%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO187+3 : TPTP v9.0.0. Released v4.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n002.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 05:05:49 AM UTC 2025
% Result : Theorem 0.19s 0.37s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : GEO187+3 : TPTP v9.0.0. Released v4.0.0.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n002.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 14:42:01 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.37 % SZS status Theorem
% 0.19/0.37 % SZS output start Proof
% 0.19/0.37 tff(incident_point_and_line_type, type, (
% 0.19/0.37 incident_point_and_line: ( $i * $i ) > $o)).
% 0.19/0.37 tff(line_connecting_type, type, (
% 0.19/0.37 line_connecting: ( $i * $i ) > $i)).
% 0.19/0.37 tff(tptp_fun_Y_2_type, type, (
% 0.19/0.37 tptp_fun_Y_2: $i)).
% 0.19/0.37 tff(tptp_fun_X_3_type, type, (
% 0.19/0.37 tptp_fun_X_3: $i)).
% 0.19/0.37 tff(tptp_fun_V_0_type, type, (
% 0.19/0.37 tptp_fun_V_0: $i)).
% 0.19/0.37 tff(apart_point_and_line_type, type, (
% 0.19/0.37 apart_point_and_line: ( $i * $i ) > $o)).
% 0.19/0.37 tff(distinct_lines_type, type, (
% 0.19/0.37 distinct_lines: ( $i * $i ) > $o)).
% 0.19/0.37 tff(tptp_fun_U_1_type, type, (
% 0.19/0.37 tptp_fun_U_1: $i)).
% 0.19/0.37 tff(distinct_points_type, type, (
% 0.19/0.37 distinct_points: ( $i * $i ) > $o)).
% 0.19/0.37 tff(1,plain,
% 0.19/0.37 (^[X: $i, Y: $i] : refl((~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y))) <=> (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y))))),
% 0.19/0.37 inference(bind,[status(th)],[])).
% 0.19/0.37 tff(2,plain,
% 0.19/0.37 (![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y))) <=> ![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))),
% 0.19/0.37 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.37 tff(3,plain,
% 0.19/0.37 (![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y))) <=> ![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))),
% 0.19/0.37 inference(rewrite,[status(thm)],[])).
% 0.19/0.37 tff(4,plain,
% 0.19/0.37 (^[X: $i, Y: $i] : trans(rewrite((incident_point_and_line(X, Y) <=> (~apart_point_and_line(X, Y))) <=> (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))), rewrite((~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y))) <=> (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))), ((incident_point_and_line(X, Y) <=> (~apart_point_and_line(X, Y))) <=> (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))))),
% 0.19/0.37 inference(bind,[status(th)],[])).
% 0.19/0.37 tff(5,plain,
% 0.19/0.37 (![X: $i, Y: $i] : (incident_point_and_line(X, Y) <=> (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))),
% 0.19/0.37 inference(quant_intro,[status(thm)],[4])).
% 0.19/0.37 tff(6,axiom,(![X: $i, Y: $i] : (incident_point_and_line(X, Y) <=> (~apart_point_and_line(X, Y)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+6.ax','a4')).
% 0.19/0.37 tff(7,plain,
% 0.19/0.37 (![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[6, 5])).
% 0.19/0.37 tff(8,plain,
% 0.19/0.37 (![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.19/0.37 tff(9,plain,(
% 0.19/0.37 ![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))),
% 0.19/0.37 inference(skolemize,[status(sab)],[8])).
% 0.19/0.37 tff(10,plain,
% 0.19/0.37 (![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[9, 2])).
% 0.19/0.37 tff(11,plain,
% 0.19/0.37 ((~![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))) | (~(apart_point_and_line(V!0, line_connecting(X!3, Y!2)) <=> incident_point_and_line(V!0, line_connecting(X!3, Y!2))))),
% 0.19/0.37 inference(quant_inst,[status(thm)],[])).
% 0.19/0.37 tff(12,plain,
% 0.19/0.37 (~(apart_point_and_line(V!0, line_connecting(X!3, Y!2)) <=> incident_point_and_line(V!0, line_connecting(X!3, Y!2)))),
% 0.19/0.37 inference(unit_resolution,[status(thm)],[11, 10])).
