%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO203+2 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n010.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 05:05:55 AM UTC 2025
% Result : Theorem 0.19s 0.39s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GEO203+2 : TPTP v9.0.0. Released v3.3.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n010.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Jun 19 14:42:56 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.39 % SZS status Theorem
% 0.19/0.39 % SZS output start Proof
% 0.19/0.39 tff(distinct_points_type, type, (
% 0.19/0.39 distinct_points: ( $i * $i ) > $o)).
% 0.19/0.39 tff(intersection_point_type, type, (
% 0.19/0.39 intersection_point: ( $i * $i ) > $i)).
% 0.19/0.39 tff(tptp_fun_Z_0_type, type, (
% 0.19/0.39 tptp_fun_Z_0: $i)).
% 0.19/0.39 tff(tptp_fun_X_2_type, type, (
% 0.19/0.39 tptp_fun_X_2: $i)).
% 0.19/0.39 tff(tptp_fun_Y_1_type, type, (
% 0.19/0.39 tptp_fun_Y_1: $i)).
% 0.19/0.39 tff(apart_point_and_line_type, type, (
% 0.19/0.39 apart_point_and_line: ( $i * $i ) > $o)).
% 0.19/0.39 tff(line_connecting_type, type, (
% 0.19/0.39 line_connecting: ( $i * $i ) > $i)).
% 0.19/0.39 tff(convergent_lines_type, type, (
% 0.19/0.39 convergent_lines: ( $i * $i ) > $o)).
% 0.19/0.39 tff(distinct_lines_type, type, (
% 0.19/0.39 distinct_lines: ( $i * $i ) > $o)).
% 0.19/0.39 tff(1,plain,
% 0.19/0.39 (^[X: $i] : refl((~distinct_points(X, X)) <=> (~distinct_points(X, X)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(2,plain,
% 0.19/0.39 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.39 tff(3,plain,
% 0.19/0.39 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(4,axiom,(![X: $i] : (~distinct_points(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','apart1')).
% 0.19/0.39 tff(5,plain,
% 0.19/0.39 (![X: $i] : (~distinct_points(X, X))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.39 tff(6,plain,(
% 0.19/0.39 ![X: $i] : (~distinct_points(X, X))),
% 0.19/0.39 inference(skolemize,[status(sab)],[5])).
% 0.19/0.39 tff(7,plain,
% 0.19/0.39 (![X: $i] : (~distinct_points(X, X))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.19/0.39 tff(8,plain,
% 0.19/0.39 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(9,plain,
% 0.19/0.39 (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.19/0.39 tff(10,plain,
% 0.19/0.39 (((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1))),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(11,plain,
% 0.19/0.39 ((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[10, 9])).
% 0.19/0.39 tff(12,plain,
% 0.19/0.39 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(13,plain,
% 0.19/0.39 (~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[12, 7])).
% 0.19/0.39 tff(14,plain,
% 0.19/0.39 (((~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Y!1))) | (~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)))) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0))),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(15,plain,
% 0.19/0.39 ((~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Y!1))) | (~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[14, 13])).
% 0.19/0.39 tff(16,assumption,(apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))), introduced(assumption)).
% 0.19/0.39 tff(17,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_lines(line_connecting(intersection_point(X, Y), intersection_point(X, Z)), X)) | (~(convergent_lines(X, Y) & convergent_lines(X, Z) & distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~distinct_lines(line_connecting(intersection_point(X, Y), intersection_point(X, Z)), X)) | (~(convergent_lines(X, Y) & convergent_lines(X, Z) & distinct_points(intersection_point(X, Y), intersection_point(X, Z))))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(18,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (((convergent_lines(X, Y) & convergent_lines(X, Z)) & distinct_points(intersection_point(X, Y), intersection_point(X, Z))) => (~distinct_lines(line_connecting(intersection_point(X, Y), intersection_point(X, Z)), X)))) <=> (~![X: $i, Y: $i, Z: $i] : ((~distinct_lines(line_connecting(intersection_point(X, Y), intersection_point(X, Z)), X)) | (~(convergent_lines(X, Y) & convergent_lines(X, Z) & distinct_points(intersection_point(X, Y), intersection_point(X, Z))))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(19,axiom,(~![X: $i, Y: $i, Z: $i] : (((convergent_lines(X, Y) & convergent_lines(X, Z)) & distinct_points(intersection_point(X, Y), intersection_point(X, Z))) => (~distinct_lines(line_connecting(intersection_point(X, Y), intersection_point(X, Z)), X)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.19/0.39 tff(20,plain,
% 0.19/0.39 (~![X: $i, Y: $i, Z: $i] : ((~distinct_lines(line_connecting(intersection_point(X, Y), intersection_point(X, Z)), X)) | (~(convergent_lines(X, Y) & convergent_lines(X, Z) & distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[19, 18])).
