/2.56 | (71) line_connecting(all_0_3_3, all_0_2_2) = all_26_1_29 % 8.17/2.56 | (72) apart_point_and_line(all_0_2_2, all_26_1_29) = all_26_0_28 % 8.17/2.56 | % 8.17/2.56 | Instantiating (58) with all_28_0_30 yields: % 8.17/2.56 | (73) (all_28_0_30 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | (all_28_0_30 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | (all_28_0_30 = 0 & apart_point_and_line(all_0_3_3, all_0_0_0) = 0) | (all_28_0_30 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0) % 8.17/2.56 | % 8.17/2.56 +-Applying beta-rule and splitting (65), into two cases. % 8.17/2.56 |-Branch one: % 8.17/2.56 | (74) ~ (all_22_0_24 = 0) & apart_point_and_line(all_0_3_3, all_0_1_1) = all_22_0_24 % 8.17/2.56 | % 8.17/2.56 | Applying alpha-rule on (74) yields: % 8.17/2.56 | (75) ~ (all_22_0_24 = 0) % 8.17/2.56 | (76) apart_point_and_line(all_0_3_3, all_0_1_1) = all_22_0_24 % 8.17/2.56 | % 8.17/2.56 +-Applying beta-rule and splitting (68), into two cases. % 8.17/2.56 |-Branch one: % 8.17/2.56 | (77) ~ (all_25_0_27 = 0) & apart_point_and_line(all_0_2_2, all_0_1_1) = all_25_0_27 % 8.17/2.56 | % 8.17/2.56 | Applying alpha-rule on (77) yields: % 8.17/2.56 | (78) ~ (all_25_0_27 = 0) % 8.17/2.56 | (79) apart_point_and_line(all_0_2_2, all_0_1_1) = all_25_0_27 % 8.17/2.56 | % 8.17/2.56 | Instantiating formula (21) with all_0_3_3, all_0_2_2, all_26_1_29, all_0_1_1 and discharging atoms line_connecting(all_0_3_3, all_0_2_2) = all_26_1_29, line_connecting(all_0_3_3, all_0_2_2) = all_0_1_1, yields: % 8.17/2.56 | (80) all_26_1_29 = all_0_1_1 % 8.17/2.56 | % 8.17/2.56 | Instantiating formula (21) with all_0_3_3, all_0_2_2, all_20_1_23, all_26_1_29 and discharging atoms line_connecting(all_0_3_3, all_0_2_2) = all_26_1_29, line_connecting(all_0_3_3, all_0_2_2) = all_20_1_23, yields: % 8.17/2.56 | (81) all_26_1_29 = all_20_1_23 % 8.17/2.56 | % 8.17/2.56 | Combining equations (80,81) yields a new equation: % 8.17/2.56 | (82) all_20_1_23 = all_0_1_1 % 8.17/2.56 | % 8.17/2.56 | Combining equations (82,81) yields a new equation: % 8.17/2.56 | (80) all_26_1_29 = all_0_1_1 % 8.17/2.56 | % 8.17/2.56 | From (80) and (72) follows: % 8.17/2.56 | (84) apart_point_and_line(all_0_2_2, all_0_1_1) = all_26_0_28 % 8.17/2.56 | % 8.17/2.56 | From (82) and (64) follows: % 8.17/2.56 | (85) apart_point_and_line(all_0_3_3, all_0_1_1) = all_20_0_22 % 8.17/2.56 | % 8.17/2.56 | Instantiating formula (22) with all_0_2_2, all_0_1_1, all_26_0_28, all_25_0_27 and discharging atoms apart_point_and_line(all_0_2_2, all_0_1_1) = all_26_0_28, apart_point_and_line(all_0_2_2, all_0_1_1) = all_25_0_27, yields: % 8.17/2.56 | (86) all_26_0_28 = all_25_0_27 % 8.17/2.56 | % 8.17/2.56 | Instantiating formula (22) with all_0_3_3, all_0_1_1, all_20_0_22, all_22_0_24 and discharging atoms apart_point_and_line(all_0_3_3, all_0_1_1) = all_22_0_24, apart_point_and_line(all_0_3_3, all_0_1_1) = all_20_0_22, yields: % 8.17/2.56 | (87) all_22_0_24 = all_20_0_22 % 8.17/2.56 | % 8.17/2.56 | Equations (86) can reduce 70 to: % 8.17/2.56 | (78) ~ (all_25_0_27 = 0) % 8.17/2.56 | % 8.17/2.56 | Equations (87) can reduce 75 to: % 8.17/2.56 | (62) ~ (all_20_0_22 = 0) % 8.17/2.56 | % 8.17/2.56 | From (86) and (84) follows: % 