% 0.19/0.37 tff(13,plain,
% 0.19/0.37 ((~![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))) | (~(apart_point_and_line(Y!2, line_connecting(U!1, V!0)) <=> incident_point_and_line(Y!2, line_connecting(U!1, V!0))))),
% 0.19/0.37 inference(quant_inst,[status(thm)],[])).
% 0.19/0.37 tff(14,plain,
% 0.19/0.37 (~(apart_point_and_line(Y!2, line_connecting(U!1, V!0)) <=> incident_point_and_line(Y!2, line_connecting(U!1, V!0)))),
% 0.19/0.37 inference(unit_resolution,[status(thm)],[13, 10])).
% 0.19/0.37 tff(15,plain,
% 0.19/0.37 ((~![X: $i, Y: $i, U: $i, V: $i] : ((~(distinct_points(X, Y) & distinct_points(U, V) & incident_point_and_line(X, line_connecting(U, V)) & incident_point_and_line(Y, line_connecting(U, V)))) | (incident_point_and_line(U, line_connecting(X, Y)) & incident_point_and_line(V, line_connecting(X, Y))))) <=> (~![X: $i, Y: $i, U: $i, V: $i] : ((~(distinct_points(X, Y) & distinct_points(U, V) & incident_point_and_line(X, line_connecting(U, V)) & incident_point_and_line(Y, line_connecting(U, V)))) | (incident_point_and_line(U, line_connecting(X, Y)) & incident_point_and_line(V, line_connecting(X, Y)))))),
% 0.19/0.37 inference(rewrite,[status(thm)],[])).
% 0.19/0.37 tff(16,plain,
% 0.19/0.37 ((~![X: $i, Y: $i, U: $i, V: $i] : ((((distinct_points(X, Y) & distinct_points(U, V)) & incident_point_and_line(X, line_connecting(U, V))) & incident_point_and_line(Y, line_connecting(U, V))) => (incident_point_and_line(U, line_connecting(X, Y)) & incident_point_and_line(V, line_connecting(X, Y))))) <=> (~![X: $i, Y: $i, U: $i, V: $i] : ((~(distinct_points(X, Y) & distinct_points(U, V) & incident_point_and_line(X, line_connecting(U, V)) & incident_point_and_line(Y, line_connecting(U, V)))) | (incident_point_and_line(U, line_connecting(X, Y)) & incident_point_and_line(V, line_connecting(X, Y)))))),
% 0.19/0.37 inference(rewrite,[status(thm)],[])).
% 0.19/0.37 tff(17,axiom,(~![X: $i, Y: $i, U: $i, V: $i] : ((((distinct_points(X, Y) & distinct_points(U, V)) & incident_point_and_line(X, line_connecting(U, V))) & incident_point_and_line(Y, line_connecting(U, V))) => (incident_point_and_line(U, line_connecting(X, Y)) & incident_point_and_line(V, line_connecting(X, Y))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.19/0.37 tff(18,plain,
% 0.19/0.37 (~![X: $i, Y: $i, U: $i, V: $i] : ((~(distinct_points(X, Y) & distinct_points(U, V) & incident_point_and_line(X, line_connecting(U, V)) & incident_point_and_line(Y, line_connecting(U, V)))) | (incident_point_and_line(U, line_connecting(X, Y)) & incident_point_and_line(V, line_connecting(X, Y))))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.19/0.37 tff(19,plain,
% 0.19/0.37 (~![X: $i, Y: $i, U: $i, V: $i] : ((~(distinct_points(X, Y) & distinct_points(U, V) & incident_point_and_line(X, line_connecting(U, V)) & incident_point_and_line(Y, line_connecting(U, V)))) | (incident_point_and_line(U, line_connecting(X, Y)) & incident_point_and_line(V, line_connecting(X, Y))))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[18, 15])).
% 0.19/0.37 tff(20,plain,
% 0.19/0.37 (~![X: $i, Y: $i, U: $i, V: $i] : ((~(distinct_points(X, Y) & distinct_points(U, V) & incident_point_and_line(X, line_connecting(U, V)) & incident_point_and_line(Y, line_connecting(U, V)))) | (incident_point_and_line(U, line_connecting(X, Y)) & incident_point_and_line(V, line_connecting(X, Y))))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[19, 15])).
