%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO201+2 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n004.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 : 300s
% DateTime : Sat Jun 21 05:05:54 AM UTC 2025
% Result : Theorem 0.19s 0.38s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.02/0.12 % Problem : GEO201+2 : TPTP v9.0.0. Released v3.3.0.
% 0.02/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n004.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Thu Jun 19 14:40:53 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.19/0.38 % SZS status Theorem
% 0.19/0.38 % SZS output start Proof
% 0.19/0.38 tff(apart_point_and_line_type, type, (
% 0.19/0.38 apart_point_and_line: ( $i * $i ) > $o)).
% 0.19/0.38 tff(tptp_fun_X_1_type, type, (
% 0.19/0.38 tptp_fun_X_1: $i)).
% 0.19/0.38 tff(intersection_point_type, type, (
% 0.19/0.38 intersection_point: ( $i * $i ) > $i)).
% 0.19/0.38 tff(tptp_fun_Y_0_type, type, (
% 0.19/0.38 tptp_fun_Y_0: $i)).
% 0.19/0.38 tff(convergent_lines_type, type, (
% 0.19/0.38 convergent_lines: ( $i * $i ) > $o)).
% 0.19/0.38 tff(distinct_points_type, type, (
% 0.19/0.38 distinct_points: ( $i * $i ) > $o)).
% 0.19/0.38 tff(distinct_lines_type, type, (
% 0.19/0.38 distinct_lines: ( $i * $i ) > $o)).
% 0.19/0.38 tff(1,assumption,(~convergent_lines(Y!0, X!1)), introduced(assumption)).
% 0.19/0.38 tff(2,plain,
% 0.19/0.38 (^[X: $i] : refl((~convergent_lines(X, X)) <=> (~convergent_lines(X, X)))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(3,plain,
% 0.19/0.38 (![X: $i] : (~convergent_lines(X, X)) <=> ![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[2])).
% 0.19/0.38 tff(4,plain,
% 0.19/0.38 (![X: $i] : (~convergent_lines(X, X)) <=> ![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(5,axiom,(![X: $i] : (~convergent_lines(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','apart3')).
% 0.19/0.38 tff(6,plain,
% 0.19/0.38 (![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.19/0.38 tff(7,plain,(
% 0.19/0.38 ![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.38 inference(skolemize,[status(sab)],[6])).
% 0.19/0.38 tff(8,plain,
% 0.19/0.38 (![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.19/0.38 tff(9,plain,
% 0.19/0.38 ((~![X: $i] : (~convergent_lines(X, X))) | (~convergent_lines(X!1, X!1))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(10,plain,
% 0.19/0.38 (~convergent_lines(X!1, X!1)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[9, 8])).
% 0.19/0.38 tff(11,plain,
% 0.19/0.38 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~distinct_points(intersection_point(X, Y), intersection_point(Y, X))))) <=> (~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~distinct_points(intersection_point(X, Y), intersection_point(Y, X)))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(12,plain,
% 0.19/0.38 ((~![X: $i, Y: $i] : (convergent_lines(X, Y) => (~distinct_points(intersection_point(X, Y), intersection_point(Y, X))))) <=> (~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~distinct_points(intersection_point(X, Y), intersection_point(Y, X)))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(13,axiom,(~![X: $i, Y: $i] : (convergent_lines(X, Y) => (~distinct_points(intersection_point(X, Y), intersection_point(Y, X))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.19/0.38 tff(14,plain,
% 0.19/0.38 (~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~distinct_points(intersection_point(X, Y), intersection_point(Y, X))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.19/0.38 tff(15,plain,
% 0.19/0.38 (~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~distinct_points(intersection_point(X, Y), intersection_point(Y, X))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[14, 11])).
% 0.19/0.38 tff(16,plain,
% 0.19/0.38 (~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~distinct_points(intersection_point(X, Y), intersection_point(Y, X))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[15, 11])).
% 0.19/0.38 tff(17,plain,
% 0.19/0.38 (~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~distinct_points(intersection_point(X, Y), intersection_point(Y, X))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[16, 11])).
% 0.19/0.38 tff(18,plain,(
% 0.19/0.38 ~((~convergent_lines(X!1, Y!0)) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))),
% 0.19/0.38 inference(skolemize,[status(sab)],[17])).