8.17/2.56 | (79) apart_point_and_line(all_0_2_2, all_0_1_1) = all_25_0_27 % 8.17/2.56 | % 8.17/2.56 | From (87) and (76) follows: % 8.17/2.56 | (85) apart_point_and_line(all_0_3_3, all_0_1_1) = all_20_0_22 % 8.17/2.56 | % 8.17/2.56 | Instantiating formula (10) with all_25_0_27, all_25_0_27, all_0_1_1, all_0_1_1, all_0_2_2, all_0_3_3 and discharging atoms apart_point_and_line(all_0_2_2, all_0_1_1) = all_25_0_27, distinct_points(all_0_3_3, all_0_2_2) = 0, yields: % 8.17/2.56 | (92) all_25_0_27 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0)) % 8.17/2.56 | % 8.17/2.56 | Instantiating formula (40) with all_25_0_27, all_25_0_27, all_0_0_0, all_0_1_1, all_0_2_2, all_0_2_2 and discharging atoms apart_point_and_line(all_0_2_2, all_0_1_1) = all_25_0_27, distinct_lines(all_0_1_1, all_0_0_0) = 0, yields: % 8.17/2.56 | (93) all_25_0_27 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_2_2) = v0)) % 8.17/2.56 | % 8.17/2.56 | Instantiating formula (14) with all_25_0_27, all_25_0_27, all_0_1_1, all_0_1_1, all_0_2_2, all_0_2_2 and discharging atoms apart_point_and_line(all_0_2_2, all_0_1_1) = all_25_0_27, yields: % 8.17/2.56 | (94) all_25_0_27 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_2_2) = v0)) % 8.17/2.56 | % 8.17/2.56 | Instantiating formula (40) with all_20_0_22, all_25_0_27, all_0_0_0, all_0_1_1, all_0_3_3, all_0_2_2 and discharging atoms apart_point_and_line(all_0_2_2, all_0_1_1) = all_25_0_27, apart_point_and_line(all_0_3_3, all_0_1_1) = all_20_0_22, distinct_lines(all_0_1_1, all_0_0_0) = 0, yields: % 8.17/2.56 | (95) all_25_0_27 = 0 | all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_0_0) = 0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_3_3) = v0)) % 8.17/2.56 | % 8.17/2.56 | Instantiating formula (40) with all_25_0_27, all_20_0_22, all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3 and discharging atoms apart_point_and_line(all_0_2_2, all_0_1_1) = all_25_0_27, apart_point_and_line(all_0_3_3, all_0_1_1) = all_20_0_22, distinct_lines(all_0_1_1, all_0_0_0) = 0, yields: % 8.17/2.56 | (96) all_25_0_27 = 0 | all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_0_0) = 0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_2_2) = v0)) % 8.17/2.57 | % 8.17/2.57 | Instantiating formula (14) with all_20_0_22, all_25_0_27, all_0_1_1, all_0_1_1, all_0_3_3, all_0_2_2 and discharging atoms apart_point_and_line(all_0_2_2, all_0_1_1) = all_25_0_27, apart_point_and_line(all_0_3_3, all_0_1_1) = all_20_0_22, yields: % 8.17/2.57 | (97) all_25_0_27 = 0 | all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_3_3) = v0)) % 8.17/2.57 | % 8.17/2.57 | Instantiating formula (14) with all_25_0_27, all_20_0_22, all_0_1_1, all_0_1_1, all_0_2_2, all_0_3_3 and discharging atoms apart_point_and_line(all_0_2_2, all_0_1_1) = all_25_0_27, apart_point_and_line(all_0_3_3, all_0_1_1) = all_20_0_22, yields: % 8.17/2.57 | (98) all_25_0_27 = 0 | all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_2_2) = v0)) % 8.17/2.57 | % 8.17/2.57 | Instantiating formula (30) with all_20_0_22, all_20_0_22, all_0_1_1, all_0_1_1, all_0_2_2, all_0_3_3 and discharging atoms apart_point_and_line(all_0_3_3, all_0_1_1) = all_20_0_22, distinct_points(all_0_3_3, all_0_2_2) = 