% 0.19/0.37 tff(21,plain,
% 0.19/0.37 (~![X: $i, Y: $i, U: $i, V: $i] : ((~(distinct_points(X, Y) & distinct_points(U, V) & incident_point_and_line(X, line_connecting(U, V)) & incident_point_and_line(Y, line_connecting(U, V)))) | (incident_point_and_line(U, line_connecting(X, Y)) & incident_point_and_line(V, line_connecting(X, Y))))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[20, 15])).
% 0.19/0.37 tff(22,plain,(
% 0.19/0.37 ~((~(distinct_points(X!3, Y!2) & distinct_points(U!1, V!0) & incident_point_and_line(X!3, line_connecting(U!1, V!0)) & incident_point_and_line(Y!2, line_connecting(U!1, V!0)))) | (incident_point_and_line(U!1, line_connecting(X!3, Y!2)) & incident_point_and_line(V!0, line_connecting(X!3, Y!2))))),
% 0.19/0.37 inference(skolemize,[status(sab)],[21])).
% 0.19/0.37 tff(23,plain,
% 0.19/0.37 (distinct_points(X!3, Y!2) & distinct_points(U!1, V!0) & incident_point_and_line(X!3, line_connecting(U!1, V!0)) & incident_point_and_line(Y!2, line_connecting(U!1, V!0))),
% 0.19/0.37 inference(or_elim,[status(thm)],[22])).
% 0.19/0.37 tff(24,plain,
% 0.19/0.37 (incident_point_and_line(Y!2, line_connecting(U!1, V!0))),
% 0.19/0.37 inference(and_elim,[status(thm)],[23])).
% 0.19/0.37 tff(25,plain,
% 0.19/0.37 ((apart_point_and_line(Y!2, line_connecting(U!1, V!0)) <=> incident_point_and_line(Y!2, line_connecting(U!1, V!0))) | (~apart_point_and_line(Y!2, line_connecting(U!1, V!0))) | (~incident_point_and_line(Y!2, line_connecting(U!1, V!0)))),
% 0.19/0.37 inference(tautology,[status(thm)],[])).
% 0.19/0.37 tff(26,plain,
% 0.19/0.37 ((apart_point_and_line(Y!2, line_connecting(U!1, V!0)) <=> incident_point_and_line(Y!2, line_connecting(U!1, V!0))) | (~apart_point_and_line(Y!2, line_connecting(U!1, V!0)))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[25, 24])).
% 0.19/0.38 tff(27,plain,
% 0.19/0.38 (~apart_point_and_line(Y!2, line_connecting(U!1, V!0))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[26, 14])).
% 0.19/0.38 tff(28,plain,
% 0.19/0.38 ((~![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))) | (~(apart_point_and_line(X!3, line_connecting(U!1, V!0)) <=> incident_point_and_line(X!3, line_connecting(U!1, V!0))))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(29,plain,
% 0.19/0.38 (~(apart_point_and_line(X!3, line_connecting(U!1, V!0)) <=> incident_point_and_line(X!3, line_connecting(U!1, V!0)))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[28, 10])).
% 0.19/0.38 tff(30,plain,
% 0.19/0.38 (incident_point_and_line(X!3, line_connecting(U!1, V!0))),
% 0.19/0.38 inference(and_elim,[status(thm)],[23])).
% 0.19/0.38 tff(31,plain,
% 0.19/0.38 ((apart_point_and_line(X!3, line_connecting(U!1, V!0)) <=> incident_point_and_line(X!3, line_connecting(U!1, V!0))) | (~apart_point_and_line(X!3, line_connecting(U!1, V!0))) | (~incident_point_and_line(X!3, line_connecting(U!1, V!0)))),
% 0.19/0.38 inference(tautology,[status(thm)],[])).
% 0.19/0.38 tff(32,plain,
% 0.19/0.38 ((apart_point_and_line(X!3, line_connecting(U!1, V!0)) <=> incident_point_and_line(X!3, line_connecting(U!1, V!0))) | (~apart_point_and_line(X!3, line_connecting(U!1, V!0)))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[31, 30])).
% 0.19/0.38 tff(33,plain,
% 0.19/0.38 (~apart_point_and_line(X!3, line_connecting(U!1, V!0))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[32, 29])).