% 0.19/0.38 tff(19,plain,
% 0.19/0.38 (convergent_lines(X!1, Y!0)),
% 0.19/0.38 inference(or_elim,[status(thm)],[18])).
% 0.19/0.38 tff(20,plain,
% 0.19/0.38 (^[X: $i, Y: $i, Z: $i] : refl((convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(21,plain,
% 0.19/0.38 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[20])).
% 0.19/0.38 tff(22,plain,
% 0.19/0.38 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(23,plain,
% 0.19/0.38 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((convergent_lines(X, Z) | convergent_lines(Y, Z)) <=> (convergent_lines(Y, Z) | convergent_lines(X, Z))), ((convergent_lines(X, Y) => (convergent_lines(X, Z) | convergent_lines(Y, Z))) <=> (convergent_lines(X, Y) => (convergent_lines(Y, Z) | convergent_lines(X, Z))))), rewrite((convergent_lines(X, Y) => (convergent_lines(Y, Z) | convergent_lines(X, Z))) <=> (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))), ((convergent_lines(X, Y) => (convergent_lines(X, Z) | convergent_lines(Y, Z))) <=> (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(24,plain,
% 0.19/0.38 (![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => (convergent_lines(X, Z) | convergent_lines(Y, Z))) <=> ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[23])).
% 0.19/0.38 tff(25,axiom,(![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => (convergent_lines(X, Z) | convergent_lines(Y, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','apart6')).
% 0.19/0.38 tff(26,plain,
% 0.19/0.38 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[25, 24])).
% 0.19/0.38 tff(27,plain,
% 0.19/0.38 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[26, 22])).
% 0.19/0.38 tff(28,plain,(
% 0.19/0.38 ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[27])).
% 0.19/0.38 tff(29,plain,
% 0.19/0.38 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[28, 21])).
% 0.19/0.38 tff(30,plain,
% 0.19/0.38 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!1, Y!0)) | convergent_lines(X!1, X!1) | convergent_lines(Y!0, X!1))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!1, Y!0)) | convergent_lines(X!1, X!1) | convergent_lines(Y!0, X!1))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(31,plain,
% 0.19/0.38 ((convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1) | (~convergent_lines(X!1, Y!0))) <=> ((~convergent_lines(X!1, Y!0)) | convergent_lines(X!1, X!1) | convergent_lines(Y!0, X!1))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(32,plain,
% 0.19/0.38 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1) | (~convergent_lines(X!1, Y!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!1, Y!0)) | convergent_lines(X!1, X!1) | convergent_lines(Y!0, X!1)))),
% 0.19/0.38 inference(monotonicity,[status(thm)],[31])).
% 0.19/0.38 tff(33,plain,
% 0.19/0.38 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1) | (~convergent_lines(X!1, Y!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!1, Y!0)) | convergent_lines(X!1, X!1) | convergent_lines(Y!0, X!1))),
% 0.19/0.38 inference(transitivity,[status(thm)],[32, 30])).
% 0.19/0.38 tff(34,plain,
% 0.19/0.38 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1) | (~convergent_lines(X!1, Y!0)))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(35,plain,
% 0.19/0.38 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!1, Y!0)) | convergent_lines(X!1, X!1) | convergent_lines(Y!0, X!1)),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.19/0.38 tff(36,plain,
% 0.19/0.38 ($false),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[35, 29, 19, 10, 1])).
% 0.19/0.38 tff(37,plain,(convergent_lines(Y!0, X!1)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.38 tff(38,plain,
% 0.19/0.38 (^[X: $i] : refl((~distinct_points(X, X)) <=> (~distinct_points(X, X)))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(39,plain,
% 0.19/0.38 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[38])).
% 0.19/0.38 tff(40,plain,
% 0.19/0.38 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(41,axiom,(![X: $i] : (~distinct_points(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','apart1')).
% 0.19/0.38 tff(42,plain,
% 0.19/0.38 (![X: $i] : (~distinct_points(X, X))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.19/0.38 tff(43,plain,(
% 0.19/0.38 ![X: $i] : (~distinct_points(X, X))),
% 0.19/0.38 inference(skolemize,[status(sab)],[42])).
