%------------------------------------------------------------------------------ % File : ePrincess---1.0 % Problem : GEO172+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 03:48:08 EDT 2022 % Result : Theorem 5.45s 2.00s % Output : Proof 8.82s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.12 % Problem : GEO172+1 : TPTP v8.1.0. Released v3.3.0. % 0.07/0.13 % Command : ePrincess-casc -timeout=%d %s % 0.12/0.34 % Computer : n024.cluster.edu % 0.12/0.34 % Model : x86_64 x86_64 % 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.34 % Memory : 8042.1875MB % 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.34 % CPULimit : 300 % 0.12/0.34 % WCLimit : 600 % 0.12/0.34 % DateTime : Sat Jun 18 01:28:32 EDT 2022 % 0.12/0.34 % CPUTime : % 0.19/0.58 ____ _ % 0.19/0.58 ___ / __ \_____(_)___ ________ __________ % 0.19/0.58 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/ % 0.19/0.58 / __/ ____/ / / / / / / /__/ __(__ |__ ) % 0.19/0.58 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/ % 0.19/0.58 % 0.19/0.58 A Theorem Prover for First-Order Logic % 0.19/0.59 (ePrincess v.1.0) % 0.19/0.59 % 0.19/0.59 (c) Philipp Rümmer, 2009-2015 % 0.19/0.59 (c) Peter Backeman, 2014-2015 % 0.19/0.59 (contributions by Angelo Brillout, Peter Baumgartner) % 0.19/0.59 Free software under GNU Lesser General Public License (LGPL). % 0.19/0.59 Bug reports to peter@backeman.se % 0.19/0.59 % 0.19/0.59 For more information, visit http://user.uu.se/~petba168/breu/ % 0.19/0.59 % 0.19/0.59 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ... % 0.67/0.66 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 1.59/0.97 Prover 0: Preprocessing ... % 1.97/1.10 Prover 0: Warning: ignoring some quantifiers % 1.97/1.12 Prover 0: Constructing countermodel ... % 2.82/1.35 Prover 0: gave up % 2.82/1.36 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all % 3.03/1.40 Prover 1: Preprocessing ... % 3.43/1.54 Prover 1: Constructing countermodel ... % 3.43/1.59 Prover 1: gave up % 3.43/1.59 Prover 2: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 3.43/1.61 Prover 2: Preprocessing ... % 4.11/1.74 Prover 2: Warning: ignoring some quantifiers % 4.11/1.75 Prover 2: Constructing countermodel ... % 5.45/2.00 Prover 2: proved (413ms) % 5.45/2.00 % 5.45/2.00 No countermodel exists, formula is valid % 5.45/2.00 % SZS status Theorem for theBenchmark % 5.45/2.00 % 5.45/2.00 Generating proof ... Warning: ignoring some quantifiers % 8.13/2.62 found it (size 66) % 8.13/2.62 % 8.13/2.62 % SZS output start Proof for theBenchmark % 8.13/2.62 Assumed formulas after preprocessing and simplification: % 8.13/2.62 | (0) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ? [v5] : ( ~ (v4 = 0) & ~ (v3 = 0) & intersection_point(v0, v1) = v5 & apart_point_and_line(v2, v1) = v4 & apart_point_and_line(v2, v0) = v3 & convergent_lines(v0, v1) = 0 & distinct_points(v2, v5) = 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) % 8.55/2.67 | 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: % 8.55/2.67 | (1) ~ (all_0_1_1 = 0) & ~ (all_0_2_2 = 0) & intersection_point(all_0_5_5, all_0_4_4) = all_0_0_0 & apart_point_and_line(all_0_3_3, all_0_4_4) = all_0_1_1 & apart_point_and_line(all_0_3_3, all_0_5_5) = all_0_2_2 & convergent_lines(all_0_5_5, all_0_4_4) = 0 & distinct_points(all_0_3_3, all_0_0_0) = 