0, yields: % 8.17/2.57 | (99) all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0)) % 8.17/2.57 | % 8.17/2.57 | Instantiating formula (40) with all_20_0_22, all_20_0_22, all_0_0_0, all_0_1_1, all_0_3_3, all_0_3_3 and discharging atoms apart_point_and_line(all_0_3_3, all_0_1_1) = all_20_0_22, distinct_lines(all_0_1_1, all_0_0_0) = 0, yields: % 8.17/2.57 | (100) all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_3_3, all_0_0_0) = 0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_3_3) = v0)) % 8.17/2.57 | % 8.17/2.57 | Instantiating formula (14) with all_20_0_22, all_20_0_22, all_0_1_1, all_0_1_1, all_0_3_3, all_0_3_3 and discharging atoms apart_point_and_line(all_0_3_3, all_0_1_1) = all_20_0_22, yields: % 8.17/2.57 | (101) all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_3_3) = v0)) % 8.17/2.57 | % 8.17/2.57 +-Applying beta-rule and splitting (66), into two cases. % 8.17/2.57 |-Branch one: % 8.17/2.57 | (102) ~ (all_23_0_25 = 0) & apart_point_and_line(all_0_2_2, all_0_0_0) = all_23_0_25 % 8.17/2.57 | % 8.17/2.57 | Applying alpha-rule on (102) yields: % 8.17/2.57 | (103) ~ (all_23_0_25 = 0) % 8.17/2.57 | (104) apart_point_and_line(all_0_2_2, all_0_0_0) = all_23_0_25 % 8.17/2.57 | % 8.17/2.57 +-Applying beta-rule and splitting (73), into two cases. % 8.17/2.57 |-Branch one: % 8.17/2.57 | (105) (all_28_0_30 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | (all_28_0_30 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | (all_28_0_30 = 0 & apart_point_and_line(all_0_3_3, all_0_0_0) = 0) % 8.17/2.57 | % 8.17/2.57 +-Applying beta-rule and splitting (105), into two cases. % 8.17/2.57 |-Branch one: % 8.17/2.57 | (106) (all_28_0_30 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | (all_28_0_30 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) % 8.17/2.57 | % 8.17/2.57 +-Applying beta-rule and splitting (106), into two cases. % 8.17/2.57 |-Branch one: % 8.17/2.57 | (107) all_28_0_30 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0 % 8.17/2.57 | % 8.17/2.57 | Applying alpha-rule on (107) yields: % 8.17/2.57 | (108) all_28_0_30 = 0 % 8.17/2.57 | (109) apart_point_and_line(all_0_2_2, all_0_0_0) = 0 % 8.17/2.57 | % 8.17/2.57 | Instantiating formula (22) with all_0_2_2, all_0_0_0, 0, all_23_0_25 and discharging atoms apart_point_and_line(all_0_2_2, all_0_0_0) = all_23_0_25, apart_point_and_line(all_0_2_2, all_0_0_0) = 0, yields: % 8.17/2.57 | (110) all_23_0_25 = 0 % 8.17/2.57 | % 8.17/2.57 | Equations (110) can reduce 103 to: % 8.17/2.57 | (111) $false % 8.17/2.57 | % 8.17/2.57 |-The branch is then unsatisfiable % 8.17/2.57 |-Branch two: % 8.17/2.57 | (112) all_28_0_30 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0 % 8.17/2.57 | % 8.17/2.57 | Applying alpha-rule on (112) yields: % 8.17/2.57 | (108) all_28_0_30 = 0 % 8.17/2.57 | (114) apart_point_and_line(all_0_2_2, all_0_1_1) = 0 % 8.17/2.57 | % 8.17/2.57 +-Applying beta-rule and splitting (96), into two cases. % 8.17/2.57 |-Branch one: % 8.17/2.57 | (115) all_25_0_27 = 0 % 8.17/2.57 | % 8.17/2.57 | Equations (115) can reduce 78 to: % 8.17/2.57 | (111) $false % 8.17/2.57 | % 8.17/2.57 |-The branch is then unsatisfiable % 8.17/2.57 |-Branch two: % 8.17/2.57 | (78) ~ (all_25_0_27 = 0) % 8.17/2.57 | (118) all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_0_0) = 0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_2_2) = v0)) % 8.17/2.57 | % 8.17/2.57 +-Applying beta-rule and splitting (92), into two cases. % 8.17/2.57 |-Branch one: % 8.17/2.57 | (115) all_25_0_27 = 0 % 8.17/2.57 | % 8.17/2.57 | Equations (115) can reduce 78 to: % 8.17/2.57 | (111) $false % 8.17/2.57 | % 8.17/2.57 |-The branch is then unsatisfiable % 8.17/2.57 |-Branch two: % 8.17/2.58 | (78) ~ (all_25_0_27 = 0) % 8.17/2.58 | (122) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0)) % 8.17/2.58 | % 8.17/2.58 +-Applying beta-rule and splitting (93), into two cases. % 8.17/2.58 |-Branch one: % 8.17/2.58 | (115) all_25_0_27 = 0 % 8.17/2.58 | % 8.17/2.58 | Equations (115) can reduce 78 to: % 8.17/2.58 | (111) $false % 8.17/2.58 | % 8.17/2.58 |-The branch is then unsatisfiable % 8.17/2.58 |-Branch two: % 8.17/2.58 | (78) ~ (all_25_0_27 = 0) % 8.17/2.58 | (126) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_2_2) = v0)) % 8.17/2.58 | % 8.17/2.58 +-Applying beta-rule and splitting (94), into two cases. % 8.17/2.58 |-Branch one: % 8.17/2.58 | (115) all_25_0_27 = 0 % 8.17/2.58 | % 8.17/2.58 | Equations (115) can reduce 78 to: % 8.17/2.58 | (111) $false % 8.17/2.58 | % 8.17/2.58 |-The branch is then unsatisfiable % 8.17/2.58 |-Branch two: % 8.17/2.58 | (78) ~ (all_25_0_27 = 0) % 8.17/2.58 | (130) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_2_2) = v0)) % 8.17/2.58 | % 8.17/2.58 +-Applying beta-rule and splitting (97), into two cases. % 8.17/2.58 |-Branch one: % 8.17/2.58 | (115) all_25_0_27 = 0 % 8.17/2.58 | % 8.17/2.58 | Equations (115) can reduce 78 to: % 8.17/2.58 | (111) $false % 8.17/2.58 | % 8.17/2.58 |-The branch is then unsatisfiable % 8.17/2.58 |-Branch two: % 8.17/2.58 | (78) ~ (all_25_0_27 = 0) % 8.17/2.58 | (134) all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_3_3) = v0)) % 8.17/2.58 | % 8.17/2.58 +-Applying beta-rule and splitting (98), into two cases. % 8.17/2.58 |-Branch one: % 8.17/2.58 | (115) all_25_0_27 = 0 % 8.17/2.58 | % 8.17/2.58 | Equations (115) can reduce 78 to: % 8.17/2.58 | (111) $false % 8.17/2.58 | % 8.17/2.58 |-The branch is then unsatisfiable % 8.17/2.58 |-Branch two: % 8.17/2.58 | (78) ~ (all_25_0_27 = 0) % 8.17/2.58 | (138) all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_2_2) = v0)) % 8.17/2.58 | % 8.17/2.58 +-Applying beta-rule and splitting (95), into two cases. % 8.17/2.58 |-Branch one: % 8.17/2.58 | (115) all_25_0_27 = 0 % 8.17/2.58 | % 8.17/2.58 | Equations (115) can reduce 78 to: % 8.17/2.58 | (111) $false % 8.17/2.58 | % 8.17/2.58 |-The branch is then unsatisfiable % 8.17/2.58 |-Branch two: % 8.17/2.58 | (78) ~ (all_25_0_27 = 0) % 8.17/2.58 | (142) all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_0_0) = 0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_3_3) = v0)) % 8.17/2.58 | % 8.17/2.58 | Instantiating formula (22) with all_0_2_2, all_0_1_1, 0, all_25_0_27 and discharging atoms apart_point_and_line(all_0_2_2, all_0_1_1) = all_25_0_27, apart_point_and_line(all_0_2_2, all_0_1_1) = 0, yields: % 8.17/2.58 | (115) all_25_0_27 = 0 % 8.17/2.58 | % 8.17/2.58 | Equations (115) can