% 0.19/0.39 tff(21,plain,
% 0.19/0.39 (~![X: $i, Y: $i, Z: $i] : ((~distinct_lines(line_connecting(intersection_point(X, Y), intersection_point(X, Z)), X)) | (~(convergent_lines(X, Y) & convergent_lines(X, Z) & distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[20, 17])).
% 0.19/0.39 tff(22,plain,
% 0.19/0.39 (~![X: $i, Y: $i, Z: $i] : ((~distinct_lines(line_connecting(intersection_point(X, Y), intersection_point(X, Z)), X)) | (~(convergent_lines(X, Y) & convergent_lines(X, Z) & distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[21, 17])).
% 0.19/0.39 tff(23,plain,
% 0.19/0.39 (~![X: $i, Y: $i, Z: $i] : ((~distinct_lines(line_connecting(intersection_point(X, Y), intersection_point(X, Z)), X)) | (~(convergent_lines(X, Y) & convergent_lines(X, Z) & distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[22, 17])).
% 0.19/0.39 tff(24,plain,(
% 0.19/0.39 ~((~distinct_lines(line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)), X!2)) | (~(convergent_lines(X!2, Y!1) & convergent_lines(X!2, Z!0) & distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))))),
% 0.19/0.39 inference(skolemize,[status(sab)],[23])).
% 0.19/0.39 tff(25,plain,
% 0.19/0.39 (convergent_lines(X!2, Y!1) & convergent_lines(X!2, Z!0) & distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))),
% 0.19/0.39 inference(or_elim,[status(thm)],[24])).
% 0.19/0.39 tff(26,plain,
% 0.19/0.39 (distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))),
% 0.19/0.39 inference(and_elim,[status(thm)],[25])).
% 0.19/0.39 tff(27,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : refl(((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(28,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))) <=> ![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[27])).
% 0.19/0.39 tff(29,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_points(Z, X) & distinct_points(Z, Y)) <=> (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))), (((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ((~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))))), rewrite(((~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))), (((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(30,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[29])).
% 0.19/0.39 tff(31,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(32,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y))) <=> ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))), ((distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> (distinct_points(X, Y) => ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))))), rewrite((distinct_points(X, Y) => ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))), ((distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(33,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[32])).
% 0.19/0.39 tff(34,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y))))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','con1')).
% 0.19/0.39 tff(35,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.19/0.39 tff(36,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[35, 31])).
% 0.19/0.39 tff(37,plain,(
% 0.19/0.39 ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.19/0.39 inference(skolemize,[status(sab)],[36])).
% 0.19/0.39 tff(38,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[37, 30])).
% 0.19/0.39 tff(39,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[38, 28])).
% 0.19/0.39 tff(40,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))) | (~((~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Y!1))) | (~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0))))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))) | (~((~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Y!1))) | (~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0))))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(41,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))) | (~((~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Y!1))) | (~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0))))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(42,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))) | (~((~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Y!1))) | (~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.19/0.39 tff(43,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[42, 39, 26, 16, 15])).
% 0.19/0.39 tff(44,plain,(~apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.39 tff(45,plain,
% 0.19/0.39 (convergent_lines(X!2, Z!0)),
% 0.19/0.39 inference(and_elim,[status(thm)],[25])).
% 0.19/0.39 tff(46,plain,
% 0.19/0.39 (^[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)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(47,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)))) <=> ![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)],[46])).
% 0.19/0.39 tff(48,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)))) <=> ![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(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(49,plain,
% 0.19/0.39 (^[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(50,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)],[49])).
% 0.19/0.39 tff(51,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(52,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)],[51, 50])).
% 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, 48])).
% 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(skolemize,[status(sab)],[53])).
% 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(modus_ponens,[status(thm)],[54, 47])).
% 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))))) | ((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!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))))) | (~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(57,plain,
% 0.19/0.40 ((distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~convergent_lines(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0)))) <=> ((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(58,plain,
% 0.19/0.40 ((~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))) <=> (~(apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(59,plain,
% 0.19/0.40 ((distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~convergent_lines(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2)))) <=> (distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~convergent_lines(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[58])).
% 0.19/0.40 tff(60,plain,
% 0.19/0.40 ((distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~convergent_lines(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2)))) <=> ((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[59, 57])).