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 % 8.55/2.68 | % 8.55/2.68 | Applying alpha-rule on (1) yields: % 8.55/2.68 | (2) ! [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))) % 8.55/2.68 | (3) ! [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)) % 8.55/2.69 | (4) ! [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)) % 8.55/2.69 | (5) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (convergent_lines(v3, v2) = v1) | ~ (convergent_lines(v3, v2) = v0)) % 8.55/2.69 | (6) ? [v0] : ? [v1] : ? [v2] : convergent_lines(v1, v0) = v2 % 8.55/2.69 | (7) distinct_points(all_0_3_3, all_0_0_0) = 0 % 8.55/2.69 | (8) ! [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))) % 8.55/2.69 | (9) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (apart_point_and_line(v3, v2) = v1) | ~ (apart_point_and_line(v3, v2) = v0)) % 8.55/2.69 | (10) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v1) = 0) | ~ (distinct_lines(v1, v2) = v3) | convergent_lines(v0, v2) = 0) % 8.64/2.69 | (11) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v1, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v0, v2) = 0) % 8.64/2.69 | (12) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_lines(v3, v2) = v1) | ~ (distinct_lines(v3, v2) = v0)) % 8.64/2.69 | (13) ! [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))) % 8.64/2.69 | (14) ! [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) % 8.64/2.69 | (15) ! [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))) % 8.64/2.69 | (16) ! [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))) % 8.64/2.69 | (17) ! [v0] : ! [v1] : ( ~ (distinct_points(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & line_connecting(v0, v1) = v2 & apart_point_and_line(v0, v2) = v3)) % 8.64/2.69 | (18) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v1, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v0, v2) = 0) % 8.64/2.69 | (19) ? [v0] : ? [v1] : ? [v2] : intersection_point(v1, v0) = v2 % 8.64/2.69 | (20) apart_point_and_line(all_0_3_3, all_0_4_4) = all_0_1_1 % 8.64/2.69 | (21) ! [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) % 8.64/2.69 | (22) ! [v0] : ! [v1] : ( ~ (distinct_points(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & line_connecting(v0, v1) = v2 & apart_point_and_line(v1, v2) = v3)) % 8.64/2.69 | (23) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0) % 8.64/2.69 | (24) ? [v0] : ? [v1] : ? [v2] : apart_point_and_line(v1, v0) = v2 % 8.64/2.69 | (25) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (distinct_points(v3, v2) = v1) | ~ (distinct_points(v3, v2) = v0)) % 8.64/2.69 | (26) ? [v0] : ? [v1] : ? [v2] : line_connecting(v1, v0) = v2 % 8.64/2.69 | (27) ? [v0] : ? [v1] : ? [v2] : distinct_lines(v1, v0) = v2 % 8.64/2.69 | (28) ! [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)) % 8.64/2.69 | (29) ! [v0] : ~ (convergent_lines(v0, v0) = 0) % 8.64/2.69 | (30) ! [v0] : ! [v1] : ( ~ (convergent_lines(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & intersection_point(v0, v1) = v2 & apart_point_and_line(v2, v1) = v3)) % 8.64/2.70 | (31) ! [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))) % 8.64/2.70 | (32) ! [v0] : ! [v1] : ( ~ (convergent_lines(v0, v1) = 0) | ? [v2] : ? [v3] : ( ~ (v3 = 0) & intersection_point(v0, v1) = v2 & apart_point_and_line(v2, v0) = v3)) % 8.64/2.70 | (33) ? [v0] : ? [v1] : ? [v2] : distinct_points(v1, v0) = v2 % 8.64/2.70 | (34) intersection_point(all_0_5_5, all_0_4_4) = all_0_0_0 % 8.64/2.70 | (35) ! [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))) % 8.64/2.70 | (36) ! [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) % 8.64/2.70 | (37) ~ (all_0_1_1 = 0) % 8.64/2.70 | (38) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_points(v0, v2) = v3) | ~ (distinct_points(v0, v1) = 0) | distinct_points(v1, v2) = 0) % 8.64/2.70 | (39) convergent_lines(all_0_5_5, all_0_4_4) = 0 % 8.64/2.70 | (40) ! [v0] : ~ (distinct_lines(v0, v0) = 0) % 8.64/2.70 | (41) apart_point_and_line(all_0_3_3, all_0_5_5) = all_0_2_2 % 8.64/2.70 | (42) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (line_connecting(v3, v2) = v1) | ~ (line_connecting(v3, v2) = v0)) % 8.64/2.70 | (43) ! [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))) % 8.64/2.70 | (44) ! [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))) % 8.64/2.70 | (45) ~ (all_0_2_2 = 0) % 8.64/2.70 | (46) ! [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)) % 8.64/2.70 | (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))) % 8.64/2.70 | (48) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (intersection_point(v3, v2) = v1) | ~ (intersection_point(v3, v2) = v0)) % 8.64/2.70 | (49) ! [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)) % 8.64/2.70 | (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)) % 8.64/2.70 | (51) ! [v0] : ~ (distinct_points(v0, v0) = 0) % 8.64/2.70 | (52) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (distinct_lines(v0, v2) = v3) | ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0) % 8.64/2.70 | (53) ! [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))) % 8.64/2.70 | (54) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v0, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v1, v2) = 0) % 8.64/2.70 | (55) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (convergent_lines(v1, v2) = v3) | ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v0, v2) = 0) % 8.64/2.70 | (56) ! [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) % 8.64/2.70 | % 8.64/2.70 | Instantiating formula (13) with all_0_0_0, all_0_4_4, all_0_5_5 and discharging atoms intersection_point(all_0_5_5, all_0_4_4) = all_0_0_0, yields: % 8.64/2.71 | (57) ? [v0] : (( ~ (v0 = 0) & apart_point_and_line(all_0_0_0, all_0_4_4) = v0) | ( ~ (v0 = 0) & convergent_lines(all_0_5_5, all_0_4_4) = v0)) % 8.64/2.71 | % 8.64/2.71 | Instantiating formula (2) with all_0_0_0, all_0_4_4, all_0_5_5 and discharging atoms intersection_point(all_0_5_5, all_0_4_4) = all_0_0_0, yields: % 8.64/2.71 | (58) ? [v0] : (( ~ (v0 = 0) & apart_point_and_line(all_0_0_0, all_0_5_5) = v0) | ( ~ (v0 = 0) & convergent_lines(all_0_5_5, all_0_4_4) = v0)) % 8.64/2.71 | % 8.64/2.71 | Instantiating formula (31) with all_0_1_1, all_0_1_1, all_0_4_4, all_0_4_4, all_0_3_3, all_0_3_3 and discharging atoms apart_point_and_line(all_0_3_3, all_0_4_4) = all_0_1_1, yields: % 8.64/2.71 | (59) all_0_1_1 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_3_3, all_0_4_4) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_4_4, all_0_4_4) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_3_3) = v0)) % 8.64/2.71 | % 