reduce 78 to: % 8.17/2.58 | (111) $false % 8.17/2.58 | % 8.17/2.58 |-The branch is then unsatisfiable % 8.17/2.58 |-Branch two: % 8.17/2.58 | (145) all_28_0_30 = 0 & apart_point_and_line(all_0_3_3, all_0_0_0) = 0 % 8.17/2.58 | % 8.17/2.58 | Applying alpha-rule on (145) yields: % 8.17/2.58 | (108) all_28_0_30 = 0 % 8.17/2.58 | (147) apart_point_and_line(all_0_3_3, all_0_0_0) = 0 % 8.17/2.58 | % 8.17/2.58 +-Applying beta-rule and splitting (67), into two cases. % 8.17/2.58 |-Branch one: % 8.17/2.58 | (148) ~ (all_24_0_26 = 0) & apart_point_and_line(all_0_3_3, all_0_0_0) = all_24_0_26 % 8.17/2.58 | % 8.17/2.58 | Applying alpha-rule on (148) yields: % 8.17/2.58 | (149) ~ (all_24_0_26 = 0) % 8.17/2.58 | (150) apart_point_and_line(all_0_3_3, all_0_0_0) = all_24_0_26 % 8.17/2.58 | % 8.17/2.58 | Instantiating formula (22) with all_0_3_3, all_0_0_0, 0, all_24_0_26 and discharging atoms apart_point_and_line(all_0_3_3, all_0_0_0) = all_24_0_26, apart_point_and_line(all_0_3_3, all_0_0_0) = 0, yields: % 8.17/2.58 | (151) all_24_0_26 = 0 % 8.17/2.58 | % 8.17/2.58 | Equations (151) can reduce 149 to: % 8.17/2.58 | (111) $false % 8.17/2.58 | % 8.17/2.58 |-The branch is then unsatisfiable % 8.17/2.58 |-Branch two: % 8.17/2.58 | (153) ~ (all_24_0_26 = 0) & distinct_points(all_0_2_2, all_0_3_3) = all_24_0_26 % 8.17/2.58 | % 8.17/2.58 | Applying alpha-rule on (153) yields: % 8.17/2.58 | (149) ~ (all_24_0_26 = 0) % 8.17/2.58 | (155) distinct_points(all_0_2_2, all_0_3_3) = all_24_0_26 % 8.17/2.58 | % 8.17/2.58 | Instantiating formula (17) with all_24_0_26, all_0_3_3, all_0_2_2, all_0_3_3 and discharging atoms distinct_points(all_0_2_2, all_0_3_3) = all_24_0_26, distinct_points(all_0_3_3, all_0_2_2) = 0, yields: % 8.17/2.58 | (156) all_24_0_26 = 0 | distinct_points(all_0_3_3, all_0_3_3) = 0 % 8.17/2.58 | % 8.17/2.58 +-Applying beta-rule and splitting (156), into two cases. % 8.17/2.58 |-Branch one: % 8.17/2.58 | (157) distinct_points(all_0_3_3, all_0_3_3) = 0 % 8.17/2.58 | % 8.17/2.58 | Instantiating formula (3) with all_0_3_3 and discharging atoms distinct_points(all_0_3_3, all_0_3_3) = 0, yields: % 8.17/2.58 | (158) $false % 8.17/2.58 | % 8.17/2.58 |-The branch is then unsatisfiable % 8.17/2.58 |-Branch two: % 8.17/2.58 | (159) ~ (distinct_points(all_0_3_3, all_0_3_3) = 0) % 8.17/2.58 | (151) all_24_0_26 = 0 % 8.17/2.58 | % 8.17/2.58 | Equations (151) can reduce 149 to: % 8.17/2.58 | (111) $false % 8.17/2.58 | % 8.17/2.58 |-The branch is then unsatisfiable % 8.17/2.58 |-Branch two: % 8.17/2.58 | (162) all_28_0_30 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0 % 8.17/2.58 | % 8.17/2.58 | Applying alpha-rule on (162) yields: % 8.17/2.58 | (108) all_28_0_30 = 0 % 8.17/2.58 | (164) apart_point_and_line(all_0_3_3, all_0_1_1) = 0 % 8.17/2.58 | % 8.17/2.58 +-Applying beta-rule and splitting (96), into two cases. % 8.17/2.58 |-Branch one: % 8.17/2.58 | (115) all_25_0_27 = 0 % 8.17/2.58 | % 8.17/2.58 | Equations (115) can reduce 78 to: % 8.17/2.58 | (111) $false % 8.17/2.58 | % 8.17/2.58 |-The branch is then unsatisfiable % 8.17/2.58 |-Branch two: % 8.17/2.58 | (78) ~ (all_25_0_27 = 0) % 8.17/2.58 | (118) all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_0_0) = 