% 0.19/0.38 tff(34,plain,
% 0.19/0.38 (distinct_points(X!3, Y!2)),
% 0.19/0.38 inference(and_elim,[status(thm)],[23])).
% 0.19/0.38 tff(35,plain,
% 0.19/0.38 (^[X: $i, Y: $i] : refl(((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(36,plain,
% 0.19/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[35])).
% 0.19/0.38 tff(37,plain,
% 0.19/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(38,plain,
% 0.19/0.38 (^[X: $i, Y: $i] : rewrite((distinct_points(X, Y) => (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(39,plain,
% 0.19/0.38 (![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[38])).
% 0.19/0.38 tff(40,axiom,(![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(X, line_connecting(X, Y))))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ci1')).
% 0.19/0.38 tff(41,plain,
% 0.19/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.19/0.38 tff(42,plain,
% 0.19/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[41, 37])).
% 0.19/0.38 tff(43,plain,(
% 0.19/0.38 ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.38 inference(skolemize,[status(sab)],[42])).
% 0.19/0.38 tff(44,plain,
% 0.19/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[43, 36])).
% 0.19/0.38 tff(45,plain,
% 0.19/0.38 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(X!3, Y!2)) | (~apart_point_and_line(X!3, line_connecting(X!3, Y!2))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(X!3, Y!2)) | (~apart_point_and_line(X!3, line_connecting(X!3, Y!2))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(46,plain,
% 0.19/0.38 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(X!3, Y!2)) | (~apart_point_and_line(X!3, line_connecting(X!3, Y!2))))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(47,plain,
% 0.19/0.38 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(X!3, Y!2)) | (~apart_point_and_line(X!3, line_connecting(X!3, Y!2)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.19/0.38 tff(48,plain,
% 0.19/0.38 (~apart_point_and_line(X!3, line_connecting(X!3, Y!2))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[47, 44, 34])).
% 0.19/0.38 tff(49,plain,
% 0.19/0.38 (^[X: $i, Y: $i] : refl(((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(50,plain,
% 0.19/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[49])).
% 0.19/0.38 tff(51,plain,
% 0.19/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(52,plain,
% 0.19/0.38 (^[X: $i, Y: $i] : rewrite((distinct_points(X, Y) => (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(53,plain,
% 0.19/0.38 (![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[52])).
% 0.19/0.38 tff(54,axiom,(![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(Y, line_connecting(X, Y))))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ci2')).
% 0.19/0.38 tff(55,plain,
% 0.19/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.19/0.38 tff(56,plain,
% 0.19/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[55, 51])).
% 0.19/0.38 tff(57,plain,(
% 0.19/0.38 ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.38 inference(skolemize,[status(sab)],[56])).
% 0.19/0.38 tff(58,plain,
% 0.19/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[57, 50])).
% 0.19/0.38 tff(59,plain,
% 0.19/0.38 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(X!3, Y!2)) | (~apart_point_and_line(Y!2, line_connecting(X!3, Y!2))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(X!3, Y!2)) | (~apart_point_and_line(Y!2, line_connecting(X!3, Y!2))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(60,plain,
% 0.19/0.38 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(X!3, Y!2)) | (~apart_point_and_line(Y!2, line_connecting(X!3, Y!2))))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(61,plain,
% 0.19/0.38 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(X!3, Y!2)) | (~apart_point_and_line(Y!2, line_connecting(X!3, Y!2)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[60, 59])).
% 0.19/0.38 tff(62,plain,
% 0.19/0.38 (~apart_point_and_line(Y!2, line_connecting(X!3, Y!2))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[61, 58, 34])).
% 0.19/0.38 tff(63,plain,
% 0.19/0.38 (^[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_lines(U, V)) | (~distinct_points(X, Y))) <=> (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_lines(U, V)) | (~distinct_points(X, Y))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(64,plain,
% 0.19/0.38 (![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_lines(U, V)) | (~distinct_points(X, Y))) <=> ![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_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[63])).