% 0.19/0.38 tff(44,plain,
% 0.19/0.38 (![X: $i] : (~distinct_points(X, X))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[43, 39])).
% 0.19/0.38 tff(45,plain,
% 0.19/0.38 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(Y!0, X!1)))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(46,plain,
% 0.19/0.38 (~distinct_points(intersection_point(Y!0, X!1), intersection_point(Y!0, X!1))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[45, 44])).
% 0.19/0.38 tff(47,plain,
% 0.19/0.38 (^[X: $i, Y: $i, Z: $i] : refl((distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)))) <=> (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(48,plain,
% 0.19/0.38 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[47])).
% 0.19/0.38 tff(49,plain,
% 0.19/0.38 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(50,plain,
% 0.19/0.38 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(trans(monotonicity(rewrite((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) <=> (apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))), (((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y))) <=> ((apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)) => distinct_points(Z, intersection_point(X, Y))))), rewrite(((apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)) => distinct_points(Z, intersection_point(X, Y))) <=> ((~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))) | distinct_points(Z, intersection_point(X, Y)))), (((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y))) <=> ((~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))) | distinct_points(Z, intersection_point(X, Y))))), ((convergent_lines(X, Y) => ((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y)))) <=> (convergent_lines(X, Y) => ((~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))) | distinct_points(Z, intersection_point(X, Y)))))), rewrite((convergent_lines(X, Y) => ((~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))) | distinct_points(Z, intersection_point(X, Y)))) <=> (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))), ((convergent_lines(X, Y) => ((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y)))) <=> (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(51,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => ((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y)))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[50])).
% 0.19/0.39 tff(52,axiom,(![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => ((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y))))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','con2')).
% 0.19/0.39 tff(53,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.19/0.39 tff(54,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[53, 49])).
% 0.19/0.39 tff(55,plain,(
% 0.19/0.39 ![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.19/0.39 inference(skolemize,[status(sab)],[54])).
% 0.19/0.39 tff(56,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[55, 48])).
% 0.19/0.39 tff(57,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(Y!0, X!1), intersection_point(Y!0, X!1)) | (~convergent_lines(Y!0, X!1)) | (~(apart_point_and_line(intersection_point(Y!0, X!1), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | distinct_points(intersection_point(Y!0, X!1), intersection_point(Y!0, X!1)) | (~convergent_lines(Y!0, X!1)) | (~(apart_point_and_line(intersection_point(Y!0, X!1), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(58,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(Y!0, X!1), intersection_point(Y!0, X!1)) | (~convergent_lines(Y!0, X!1)) | (~(apart_point_and_line(intersection_point(Y!0, X!1), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(59,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | distinct_points(intersection_point(Y!0, X!1), intersection_point(Y!0, X!1)) | (~convergent_lines(Y!0, X!1)) | (~(apart_point_and_line(intersection_point(Y!0, X!1), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.19/0.39 tff(60,plain,
% 0.19/0.39 ((~convergent_lines(Y!0, X!1)) | (~(apart_point_and_line(intersection_point(Y!0, X!1), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0)))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[59, 56, 46])).
% 0.19/0.39 tff(61,plain,
% 0.19/0.39 (~(apart_point_and_line(intersection_point(Y!0, X!1), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[60, 37])).
% 0.19/0.39 tff(62,plain,
% 0.19/0.39 ((apart_point_and_line(intersection_point(Y!0, X!1), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0)) | (~apart_point_and_line(intersection_point(Y!0, X!1), X!1))),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(63,plain,
% 0.19/0.39 (~apart_point_and_line(intersection_point(Y!0, X!1), X!1)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[62, 61])).
% 0.19/0.39 tff(64,plain,
% 0.19/0.39 ((apart_point_and_line(intersection_point(Y!0, X!1), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0)) | (~apart_point_and_line(intersection_point(Y!0, X!1), Y!0))),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(65,plain,
% 0.19/0.39 (~apart_point_and_line(intersection_point(Y!0, X!1), Y!0)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[64, 61])).
% 0.19/0.39 tff(66,assumption,(~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))), introduced(assumption)).
% 0.19/0.39 tff(67,plain,
% 0.19/0.39 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(68,plain,
% 0.19/0.39 (~distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[67, 44])).