0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_2_2) = v0)) % 8.17/2.59 | % 8.17/2.59 +-Applying beta-rule and splitting (99), into two cases. % 8.17/2.59 |-Branch one: % 8.17/2.59 | (169) all_20_0_22 = 0 % 8.17/2.59 | % 8.17/2.59 | Equations (169) can reduce 62 to: % 8.17/2.59 | (111) $false % 8.17/2.59 | % 8.17/2.59 |-The branch is then unsatisfiable % 8.17/2.59 |-Branch two: % 8.17/2.59 | (62) ~ (all_20_0_22 = 0) % 8.17/2.59 | (172) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0)) % 8.17/2.59 | % 8.17/2.59 +-Applying beta-rule and splitting (100), into two cases. % 8.17/2.59 |-Branch one: % 8.17/2.59 | (169) all_20_0_22 = 0 % 8.17/2.59 | % 8.17/2.59 | Equations (169) can reduce 62 to: % 8.17/2.59 | (111) $false % 8.17/2.59 | % 8.17/2.59 |-The branch is then unsatisfiable % 8.17/2.59 |-Branch two: % 8.17/2.59 | (62) ~ (all_20_0_22 = 0) % 8.17/2.59 | (176) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_3_3, all_0_0_0) = 0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_3_3) = v0)) % 8.17/2.59 | % 8.17/2.59 +-Applying beta-rule and splitting (101), into two cases. % 8.17/2.59 |-Branch one: % 8.17/2.59 | (169) all_20_0_22 = 0 % 8.17/2.59 | % 8.17/2.59 | Equations (169) can reduce 62 to: % 8.17/2.59 | (111) $false % 8.17/2.59 | % 8.17/2.59 |-The branch is then unsatisfiable % 8.17/2.59 |-Branch two: % 8.17/2.59 | (62) ~ (all_20_0_22 = 0) % 8.17/2.59 | (180) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_3_3) = v0)) % 8.17/2.59 | % 8.17/2.59 +-Applying beta-rule and splitting (92), into two cases. % 8.17/2.59 |-Branch one: % 8.17/2.59 | (115) all_25_0_27 = 0 % 8.17/2.59 | % 8.17/2.59 | Equations (115) can reduce 78 to: % 8.17/2.59 | (111) $false % 8.17/2.59 | % 8.17/2.59 |-The branch is then unsatisfiable % 8.17/2.59 |-Branch two: % 8.17/2.59 | (78) ~ (all_25_0_27 = 0) % 8.17/2.59 | (122) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0)) % 8.17/2.59 | % 8.17/2.59 +-Applying beta-rule and splitting (93), into two cases. % 8.17/2.59 |-Branch one: % 8.17/2.59 | (115) all_25_0_27 = 0 % 8.17/2.59 | % 8.17/2.59 | Equations (115) can reduce 78 to: % 8.17/2.59 | (111) $false % 8.17/2.59 | % 8.17/2.59 |-The branch is then unsatisfiable % 8.17/2.59 |-Branch two: % 8.17/2.59 | (78) ~ (all_25_0_27 = 0) % 8.17/2.59 | (126) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_2_2) = v0)) % 8.61/2.59 | % 8.61/2.59 +-Applying beta-rule and splitting (94), into two cases. % 8.61/2.59 |-Branch one: % 8.61/2.59 | (115) all_25_0_27 = 0 % 8.61/2.59 | % 8.61/2.59 | Equations (115) can reduce 78 to: % 8.61/2.59 | (111) $false % 8.61/2.59 | % 8.61/2.59 |-The branch is then unsatisfiable % 8.61/2.59 |-Branch two: % 8.61/2.59 | (78) ~ (all_25_0_27 = 0) % 8.61/2.59 | (130) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_2_2) = v0)) % 8.61/2.59 | % 8.61/2.59 +-Applying beta-rule and splitting (97), into two cases. % 8.61/2.59 |-Branch one: % 8.61/2.59 | (115) all_25_0_27 = 0 % 8.61/2.59 | % 8.61/2.59 | Equations (115) can reduce 78 to: % 8.61/2.59 | (111) $false % 8.61/2.59 | % 8.61/2.59 |-The branch is then unsatisfiable % 8.61/2.59 |-Branch two: % 8.61/2.59 | (78) ~ (all_25_0_27 = 0) % 8.61/2.59 | (134) all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_3_3) = v0)) % 