% 0.19/0.38 tff(65,plain,
% 0.19/0.38 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(rewrite((distinct_points(X, Y) & distinct_lines(U, V)) <=> (~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> (~(~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))))), rewrite((~(~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))) <=> ((~distinct_lines(U, V)) | (~distinct_points(X, Y)))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> ((~distinct_lines(U, V)) | (~distinct_points(X, Y))))), ((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_lines(U, V)) | (~distinct_points(X, Y)))))), 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_lines(U, V)) | (~distinct_points(X, Y)))) <=> (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_lines(U, V)) | (~distinct_points(X, Y)))), ((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_lines(U, V)) | (~distinct_points(X, Y)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(66,plain,
% 0.19/0.38 (![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_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[65])).
% 0.19/0.38 tff(67,plain,
% 0.19/0.38 (![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))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(68,plain,
% 0.19/0.38 (^[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))))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(69,plain,
% 0.19/0.38 (![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))))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[68])).
% 0.19/0.38 tff(70,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/GEO006+0.ax','cu1')).
% 0.19/0.38 tff(71,plain,
% 0.19/0.38 (![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))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[70, 69])).
% 0.19/0.38 tff(72,plain,
% 0.19/0.38 (![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))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[71, 67])).
% 0.19/0.38 tff(73,plain,(
% 0.19/0.38 ![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))))),
% 0.19/0.38 inference(skolemize,[status(sab)],[72])).
% 0.19/0.38 tff(74,plain,
% 0.19/0.38 (![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_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[73, 66])).
% 0.19/0.38 tff(75,plain,
% 0.19/0.38 (![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_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[74, 64])).
% 0.19/0.38 tff(76,plain,
% 0.19/0.38 (((~![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_lines(U, V)) | (~distinct_points(X, Y)))) | ((~distinct_points(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(U!1, V!0)) | apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | (~distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!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_lines(U, V)) | (~distinct_points(X, Y)))) | (~distinct_points(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(U!1, V!0)) | apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | (~distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(77,plain,
% 0.19/0.38 ((apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(U!1, V!0)) | apart_point_and_line(X!3, line_connecting(X!3, Y!2)) | (~distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0))) | (~distinct_points(X!3, Y!2))) <=> ((~distinct_points(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(U!1, V!0)) | apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | (~distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(78,plain,
% 0.19/0.38 (((~![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_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(U!1, V!0)) | apart_point_and_line(X!3, line_connecting(X!3, Y!2)) | (~distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0))) | (~distinct_points(X!3, Y!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_lines(U, V)) | (~distinct_points(X, Y)))) | ((~distinct_points(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(U!1, V!0)) | apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | (~distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)))))),
% 0.19/0.38 inference(monotonicity,[status(thm)],[77])).
% 0.19/0.38 tff(79,plain,
% 0.19/0.38 (((~![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_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(U!1, V!0)) | apart_point_and_line(X!3, line_connecting(X!3, Y!2)) | (~distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0))) | (~distinct_points(X!3, Y!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_lines(U, V)) | (~distinct_points(X, Y)))) | (~distinct_points(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(U!1, V!0)) | apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | (~distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0))))),
% 0.19/0.38 inference(transitivity,[status(thm)],[78, 76])).
% 0.19/0.38 tff(80,plain,
% 0.19/0.38 ((~![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_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(U!1, V!0)) | apart_point_and_line(X!3, line_connecting(X!3, Y!2)) | (~distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0))) | (~distinct_points(X!3, Y!2)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(81,plain,
% 0.19/0.39 ((~![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_lines(U, V)) | (~distinct_points(X, Y)))) | (~distinct_points(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(X!3, Y!2)) | apart_point_and_line(X!3, line_connecting(U!1, V!0)) | apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | (~distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[80, 79])).
% 0.19/0.39 tff(82,plain,
% 0.19/0.39 (~distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[81, 75, 34, 62, 48, 33, 27])).
% 0.19/0.39 tff(83,plain,
% 0.19/0.39 (distinct_points(U!1, V!0)),
% 0.19/0.39 inference(and_elim,[status(thm)],[23])).
% 0.19/0.39 tff(84,plain,
% 0.19/0.39 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(U!1, V!0))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(U!1, V!0))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(85,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(U!1, V!0))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(86,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(U!1, V!0)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.19/0.39 tff(87,plain,
% 0.19/0.39 (~apart_point_and_line(V!0, line_connecting(U!1, V!0))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[86, 58, 83])).
