%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO224+3 : TPTP v9.0.0. Released v4.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n016.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 05:06:02 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
% 0.00/0.12 % Problem : GEO224+3 : TPTP v9.0.0. Released v4.0.0.
% 0.00/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n016.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 14:42:47 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_lines_type, type, (
% 0.19/0.39 distinct_lines: ( $i * $i ) > $o)).
% 0.19/0.39 tff(parallel_through_point_type, type, (
% 0.19/0.39 parallel_through_point: ( $i * $i ) > $i)).
% 0.19/0.39 tff(tptp_fun_X_1_type, type, (
% 0.19/0.39 tptp_fun_X_1: $i)).
% 0.19/0.39 tff(tptp_fun_Y_0_type, type, (
% 0.19/0.39 tptp_fun_Y_0: $i)).
% 0.19/0.39 tff(convergent_lines_type, type, (
% 0.19/0.39 convergent_lines: ( $i * $i ) > $o)).
% 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(incident_point_and_line_type, type, (
% 0.19/0.39 incident_point_and_line: ( $i * $i ) > $o)).
% 0.19/0.39 tff(point_type, type, (
% 0.19/0.39 point: $i > $o)).
% 0.19/0.39 tff(line_type, type, (
% 0.19/0.39 line: $i > $o)).
% 0.19/0.39 tff(1,plain,
% 0.19/0.39 (^[X: $i, Y: $i] : refl((~convergent_lines(parallel_through_point(Y, X), Y)) <=> (~convergent_lines(parallel_through_point(Y, X), Y)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(2,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y)) <=> ![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.39 tff(3,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y)) <=> ![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(4,axiom,(![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+2.ax','cp1')).
% 0.19/0.39 tff(5,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.39 tff(6,plain,(
% 0.19/0.39 ![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y))),
% 0.19/0.39 inference(skolemize,[status(sab)],[5])).
% 0.19/0.39 tff(7,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.19/0.39 tff(8,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y))) | (~convergent_lines(parallel_through_point(Y!0, X!1), Y!0))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(9,plain,
% 0.19/0.39 (~convergent_lines(parallel_through_point(Y!0, X!1), Y!0)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.19/0.39 tff(10,plain,
% 0.19/0.39 (^[X: $i, Y: $i] : refl(((~distinct_lines(X, Y)) | convergent_lines(X, Y)) <=> ((~distinct_lines(X, Y)) | convergent_lines(X, Y)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(11,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[10])).
% 0.19/0.39 tff(12,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(13,plain,
% 0.19/0.39 (^[X: $i, Y: $i] : rewrite((distinct_lines(X, Y) => convergent_lines(X, Y)) <=> ((~distinct_lines(X, Y)) | convergent_lines(X, Y)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(14,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (distinct_lines(X, Y) => convergent_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[13])).
% 0.19/0.39 tff(15,axiom,(![X: $i, Y: $i] : (distinct_lines(X, Y) => convergent_lines(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+1.ax','p1')).
% 0.19/0.39 tff(16,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.19/0.39 tff(17,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.19/0.39 tff(18,plain,(
% 0.19/0.39 ![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.19/0.39 inference(skolemize,[status(sab)],[17])).
% 0.19/0.39 tff(19,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[18, 11])).
% 0.19/0.39 tff(20,plain,
% 0.19/0.39 (((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | ((~distinct_lines(parallel_through_point(Y!0, X!1), Y!0)) | convergent_lines(parallel_through_point(Y!0, X!1), Y!0))) <=> ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | (~distinct_lines(parallel_through_point(Y!0, X!1), Y!0)) | convergent_lines(parallel_through_point(Y!0, X!1), Y!0))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(21,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | ((~distinct_lines(parallel_through_point(Y!0, X!1), Y!0)) | convergent_lines(parallel_through_point(Y!0, X!1), Y!0))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(22,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | (~distinct_lines(parallel_through_point(Y!0, X!1), Y!0)) | convergent_lines(parallel_through_point(Y!0, X!1), Y!0)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.19/0.39 tff(23,plain,
% 0.19/0.39 (~distinct_lines(parallel_through_point(Y!0, X!1), Y!0)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[22, 19, 9])).
% 0.19/0.39 tff(24,plain,
% 0.19/0.39 (^[X: $i] : refl((~convergent_lines(X, X)) <=> (~convergent_lines(X, X)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(25,plain,
% 0.19/0.39 (![X: $i] : (~convergent_lines(X, X)) <=> ![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[24])).
% 0.19/0.39 tff(26,plain,
% 0.19/0.39 (![X: $i] : (~convergent_lines(X, X)) <=> ![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(27,axiom,(![X: $i] : (~convergent_lines(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','apart3')).
% 0.19/0.39 tff(28,plain,
% 0.19/0.39 (![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.19/0.39 tff(29,plain,(
% 0.19/0.39 ![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.39 inference(skolemize,[status(sab)],[28])).