8.61/2.59 | % 8.61/2.59 +-Applying beta-rule and splitting (98), into two cases. % 8.61/2.59 |-Branch one: % 8.61/2.59 | (115) all_25_0_27 = 0 % 8.61/2.59 | % 8.61/2.59 | Equations (115) can reduce 78 to: % 8.61/2.59 | (111) $false % 8.61/2.59 | % 8.61/2.59 |-The branch is then unsatisfiable % 8.61/2.59 |-Branch two: % 8.61/2.59 | (78) ~ (all_25_0_27 = 0) % 8.61/2.59 | (138) all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_2_2) = v0)) % 8.61/2.59 | % 8.61/2.59 +-Applying beta-rule and splitting (95), into two cases. % 8.61/2.59 |-Branch one: % 8.61/2.59 | (115) all_25_0_27 = 0 % 8.61/2.59 | % 8.61/2.59 | Equations (115) can reduce 78 to: % 8.61/2.59 | (111) $false % 8.61/2.59 | % 8.61/2.59 |-The branch is then unsatisfiable % 8.61/2.59 |-Branch two: % 8.61/2.59 | (78) ~ (all_25_0_27 = 0) % 8.61/2.59 | (142) all_20_0_22 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_0_0) = 0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_3_3) = v0)) % 8.61/2.59 | % 8.61/2.59 +-Applying beta-rule and splitting (118), into two cases. % 8.61/2.59 |-Branch one: % 8.61/2.59 | (169) all_20_0_22 = 0 % 8.61/2.59 | % 8.61/2.59 | Equations (169) can reduce 62 to: % 8.61/2.59 | (111) $false % 8.61/2.59 | % 8.61/2.59 |-The branch is then unsatisfiable % 8.61/2.59 |-Branch two: % 8.61/2.59 | (62) ~ (all_20_0_22 = 0) % 8.61/2.59 | (208) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_0_0) = 0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_2_2) = v0)) % 8.61/2.59 | % 8.61/2.59 +-Applying beta-rule and splitting (134), into two cases. % 8.61/2.59 |-Branch one: % 8.61/2.59 | (169) all_20_0_22 = 0 % 8.61/2.59 | % 8.61/2.59 | Equations (169) can reduce 62 to: % 8.61/2.59 | (111) $false % 8.61/2.59 | % 8.61/2.59 |-The branch is then unsatisfiable % 8.61/2.59 |-Branch two: % 8.61/2.59 | (62) ~ (all_20_0_22 = 0) % 8.61/2.59 | (212) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_3_3) = v0)) % 8.61/2.59 | % 8.61/2.59 +-Applying beta-rule and splitting (138), into two cases. % 8.61/2.59 |-Branch one: % 8.61/2.59 | (169) all_20_0_22 = 0 % 8.61/2.59 | % 8.61/2.59 | Equations (169) can reduce 62 to: % 8.61/2.59 | (111) $false % 8.61/2.59 | % 8.61/2.59 |-The branch is then unsatisfiable % 8.61/2.59 |-Branch two: % 8.61/2.59 | (62) ~ (all_20_0_22 = 0) % 8.61/2.59 | (216) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_1_1) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_1_1) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_1_1, all_0_1_1) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_2_2) = v0)) % 8.61/2.59 | % 8.61/2.59 +-Applying beta-rule and splitting (142), into two cases. % 8.61/2.59 |-Branch one: % 8.61/2.59 | (169) all_20_0_22 = 0 % 8.61/2.59 | % 8.61/2.59 | Equations (169) can reduce 62 to: % 8.61/2.59 | (111) $false % 8.61/2.59 | % 8.61/2.59 |-The branch is then unsatisfiable % 8.61/2.59 |-Branch two: % 8.61/2.59 | (62) ~ (all_20_0_22 = 0) % 8.61/2.59 | (220) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_2_2, all_0_0_0) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_0_0) = 0) | ( ~ (v0 = 0) & distinct_points(all_0_2_2, all_0_3_3) = v0)) % 8.61/2.59 | % 8.61/2.59 | Instantiating formula (22) with