% 0.19/0.39 tff(88,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : refl((distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(89,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[88])).
% 0.19/0.39 tff(90,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(91,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_lines(Y, Z) | apart_point_and_line(X, Z)) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z))), ((apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> (apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))))), rewrite((apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))), ((apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(92,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[91])).
% 0.19/0.39 tff(93,axiom,(![X: $i, Y: $i, Z: $i] : (apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ceq2')).
% 0.19/0.39 tff(94,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.19/0.39 tff(95,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[94, 90])).
% 0.19/0.39 tff(96,plain,(
% 0.19/0.39 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[95])).
% 0.19/0.39 tff(97,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[96, 89])).
% 0.19/0.39 tff(98,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (apart_point_and_line(V!0, line_connecting(U!1, V!0)) | distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(X!3, Y!2))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(V!0, line_connecting(U!1, V!0)) | distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(X!3, Y!2))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(99,plain,
% 0.19/0.39 ((distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | apart_point_and_line(V!0, line_connecting(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(X!3, Y!2)))) <=> (apart_point_and_line(V!0, line_connecting(U!1, V!0)) | distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(X!3, Y!2))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(100,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | apart_point_and_line(V!0, line_connecting(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(X!3, Y!2))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (apart_point_and_line(V!0, line_connecting(U!1, V!0)) | distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(X!3, Y!2)))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[99])).
% 0.19/0.39 tff(101,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | apart_point_and_line(V!0, line_connecting(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(X!3, Y!2))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(V!0, line_connecting(U!1, V!0)) | distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(X!3, Y!2))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[100, 98])).
% 0.19/0.39 tff(102,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | apart_point_and_line(V!0, line_connecting(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(X!3, Y!2))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(103,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(V!0, line_connecting(U!1, V!0)) | distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(X!3, Y!2)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.19/0.39 tff(104,plain,
% 0.19/0.39 (~apart_point_and_line(V!0, line_connecting(X!3, Y!2))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[103, 97, 87, 82])).
% 0.19/0.39 tff(105,plain,
% 0.19/0.39 ((apart_point_and_line(V!0, line_connecting(X!3, Y!2)) <=> incident_point_and_line(V!0, line_connecting(X!3, Y!2))) | apart_point_and_line(V!0, line_connecting(X!3, Y!2)) | incident_point_and_line(V!0, line_connecting(X!3, Y!2))),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(106,plain,
% 0.19/0.39 ((apart_point_and_line(V!0, line_connecting(X!3, Y!2)) <=> incident_point_and_line(V!0, line_connecting(X!3, Y!2))) | incident_point_and_line(V!0, line_connecting(X!3, Y!2))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[105, 104])).
% 0.19/0.39 tff(107,plain,
% 0.19/0.39 (incident_point_and_line(V!0, line_connecting(X!3, Y!2))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[106, 12])).
% 0.19/0.39 tff(108,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))) | (~(apart_point_and_line(U!1, line_connecting(X!3, Y!2)) <=> incident_point_and_line(U!1, line_connecting(X!3, Y!2))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(109,plain,
% 0.19/0.39 (~(apart_point_and_line(U!1, line_connecting(X!3, Y!2)) <=> incident_point_and_line(U!1, line_connecting(X!3, Y!2)))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[108, 10])).
% 0.19/0.39 tff(110,plain,
% 0.19/0.39 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(U!1, V!0))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(U!1, V!0))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(111,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(U!1, V!0))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(112,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(U!1, V!0)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[111, 110])).
% 0.19/0.39 tff(113,plain,
% 0.19/0.39 (~apart_point_and_line(U!1, line_connecting(U!1, V!0))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[112, 44, 83])).
% 0.19/0.39 tff(114,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (apart_point_and_line(U!1, line_connecting(U!1, V!0)) | distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(X!3, Y!2))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(U!1, line_connecting(U!1, V!0)) | distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(X!3, Y!2))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(115,plain,
% 0.19/0.39 ((distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | apart_point_and_line(U!1, line_connecting(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(X!3, Y!2)))) <=> (apart_point_and_line(U!1, line_connecting(U!1, V!0)) | distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(X!3, Y!2))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(116,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | apart_point_and_line(U!1, line_connecting(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(X!3, Y!2))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (apart_point_and_line(U!1, line_connecting(U!1, V!0)) | distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(X!3, Y!2)))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[115])).