% 0.19/0.40 tff(61,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!2, Z!0), intersection_point(X!2, Z!0)) | (~convergent_lines(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))))) <=> ((~![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!2, Z!0)) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0)))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[60])).
% 0.19/0.40 tff(62,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!2, Z!0), intersection_point(X!2, Z!0)) | (~convergent_lines(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))))) <=> ((~![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!2, Z!0)) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[61, 56])).
% 0.19/0.40 tff(63,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!2, Z!0), intersection_point(X!2, Z!0)) | (~convergent_lines(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(64,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!2, Z!0)) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[63, 62])).
% 0.19/0.40 tff(65,plain,
% 0.19/0.40 (~(apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[64, 55, 45, 13])).
% 0.19/0.40 tff(66,plain,
% 0.19/0.40 ((apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0)) | (~apart_point_and_line(intersection_point(X!2, Z!0), X!2))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(67,plain,
% 0.19/0.40 (~apart_point_and_line(intersection_point(X!2, Z!0), X!2)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[66, 65])).
% 0.19/0.40 tff(68,plain,
% 0.19/0.40 (convergent_lines(X!2, Y!1)),
% 0.19/0.40 inference(and_elim,[status(thm)],[25])).
% 0.19/0.40 tff(69,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!2, Y!1)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!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!2, Y!1)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(70,plain,
% 0.19/0.40 ((distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1)))) <=> ((~convergent_lines(X!2, Y!1)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(71,plain,
% 0.19/0.40 ((~(apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2))) <=> (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(72,plain,
% 0.19/0.40 ((distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2)))) <=> (distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[71])).
% 0.19/0.40 tff(73,plain,
% 0.19/0.40 ((distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2)))) <=> ((~convergent_lines(X!2, Y!1)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[72, 70])).
% 0.19/0.40 tff(74,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!2, Y!1), intersection_point(X!2, Y!1)) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2))))) <=> ((~![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!2, Y!1)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1)))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[73])).
% 0.19/0.40 tff(75,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!2, Y!1), intersection_point(X!2, Y!1)) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2))))) <=> ((~![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!2, Y!1)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[74, 69])).
% 0.19/0.40 tff(76,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!2, Y!1), intersection_point(X!2, Y!1)) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2))))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(77,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!2, Y!1)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.19/0.40 tff(78,plain,
% 0.19/0.40 (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[77, 55, 68, 9])).
% 0.19/0.40 tff(79,plain,
% 0.19/0.40 ((apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), X!2))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(80,plain,
% 0.19/0.40 (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[79, 78])).
% 0.19/0.40 tff(81,plain,
% 0.19/0.40 (distinct_lines(line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)), X!2)),
% 0.19/0.40 inference(or_elim,[status(thm)],[24])).
% 0.19/0.40 tff(82,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(83,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)],[82])).
% 0.19/0.40 tff(84,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(85,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)],[84])).
% 0.19/0.40 tff(86,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(87,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(88,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)],[87])).
% 0.19/0.40 tff(89,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(90,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)],[89, 88])).
% 0.19/0.40 tff(91,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)],[90, 86])).
% 0.19/0.40 tff(92,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)],[91])).
% 0.19/0.40 tff(93,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)],[92, 85])).
% 0.19/0.40 tff(94,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)],[93, 83])).
% 0.19/0.40 tff(95,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_lines(line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)), X!2)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!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_lines(line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)), X!2)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(96,plain,
% 0.19/0.41 ((apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~distinct_lines(line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)), X!2))) <=> ((~distinct_lines(line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)), X!2)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(97,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!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~distinct_lines(line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)), X!2)))) <=> ((~![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_lines(line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)), X!2)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[96])).
% 0.19/0.41 tff(98,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!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~distinct_lines(line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)), X!2)))) <=> ((~![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_lines(line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)), X!2)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))),
% 0.19/0.41 inference(transitivity,[status(thm)],[97, 95])).
% 0.19/0.41 tff(99,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!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~distinct_lines(line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)), X!2)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(100,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_lines(line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)), X!2)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.19/0.41 tff(101,plain,
% 0.19/0.41 (apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[100, 94, 81, 26, 80, 67, 44])).
% 0.19/0.41 tff(102,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))) | (~((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))) | (~((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(103,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))) | (~((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(104,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | (~apart_point_and_line(intersection_point(X!2, Y!1), line_connecting(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))) | (~((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[103, 102])).
% 0.19/0.41 tff(105,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[104, 39, 26, 101, 11])).
% 0.19/0.41 % SZS output end Proof
% 0.19/0.41 % E exiting
%------------------------------------------------------------------------------