% 0.19/0.39 tff(69,plain,
% 0.19/0.39 (distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))),
% 0.19/0.39 inference(or_elim,[status(thm)],[18])).
% 0.19/0.39 tff(70,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : refl((distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(71,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[70])).
% 0.19/0.39 tff(72,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(73,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_points(X, Z) | distinct_points(Y, Z)) <=> (distinct_points(Y, Z) | distinct_points(X, Z))), ((distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> (distinct_points(X, Y) => (distinct_points(Y, Z) | distinct_points(X, Z))))), rewrite((distinct_points(X, Y) => (distinct_points(Y, Z) | distinct_points(X, Z))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))), ((distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(74,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[73])).
% 0.19/0.39 tff(75,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','apart4')).
% 0.19/0.39 tff(76,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.19/0.39 tff(77,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[76, 72])).
% 0.19/0.39 tff(78,plain,(
% 0.19/0.39 ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[77])).
% 0.19/0.39 tff(79,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[78, 71])).
% 0.19/0.39 tff(80,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | ((~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(81,plain,
% 0.19/0.39 ((distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))) <=> ((~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(82,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | ((~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[81])).
% 0.19/0.39 tff(83,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)))),
% 0.19/0.39 inference(transitivity,[status(thm)],[82, 80])).
% 0.19/0.39 tff(84,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(85,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.19/0.39 tff(86,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[85, 79, 69, 68, 66])).
% 0.19/0.39 tff(87,plain,(distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.39 tff(88,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | ((~convergent_lines(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(89,plain,
% 0.19/0.39 ((distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~convergent_lines(X!1, Y!0)) | (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1)))) <=> ((~convergent_lines(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(90,plain,
% 0.19/0.39 ((~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1))) <=> (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(91,plain,
% 0.19/0.39 ((distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~convergent_lines(X!1, Y!0)) | (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1)))) <=> (distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~convergent_lines(X!1, Y!0)) | (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[90])).
% 0.19/0.39 tff(92,plain,
% 0.19/0.39 ((distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~convergent_lines(X!1, Y!0)) | (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1)))) <=> ((~convergent_lines(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[91, 89])).
% 0.19/0.39 tff(93,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~convergent_lines(X!1, Y!0)) | (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | ((~convergent_lines(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1)))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[92])).
% 0.19/0.39 tff(94,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~convergent_lines(X!1, Y!0)) | (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[93, 88])).
% 0.19/0.40 tff(95,plain,
% 0.19/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~convergent_lines(X!1, Y!0)) | (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1))))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(96,plain,
% 0.19/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[95, 94])).
% 0.19/0.40 tff(97,plain,
% 0.19/0.40 (~(apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[96, 56, 19, 68])).
% 0.19/0.40 tff(98,plain,
% 0.19/0.40 ((apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1)) | (~apart_point_and_line(intersection_point(X!1, Y!0), X!1))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(99,plain,
% 0.19/0.40 (~apart_point_and_line(intersection_point(X!1, Y!0), X!1)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[98, 97])).
% 0.19/0.40 tff(100,plain,
% 0.19/0.40 ((apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1)) | (~apart_point_and_line(intersection_point(X!1, Y!0), Y!0))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(101,plain,
% 0.19/0.40 (~apart_point_and_line(intersection_point(X!1, Y!0), Y!0)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[100, 97])).
% 0.19/0.40 tff(102,plain,
% 0.19/0.40 (^[X: $i, Y: $i] : refl(((~convergent_lines(X, Y)) | distinct_lines(X, Y)) <=> ((~convergent_lines(X, Y)) | distinct_lines(X, Y)))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(103,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[102])).
% 0.19/0.40 tff(104,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(105,plain,
% 0.19/0.40 (^[X: $i, Y: $i] : rewrite((convergent_lines(X, Y) => distinct_lines(X, Y)) <=> ((~convergent_lines(X, Y)) | distinct_lines(X, Y)))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(106,plain,
% 0.19/0.40 (![X: $i, Y: $i] : (convergent_lines(X, Y) => distinct_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[105])).