% 0.19/0.39 tff(30,plain,
% 0.19/0.39 (![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[29, 25])).
% 0.19/0.39 tff(31,plain,
% 0.19/0.39 ((~![X: $i] : (~convergent_lines(X, X))) | (~convergent_lines(Y!0, Y!0))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(32,plain,
% 0.19/0.39 (~convergent_lines(Y!0, Y!0)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[31, 30])).
% 0.19/0.39 tff(33,plain,
% 0.19/0.39 (((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | ((~distinct_lines(Y!0, Y!0)) | convergent_lines(Y!0, Y!0))) <=> ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | (~distinct_lines(Y!0, Y!0)) | convergent_lines(Y!0, Y!0))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(34,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | ((~distinct_lines(Y!0, Y!0)) | convergent_lines(Y!0, Y!0))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(35,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | (~distinct_lines(Y!0, Y!0)) | convergent_lines(Y!0, Y!0)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.19/0.39 tff(36,plain,
% 0.19/0.39 (~distinct_lines(Y!0, Y!0)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[35, 19, 32])).
% 0.19/0.39 tff(37,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : refl((distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y))) <=> (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(38,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[37])).
% 0.19/0.39 tff(39,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(40,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_lines(X, Z) | distinct_lines(Y, Z)) <=> (distinct_lines(Y, Z) | distinct_lines(X, Z))), ((distinct_lines(X, Y) => (distinct_lines(X, Z) | distinct_lines(Y, Z))) <=> (distinct_lines(X, Y) => (distinct_lines(Y, Z) | distinct_lines(X, Z))))), rewrite((distinct_lines(X, Y) => (distinct_lines(Y, Z) | distinct_lines(X, Z))) <=> (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))), ((distinct_lines(X, Y) => (distinct_lines(X, Z) | distinct_lines(Y, Z))) <=> (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(41,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(X, Y) => (distinct_lines(X, Z) | distinct_lines(Y, Z))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[40])).
% 0.19/0.39 tff(42,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_lines(X, Y) => (distinct_lines(X, Z) | distinct_lines(Y, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','apart5')).
% 0.19/0.39 tff(43,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.19/0.39 tff(44,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[43, 39])).
% 0.19/0.39 tff(45,plain,(
% 0.19/0.39 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[44])).
% 0.19/0.39 tff(46,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[45, 38])).
% 0.19/0.39 tff(47,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(Y!0, Y!0) | distinct_lines(parallel_through_point(Y!0, X!1), Y!0) | (~distinct_lines(Y!0, parallel_through_point(Y!0, X!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | distinct_lines(Y!0, Y!0) | distinct_lines(parallel_through_point(Y!0, X!1), Y!0) | (~distinct_lines(Y!0, parallel_through_point(Y!0, X!1))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(48,plain,
% 0.19/0.39 ((distinct_lines(parallel_through_point(Y!0, X!1), Y!0) | distinct_lines(Y!0, Y!0) | (~distinct_lines(Y!0, parallel_through_point(Y!0, X!1)))) <=> (distinct_lines(Y!0, Y!0) | distinct_lines(parallel_through_point(Y!0, X!1), Y!0) | (~distinct_lines(Y!0, parallel_through_point(Y!0, X!1))))),
% 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] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(parallel_through_point(Y!0, X!1), Y!0) | distinct_lines(Y!0, Y!0) | (~distinct_lines(Y!0, parallel_through_point(Y!0, X!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(Y!0, Y!0) | distinct_lines(parallel_through_point(Y!0, X!1), Y!0) | (~distinct_lines(Y!0, parallel_through_point(Y!0, X!1)))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[48])).
% 0.19/0.39 tff(50,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(parallel_through_point(Y!0, X!1), Y!0) | distinct_lines(Y!0, Y!0) | (~distinct_lines(Y!0, parallel_through_point(Y!0, X!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | distinct_lines(Y!0, Y!0) | distinct_lines(parallel_through_point(Y!0, X!1), Y!0) | (~distinct_lines(Y!0, parallel_through_point(Y!0, X!1))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[49, 47])).
% 0.19/0.39 tff(51,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(parallel_through_point(Y!0, X!1), Y!0) | distinct_lines(Y!0, Y!0) | (~distinct_lines(Y!0, parallel_through_point(Y!0, X!1))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(52,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | distinct_lines(Y!0, Y!0) | distinct_lines(parallel_through_point(Y!0, X!1), Y!0) | (~distinct_lines(Y!0, parallel_through_point(Y!0, X!1)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.19/0.39 tff(53,plain,
% 0.19/0.39 (~distinct_lines(Y!0, parallel_through_point(Y!0, X!1))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[52, 46, 36, 23])).
% 0.19/0.39 tff(54,plain,
% 0.19/0.39 (^[X: $i, Y: $i] : refl((~apart_point_and_line(X, parallel_through_point(Y, X))) <=> (~apart_point_and_line(X, parallel_through_point(Y, X))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(55,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X))) <=> ![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[54])).