all_0_3_3, all_0_1_1, 0, all_20_0_22 and discharging atoms apart_point_and_line(all_0_3_3, all_0_1_1) = all_20_0_22, apart_point_and_line(all_0_3_3, all_0_1_1) = 0, yields: % 8.61/2.59 | (169) all_20_0_22 = 0 % 8.61/2.59 | % 8.61/2.59 | Equations (169) can reduce 62 to: % 8.61/2.59 | (111) $false % 8.61/2.59 | % 8.61/2.59 |-The branch is then unsatisfiable % 8.61/2.59 |-Branch two: % 8.61/2.59 | (223) ~ (all_23_0_25 = 0) & distinct_points(all_0_2_2, all_0_3_3) = all_23_0_25 % 8.61/2.60 | % 8.61/2.60 | Applying alpha-rule on (223) yields: % 8.61/2.60 | (103) ~ (all_23_0_25 = 0) % 8.61/2.60 | (225) distinct_points(all_0_2_2, all_0_3_3) = all_23_0_25 % 8.61/2.60 | % 8.61/2.60 | Instantiating formula (17) with all_23_0_25, all_0_3_3, all_0_2_2, all_0_3_3 and discharging atoms distinct_points(all_0_2_2, all_0_3_3) = all_23_0_25, distinct_points(all_0_3_3, all_0_2_2) = 0, yields: % 8.61/2.60 | (226) all_23_0_25 = 0 | distinct_points(all_0_3_3, all_0_3_3) = 0 % 8.61/2.60 | % 8.61/2.60 +-Applying beta-rule and splitting (226), into two cases. % 8.61/2.60 |-Branch one: % 8.61/2.60 | (157) distinct_points(all_0_3_3, all_0_3_3) = 0 % 8.61/2.60 | % 8.61/2.60 | Instantiating formula (3) with all_0_3_3 and discharging atoms distinct_points(all_0_3_3, all_0_3_3) = 0, yields: % 8.61/2.60 | (158) $false % 8.61/2.60 | % 8.61/2.60 |-The branch is then unsatisfiable % 8.61/2.60 |-Branch two: % 8.61/2.60 | (159) ~ (distinct_points(all_0_3_3, all_0_3_3) = 0) % 8.61/2.60 | (110) all_23_0_25 = 0 % 8.61/2.60 | % 8.61/2.60 | Equations (110) can reduce 103 to: % 8.61/2.60 | (111) $false % 8.61/2.60 | % 8.61/2.60 |-The branch is then unsatisfiable % 8.61/2.60 |-Branch two: % 8.61/2.60 | (232) ~ (all_25_0_27 = 0) & distinct_points(all_0_3_3, all_0_2_2) = all_25_0_27 % 8.61/2.60 | % 8.61/2.60 | Applying alpha-rule on (232) yields: % 8.61/2.60 | (78) ~ (all_25_0_27 = 0) % 8.61/2.60 | (234) distinct_points(all_0_3_3, all_0_2_2) = all_25_0_27 % 8.61/2.60 | % 8.61/2.60 | Instantiating formula (11) with all_0_3_3, all_0_2_2, all_25_0_27, 0 and discharging atoms distinct_points(all_0_3_3, all_0_2_2) = all_25_0_27, distinct_points(all_0_3_3, all_0_2_2) = 0, yields: % 8.61/2.60 | (115) all_25_0_27 = 0 % 8.61/2.60 | % 8.61/2.60 | Equations (115) can reduce 78 to: % 8.61/2.60 | (111) $false % 8.61/2.60 | % 8.61/2.60 |-The branch is then unsatisfiable % 8.61/2.60 |-Branch two: % 8.61/2.60 | (237) ~ (all_22_0_24 = 0) & distinct_points(all_0_3_3, all_0_2_2) = all_22_0_24 % 8.61/2.60 | % 8.61/2.60 | Applying alpha-rule on (237) yields: % 8.61/2.60 | (75) ~ (all_22_0_24 = 0) % 8.61/2.60 | (239) distinct_points(all_0_3_3, all_0_2_2) = all_22_0_24 % 8.61/2.60 | % 8.61/2.60 | Instantiating formula (11) with all_0_3_3, all_0_2_2, all_22_0_24, 0 and discharging atoms distinct_points(all_0_3_3, all_0_2_2) = all_22_0_24, distinct_points(all_0_3_3, all_0_2_2) = 0, yields: % 8.61/2.60 | (240) all_22_0_24 = 0 % 8.61/2.60 | % 8.61/2.60 | Equations (240) can reduce 75 to: % 8.61/2.60 | (111) $false % 8.61/2.60 | % 8.61/2.60 |-The branch is then unsatisfiable % 8.61/2.60 % SZS output end Proof for theBenchmark % 8.61/2.60 % 8.61/2.60 2017ms %------------------------------------------------------------------------------