% 0.19/0.40 tff(107,axiom,(![X: $i, Y: $i] : (convergent_lines(X, Y) => distinct_lines(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','ceq3')).
% 0.19/0.40 tff(108,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.19/0.40 tff(109,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[108, 104])).
% 0.19/0.40 tff(110,plain,(
% 0.19/0.40 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.19/0.40 inference(skolemize,[status(sab)],[109])).
% 0.19/0.40 tff(111,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[110, 103])).
% 0.19/0.40 tff(112,plain,
% 0.19/0.40 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | ((~convergent_lines(X!1, Y!0)) | distinct_lines(X!1, Y!0))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | (~convergent_lines(X!1, Y!0)) | distinct_lines(X!1, Y!0))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(113,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | ((~convergent_lines(X!1, Y!0)) | distinct_lines(X!1, Y!0))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(114,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | (~convergent_lines(X!1, Y!0)) | distinct_lines(X!1, Y!0)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[113, 112])).
% 0.19/0.40 tff(115,plain,
% 0.19/0.40 (distinct_lines(X!1, Y!0)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[114, 111, 19])).
% 0.19/0.40 tff(116,plain,
% 0.19/0.40 (^[X: $i, Y: $i, U: $i, V: $i] : refl((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(117,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[116])).
% 0.19/0.40 tff(118,plain,
% 0.19/0.40 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(rewrite((distinct_points(X, Y) & distinct_lines(U, V)) <=> (~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> (~(~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))))), rewrite((~(~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))) <=> ((~distinct_points(X, Y)) | (~distinct_lines(U, V)))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> ((~distinct_points(X, Y)) | (~distinct_lines(U, V))))), ((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | ((~distinct_points(X, Y)) | (~distinct_lines(U, V)))))), rewrite((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | ((~distinct_points(X, Y)) | (~distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))), ((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(119,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[118])).
% 0.19/0.40 tff(120,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(121,plain,
% 0.19/0.40 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite((apart_point_and_line(X, U) | apart_point_and_line(X, V)) <=> (apart_point_and_line(X, V) | apart_point_and_line(X, U))), (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) <=> ((apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, U)))), rewrite(((apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, U)) <=> (apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))), (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) <=> (apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)))), ((((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)) <=> ((apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, V)))), rewrite(((apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, V)) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))), ((((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)))), (((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> ((distinct_points(X, Y) & distinct_lines(U, V)) => (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))))), rewrite(((distinct_points(X, Y) & distinct_lines(U, V)) => (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))), (((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(122,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : ((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[121])).
% 0.19/0.40 tff(123,axiom,(![X: $i, Y: $i, U: $i, V: $i] : ((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','cu1')).
% 0.19/0.40 tff(124,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[123, 122])).
% 0.19/0.40 tff(125,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[124, 120])).
% 0.19/0.40 tff(126,plain,(
% 0.19/0.40 ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.40 inference(skolemize,[status(sab)],[125])).
% 0.19/0.40 tff(127,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[126, 119])).
% 0.19/0.40 tff(128,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[127, 117])).
% 0.19/0.40 tff(129,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | ((~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, Y!0)) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), X!1))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, Y!0)) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), X!1))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(130,plain,
% 0.19/0.40 ((apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, Y!0))) <=> ((~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, Y!0)) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), X!1))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(131,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, Y!0)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | ((~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, Y!0)) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), X!1)))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[130])).
% 0.19/0.40 tff(132,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, Y!0)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, Y!0)) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), X!1))),
% 0.19/0.41 inference(transitivity,[status(thm)],[131, 129])).
% 0.19/0.41 tff(133,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, Y!0)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(134,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, Y!0)) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), X!1)),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[133, 132])).
% 0.19/0.41 tff(135,plain,
% 0.19/0.41 ((~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), X!1)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[134, 128, 115])).
% 0.19/0.41 tff(136,plain,
% 0.19/0.41 ((~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), X!1)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[135, 101, 99])).
% 0.19/0.41 tff(137,plain,
% 0.19/0.41 (apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), X!1)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[136, 87])).
% 0.19/0.41 tff(138,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[137, 65, 63])).
% 0.19/0.41 % SZS output end Proof
% 0.19/0.41 % E exiting
%------------------------------------------------------------------------------