8.64/2.71 | Instantiating formula (31) with all_0_1_1, all_0_2_2, all_0_4_4, all_0_5_5, all_0_3_3, all_0_3_3 and discharging atoms apart_point_and_line(all_0_3_3, all_0_4_4) = all_0_1_1, apart_point_and_line(all_0_3_3, all_0_5_5) = all_0_2_2, yields: % 8.64/2.71 | (60) all_0_1_1 = 0 | all_0_2_2 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_3_3, all_0_4_4) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_5_5) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_5_5, all_0_4_4) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_3_3) = v0)) % 8.64/2.71 | % 8.64/2.71 | Instantiating formula (31) with all_0_2_2, all_0_2_2, all_0_5_5, all_0_5_5, all_0_3_3, all_0_3_3 and discharging atoms apart_point_and_line(all_0_3_3, all_0_5_5) = all_0_2_2, yields: % 8.64/2.71 | (61) all_0_2_2 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_3_3, all_0_5_5) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_5_5, all_0_5_5) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_3_3) = v0)) % 8.64/2.71 | % 8.64/2.71 | Instantiating formula (8) with all_0_1_1, all_0_1_1, all_0_4_4, all_0_4_4, all_0_0_0, all_0_3_3 and discharging atoms apart_point_and_line(all_0_3_3, all_0_4_4) = all_0_1_1, distinct_points(all_0_3_3, all_0_0_0) = 0, yields: % 8.64/2.71 | (62) all_0_1_1 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_0_0, all_0_4_4) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_4_4, all_0_4_4) = v0)) % 8.64/2.71 | % 8.64/2.71 | Instantiating formula (8) with all_0_2_2, all_0_1_1, all_0_5_5, all_0_4_4, all_0_0_0, all_0_3_3 and discharging atoms apart_point_and_line(all_0_3_3, all_0_4_4) = all_0_1_1, apart_point_and_line(all_0_3_3, all_0_5_5) = all_0_2_2, distinct_points(all_0_3_3, all_0_0_0) = 0, yields: % 8.64/2.71 | (63) all_0_1_1 = 0 | all_0_2_2 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_0_0, all_0_4_4) = 0) | (v0 = 0 & apart_point_and_line(all_0_0_0, all_0_5_5) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_4_4, all_0_5_5) = v0)) % 8.64/2.71 | % 8.64/2.71 | Instantiating formula (8) with all_0_2_2, all_0_2_2, all_0_5_5, all_0_5_5, all_0_0_0, all_0_3_3 and discharging atoms apart_point_and_line(all_0_3_3, all_0_5_5) = all_0_2_2, distinct_points(all_0_3_3, all_0_0_0) = 0, yields: % 8.64/2.71 | (64) all_0_2_2 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_0_0, all_0_5_5) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_5_5, all_0_5_5) = v0)) % 8.64/2.71 | % 8.64/2.71 | Instantiating (58) with all_28_0_32 yields: % 8.64/2.71 | (65) ( ~ (all_28_0_32 = 0) & apart_point_and_line(all_0_0_0, all_0_5_5) = all_28_0_32) | ( ~ (all_28_0_32 = 0) & convergent_lines(all_0_5_5, all_0_4_4) = all_28_0_32) % 8.64/2.71 | % 8.64/2.71 | Instantiating (57) with all_29_0_33 yields: % 8.64/2.71 | (66) ( ~ (all_29_0_33 = 0) & apart_point_and_line(all_0_0_0, all_0_4_4) = all_29_0_33) | ( ~ (all_29_0_33 = 0) & convergent_lines(all_0_5_5, all_0_4_4) = all_29_0_33) % 8.64/2.71 | % 8.64/2.71 +-Applying beta-rule and splitting (65), into two cases. % 8.64/2.71 |-Branch one: % 8.64/2.71 | (67) ~ (all_28_0_32 = 0) & apart_point_and_line(all_0_0_0, all_0_5_5) = all_28_0_32 % 8.64/2.71 | % 8.64/2.71 | Applying alpha-rule on (67) yields: % 8.64/2.71 | (68) ~ (all_28_0_32 = 0) % 8.64/2.71 | (69) apart_point_and_line(all_0_0_0, all_0_5_5) = all_28_0_32 % 