% 0.19/0.39 tff(117,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | apart_point_and_line(U!1, line_connecting(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(X!3, Y!2))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(U!1, line_connecting(U!1, V!0)) | distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(X!3, Y!2))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[116, 114])).
% 0.19/0.39 tff(118,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | apart_point_and_line(U!1, line_connecting(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(X!3, Y!2))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(119,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(U!1, line_connecting(U!1, V!0)) | distinct_lines(line_connecting(X!3, Y!2), line_connecting(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(X!3, Y!2)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[118, 117])).
% 0.19/0.39 tff(120,plain,
% 0.19/0.39 (~apart_point_and_line(U!1, line_connecting(X!3, Y!2))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[119, 97, 113, 82])).
% 0.19/0.39 tff(121,plain,
% 0.19/0.39 ((apart_point_and_line(U!1, line_connecting(X!3, Y!2)) <=> incident_point_and_line(U!1, line_connecting(X!3, Y!2))) | apart_point_and_line(U!1, line_connecting(X!3, Y!2)) | incident_point_and_line(U!1, line_connecting(X!3, Y!2))),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(122,plain,
% 0.19/0.39 ((apart_point_and_line(U!1, line_connecting(X!3, Y!2)) <=> incident_point_and_line(U!1, line_connecting(X!3, Y!2))) | incident_point_and_line(U!1, line_connecting(X!3, Y!2))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[121, 120])).
% 0.19/0.39 tff(123,plain,
% 0.19/0.39 (incident_point_and_line(U!1, line_connecting(X!3, Y!2))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[122, 109])).
% 0.19/0.39 tff(124,plain,
% 0.19/0.39 ((~(~((~incident_point_and_line(U!1, line_connecting(X!3, Y!2))) | (~incident_point_and_line(V!0, line_connecting(X!3, Y!2)))))) <=> ((~incident_point_and_line(U!1, line_connecting(X!3, Y!2))) | (~incident_point_and_line(V!0, line_connecting(X!3, Y!2))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(125,plain,
% 0.19/0.39 ((incident_point_and_line(U!1, line_connecting(X!3, Y!2)) & incident_point_and_line(V!0, line_connecting(X!3, Y!2))) <=> (~((~incident_point_and_line(U!1, line_connecting(X!3, Y!2))) | (~incident_point_and_line(V!0, line_connecting(X!3, Y!2)))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(126,plain,
% 0.19/0.39 ((~(incident_point_and_line(U!1, line_connecting(X!3, Y!2)) & incident_point_and_line(V!0, line_connecting(X!3, Y!2)))) <=> (~(~((~incident_point_and_line(U!1, line_connecting(X!3, Y!2))) | (~incident_point_and_line(V!0, line_connecting(X!3, Y!2))))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[125])).
% 0.19/0.39 tff(127,plain,
% 0.19/0.39 ((~(incident_point_and_line(U!1, line_connecting(X!3, Y!2)) & incident_point_and_line(V!0, line_connecting(X!3, Y!2)))) <=> ((~incident_point_and_line(U!1, line_connecting(X!3, Y!2))) | (~incident_point_and_line(V!0, line_connecting(X!3, Y!2))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[126, 124])).
% 0.19/0.39 tff(128,plain,
% 0.19/0.39 (~(incident_point_and_line(U!1, line_connecting(X!3, Y!2)) & incident_point_and_line(V!0, line_connecting(X!3, Y!2)))),
% 0.19/0.39 inference(or_elim,[status(thm)],[22])).
% 0.19/0.40 tff(129,plain,
% 0.19/0.40 ((~incident_point_and_line(U!1, line_connecting(X!3, Y!2))) | (~incident_point_and_line(V!0, line_connecting(X!3, Y!2)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[128, 127])).
% 0.19/0.40 tff(130,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[129, 123, 107])).
% 0.19/0.40 % SZS output end Proof
% 0.19/0.40 % E exiting
%------------------------------------------------------------------------------