8.64/2.71 | % 8.64/2.71 +-Applying beta-rule and splitting (66), into two cases. % 8.64/2.71 |-Branch one: % 8.64/2.71 | (70) ~ (all_29_0_33 = 0) & apart_point_and_line(all_0_0_0, all_0_4_4) = all_29_0_33 % 8.64/2.71 | % 8.64/2.71 | Applying alpha-rule on (70) yields: % 8.64/2.71 | (71) ~ (all_29_0_33 = 0) % 8.64/2.71 | (72) apart_point_and_line(all_0_0_0, all_0_4_4) = all_29_0_33 % 8.64/2.71 | % 8.64/2.71 +-Applying beta-rule and splitting (62), into two cases. % 8.64/2.71 |-Branch one: % 8.64/2.71 | (73) all_0_1_1 = 0 % 8.64/2.71 | % 8.64/2.71 | Equations (73) can reduce 37 to: % 8.64/2.71 | (74) $false % 8.64/2.71 | % 8.64/2.71 |-The branch is then unsatisfiable % 8.64/2.71 |-Branch two: % 8.64/2.71 | (37) ~ (all_0_1_1 = 0) % 8.64/2.71 | (76) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_0_0, all_0_4_4) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_4_4, all_0_4_4) = v0)) % 8.64/2.71 | % 8.64/2.71 +-Applying beta-rule and splitting (64), into two cases. % 8.64/2.71 |-Branch one: % 8.64/2.71 | (77) all_0_2_2 = 0 % 8.64/2.71 | % 8.64/2.71 | Equations (77) can reduce 45 to: % 8.64/2.71 | (74) $false % 8.64/2.71 | % 8.64/2.71 |-The branch is then unsatisfiable % 8.64/2.71 |-Branch two: % 8.64/2.71 | (45) ~ (all_0_2_2 = 0) % 8.64/2.71 | (80) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_0_0, all_0_5_5) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_5_5, all_0_5_5) = v0)) % 8.64/2.71 | % 8.64/2.71 +-Applying beta-rule and splitting (59), into two cases. % 8.64/2.71 |-Branch one: % 8.64/2.71 | (73) all_0_1_1 = 0 % 8.64/2.71 | % 8.64/2.71 | Equations (73) can reduce 37 to: % 8.64/2.71 | (74) $false % 8.64/2.71 | % 8.64/2.71 |-The branch is then unsatisfiable % 8.64/2.71 |-Branch two: % 8.64/2.71 | (37) ~ (all_0_1_1 = 0) % 8.64/2.71 | (84) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_3_3, all_0_4_4) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_4_4, all_0_4_4) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_3_3) = v0)) % 8.64/2.71 | % 8.64/2.71 +-Applying beta-rule and splitting (61), into two cases. % 8.64/2.71 |-Branch one: % 8.64/2.71 | (77) all_0_2_2 = 0 % 8.64/2.71 | % 8.64/2.71 | Equations (77) can reduce 45 to: % 8.64/2.71 | (74) $false % 8.64/2.71 | % 8.64/2.71 |-The branch is then unsatisfiable % 8.64/2.71 |-Branch two: % 8.64/2.71 | (45) ~ (all_0_2_2 = 0) % 8.64/2.72 | (88) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_3_3, all_0_5_5) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_5_5, all_0_5_5) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_3_3) = v0)) % 8.64/2.72 | % 8.64/2.72 +-Applying beta-rule and splitting (60), into two cases. % 8.64/2.72 |-Branch one: % 8.64/2.72 | (73) all_0_1_1 = 0 % 8.64/2.72 | % 8.64/2.72 | Equations (73) can reduce 37 to: % 8.64/2.72 | (74) $false % 8.64/2.72 | % 8.64/2.72 |-The branch is then unsatisfiable % 8.64/2.72 |-Branch two: % 8.64/2.72 | (37) ~ (all_0_1_1 = 0) % 8.64/2.72 | (92) all_0_2_2 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_3_3, all_0_4_4) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_5_5) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_5_5, all_0_4_4) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_3_3) = v0)) % 8.64/2.72 | % 8.64/2.72 +-Applying beta-rule and splitting (63), into two cases. % 8.64/2.72 |-Branch one: % 8.64/2.72 | (73) all_0_1_1 = 0 % 8.64/2.72 | % 8.64/2.72 | Equations (73) can reduce 37 to: % 8.64/2.72 | (74) $false % 8.64/2.72 | % 8.64/2.72 |-The branch is then unsatisfiable % 8.64/2.72 |-Branch two: % 8.64/2.72 | (37) ~ (all_0_1_1 = 0) % 8.64/2.72 | (96) all_0_2_2 = 0 | ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_0_0, all_0_4_4) = 0) | (v0 = 0 & apart_point_and_line(all_0_0_0, all_0_5_5) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_4_4, all_0_5_5) = v0)) % 8.79/2.72 | % 8.79/2.72 +-Applying beta-rule and splitting (92), into two cases. % 8.79/2.72 |-Branch one: % 8.79/2.72 | (77) all_0_2_2 = 0 % 8.79/2.72 | % 8.79/2.72 | Equations (77) can reduce 45 to: % 8.79/2.72 | (74) $false % 8.79/2.72 | % 8.79/2.72 |-The branch is then unsatisfiable % 8.79/2.72 |-Branch two: % 8.79/2.72 | (45) ~ (all_0_2_2 = 0) % 8.79/2.72 | (100) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_3_3, all_0_4_4) = 0) | (v0 = 0 & apart_point_and_line(all_0_3_3, all_0_5_5) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_5_5, all_0_4_4) = v0) | ( ~ (v0 = 0) & distinct_points(all_0_3_3, all_0_3_3) = v0)) % 8.79/2.72 | % 8.79/2.72 +-Applying beta-rule and splitting (96), into two cases. % 8.79/2.72 |-Branch one: % 8.79/2.72 | (77) all_0_2_2 = 0 % 8.79/2.72 | % 8.79/2.72 | Equations (77) can reduce 45 to: % 8.79/2.72 | (74) $false % 8.79/2.72 | % 8.79/2.72 |-The branch is then unsatisfiable % 8.79/2.72 |-Branch two: % 8.79/2.72 | (45) ~ (all_0_2_2 = 0) % 8.79/2.72 | (104) ? [v0] : ((v0 = 0 & apart_point_and_line(all_0_0_0, all_0_4_4) = 0) | (v0 = 0 & apart_point_and_line(all_0_0_0, all_0_5_5) = 0) | ( ~ (v0 = 0) & distinct_lines(all_0_4_4, all_0_5_5) = v0)) % 8.79/2.72 | % 8.79/2.72 | Instantiating (104) with all_83_0_51 yields: % 8.79/2.72 | (105) (all_83_0_51 = 0 & apart_point_and_line(all_0_0_0, all_0_4_4) = 0) | (all_83_0_51 = 0 & apart_point_and_line(all_0_0_0, all_0_5_5) = 0) | ( ~ (all_83_0_51 = 0) & distinct_lines(all_0_4_4, all_0_5_5) = all_83_0_51) % 8.79/2.72 | % 8.79/2.72 +-Applying beta-rule and splitting (105), into two cases. % 8.79/2.72 |-Branch one: % 8.79/2.72 | (106) (all_83_0_51 = 0 & apart_point_and_line(all_0_0_0, all_0_4_4) = 0) | (all_83_0_51 = 0 & apart_point_and_line(all_0_0_0, all_0_5_5) = 0) % 8.79/2.72 | % 8.79/2.72 +-Applying beta-rule and splitting (106), into two cases. % 8.79/2.72 |-Branch one: % 8.79/2.72 | (107) all_83_0_51 = 0 & apart_point_and_line(all_0_0_0, all_0_4_4) = 0 % 8.79/2.72 | % 8.79/2.72 | Applying alpha-rule on (107) yields: % 8.79/2.72 | (108) all_83_0_51 = 0 % 8.79/2.72 | (109) apart_point_and_line(all_0_0_0, all_0_4_4) = 0 % 8.79/2.72 | % 8.79/2.72 | Instantiating formula (9) with all_0_0_0, all_0_4_4, 0, all_29_0_33 and discharging atoms apart_point_and_line(all_0_0_0, all_0_4_4) = all_29_0_33, apart_point_and_line(all_0_0_0, all_0_4_4) = 0, yields: % 8.79/2.72 | (110) all_29_0_33 = 0 % 8.79/2.72 | % 8.79/2.72 | Equations (110) can reduce 71 to: % 8.79/2.72 | (74) $false % 8.79/2.72 | % 8.79/2.72 |-The branch is then unsatisfiable % 8.79/2.72 |-Branch two: % 8.79/2.72 | (112) all_83_0_51 = 0 & apart_point_and_line(all_0_0_0, all_0_5_5) = 0 % 8.79/2.72 | % 8.79/2.72 | Applying alpha-rule on (112) yields: % 8.79/2.72 | (108) all_83_0_51 = 0 % 8.79/2.72 | (114) apart_point_and_line(all_0_0_0, all_0_5_5) = 0 % 8.79/2.72 | % 8.79/2.72 | Instantiating formula (9) with all_0_0_0, all_0_5_5, 0, all_28_0_32 and discharging atoms apart_point_and_line(all_0_0_0, all_0_5_5) = all_28_0_32, apart_point_and_line(all_0_0_0, all_0_5_5) = 0, yields: % 8.79/2.72 | (115) all_28_0_32 = 0 % 8.79/2.72 | % 8.79/2.72 | Equations (115) can reduce 68 to: % 8.79/2.72 | (74) $false % 8.79/2.72 | % 8.79/2.72 |-The branch is then unsatisfiable % 8.79/2.72 |-Branch two: % 8.79/2.72 | (117) ~ (all_83_0_51 = 0) & distinct_lines(all_0_4_4, all_0_5_5) = all_83_0_51 % 8.79/2.72 | % 8.79/2.72 | Applying alpha-rule on (117) yields: % 8.79/2.72 | (118) ~ (all_83_0_51 = 0) % 8.79/2.72 | (119) distinct_lines(all_0_4_4, all_0_5_5) = all_83_0_51 % 8.79/2.72 | % 8.79/2.72 | Instantiating formula (10) with all_83_0_51, all_0_5_5, all_0_4_4, all_0_5_5 and discharging atoms convergent_lines(all_0_5_5, all_0_4_4) = 0, distinct_lines(all_0_4_4, all_0_5_5) = all_83_0_51, yields: % 8.79/2.72 | (120) all_83_0_51 = 0 | convergent_lines(all_0_5_5, all_0_5_5) = 0 % 8.79/2.72 | % 8.79/2.72 +-Applying beta-rule and splitting (120), into two cases. % 8.79/2.72 |-Branch one: % 8.79/2.72 | (121) convergent_lines(all_0_5_5, all_0_5_5) = 0 % 8.79/2.72 | % 8.79/2.72 | Instantiating formula (29) with all_0_5_5 and discharging atoms convergent_lines(all_0_5_5, all_0_5_5) = 0, yields: % 8.79/2.72 | (122) $false % 8.79/2.72 | % 8.79/2.72 |-The branch is then unsatisfiable % 8.79/2.72 |-Branch two: % 8.79/2.72 | (123) ~ (convergent_lines(all_0_5_5, all_0_5_5) = 0) % 8.79/2.72 | (108) all_83_0_51 = 0 % 8.79/2.72 | % 8.79/2.72 | Equations (108) can reduce 118 to: % 8.79/2.72 | (74) $false % 8.79/2.72 | % 8.79/2.72 |-The branch is then unsatisfiable % 8.79/2.72 |-Branch two: % 8.79/2.72 | (126) ~ (all_29_0_33 = 0) & convergent_lines(all_0_5_5, all_0_4_4) = all_29_0_33 % 8.82/2.73 | % 8.82/2.73 | Applying alpha-rule on (126) yields: % 8.82/2.73 | (71) ~ (all_29_0_33 = 0) % 8.82/2.73 | (128) convergent_lines(all_0_5_5, all_0_4_4) = all_29_0_33 % 8.82/2.73 | % 8.82/2.73 | Instantiating formula (5) with all_0_5_5, all_0_4_4, all_29_0_33, 0 and discharging atoms convergent_lines(all_0_5_5, all_0_4_4) = all_29_0_33, convergent_lines(all_0_5_5, all_0_4_4) = 0, yields: % 8.82/2.73 | (110) all_29_0_33 = 0 % 8.82/2.73 | % 8.82/2.73 | Equations (110) can reduce 71 to: % 8.82/2.73 | (74) $false % 8.82/2.73 | % 8.82/2.73 |-The branch is then unsatisfiable % 8.82/2.73 |-Branch two: % 8.82/2.73 | (131) ~ (all_28_0_32 = 0) & convergent_lines(all_0_5_5, all_0_4_4) = all_28_0_32 % 8.82/2.73 | % 8.82/2.73 | Applying alpha-rule on (131) yields: % 8.82/2.73 | (68) ~ (all_28_0_32 = 0) % 8.82/2.73 | (133) convergent_lines(all_0_5_5, all_0_4_4) = all_28_0_32 % 8.82/2.73 | % 8.82/2.73 | Instantiating formula (5) with all_0_5_5, all_0_4_4, all_28_0_32, 0 and discharging atoms convergent_lines(all_0_5_5, all_0_4_4) = all_28_0_32, convergent_lines(all_0_5_5, all_0_4_4) = 0, yields: % 8.82/2.73 | (115) all_28_0_32 = 0 % 8.82/2.73 | % 8.82/2.73 | Equations (115) can reduce 68 to: % 8.82/2.73 | (74) $false % 8.82/2.73 | % 8.82/2.73 |-The branch is then unsatisfiable % 8.82/2.73 % SZS output end Proof for theBenchmark % 8.82/2.73 % 8.82/2.73 2124ms %------------------------------------------------------------------------------