% 0.19/0.39 tff(56,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X))) <=> ![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(57,axiom,(![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+2.ax','cp2')).
% 0.19/0.39 tff(58,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.19/0.39 tff(59,plain,(
% 0.19/0.39 ![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[58])).
% 0.19/0.39 tff(60,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[59, 55])).
% 0.19/0.39 tff(61,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X)))) | (~apart_point_and_line(X!1, parallel_through_point(Y!0, X!1)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(62,plain,
% 0.19/0.39 (~apart_point_and_line(X!1, parallel_through_point(Y!0, X!1))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[61, 60])).
% 0.19/0.39 tff(63,plain,
% 0.19/0.39 (^[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))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(64,plain,
% 0.19/0.39 (![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)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[63])).
% 0.19/0.39 tff(65,plain,
% 0.19/0.39 (![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)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(66,plain,
% 0.19/0.39 (^[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)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(67,plain,
% 0.19/0.39 (![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)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[66])).
% 0.19/0.39 tff(68,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')).
% 0.19/0.39 tff(69,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[68, 67])).
% 0.19/0.39 tff(70,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[69, 65])).
% 0.19/0.39 tff(71,plain,(
% 0.19/0.39 ![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[70])).
% 0.19/0.39 tff(72,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~(apart_point_and_line(X, Y) <=> incident_point_and_line(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[71, 64])).
% 0.19/0.39 tff(73,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(X!1, Y!0) <=> incident_point_and_line(X!1, Y!0)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(74,plain,
% 0.19/0.40 (~(apart_point_and_line(X!1, Y!0) <=> incident_point_and_line(X!1, Y!0))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[73, 72])).
% 0.19/0.40 tff(75,plain,
% 0.19/0.40 (^[Z: $i] : refl(((~point(Z)) | (~incident_point_and_line(Z, Y!0))) <=> ((~point(Z)) | (~incident_point_and_line(Z, Y!0))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(76,plain,
% 0.19/0.40 (![Z: $i] : ((~point(Z)) | (~incident_point_and_line(Z, Y!0))) <=> ![Z: $i] : ((~point(Z)) | (~incident_point_and_line(Z, Y!0)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[75])).
% 0.19/0.40 tff(77,plain,
% 0.19/0.40 (^[Z: $i] : trans(monotonicity(rewrite((point(Z) & incident_point_and_line(Z, Y!0)) <=> (~((~point(Z)) | (~incident_point_and_line(Z, Y!0))))), ((~(point(Z) & incident_point_and_line(Z, Y!0))) <=> (~(~((~point(Z)) | (~incident_point_and_line(Z, Y!0))))))), rewrite((~(~((~point(Z)) | (~incident_point_and_line(Z, Y!0))))) <=> ((~point(Z)) | (~incident_point_and_line(Z, Y!0)))), ((~(point(Z) & incident_point_and_line(Z, Y!0))) <=> ((~point(Z)) | (~incident_point_and_line(Z, Y!0)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(78,plain,
% 0.19/0.40 (![Z: $i] : (~(point(Z) & incident_point_and_line(Z, Y!0))) <=> ![Z: $i] : ((~point(Z)) | (~incident_point_and_line(Z, Y!0)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[77])).
% 0.19/0.40 tff(79,plain,
% 0.19/0.40 (((point(X!1) & line(Y!0)) & ![Z: $i] : (~(point(Z) & incident_point_and_line(Z, Y!0)))) <=> (point(X!1) & line(Y!0) & ![Z: $i] : (~(point(Z) & incident_point_and_line(Z, Y!0))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(80,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((~(point(X) & line(Y))) | ?[Z: $i] : (point(Z) & incident_point_and_line(Z, Y)))) <=> (~![X: $i, Y: $i] : ((~(point(X) & line(Y))) | ?[Z: $i] : (point(Z) & incident_point_and_line(Z, Y))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(81,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((point(X) & line(Y)) => ?[Z: $i] : (point(Z) & incident_point_and_line(Z, Y)))) <=> (~![X: $i, Y: $i] : ((~(point(X) & line(Y))) | ?[Z: $i] : (point(Z) & incident_point_and_line(Z, Y))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(82,axiom,(~![X: $i, Y: $i] : ((point(X) & line(Y)) => ?[Z: $i] : (point(Z) & incident_point_and_line(Z, Y)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.19/0.40 tff(83,plain,
% 0.19/0.40 (~![X: $i, Y: $i] : ((~(point(X) & line(Y))) | ?[Z: $i] : (point(Z) & incident_point_and_line(Z, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.19/0.40 tff(84,plain,
% 0.19/0.40 (~![X: $i, Y: $i] : ((~(point(X) & line(Y))) | ?[Z: $i] : (point(Z) & incident_point_and_line(Z, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[83, 80])).
% 0.19/0.40 tff(85,plain,
% 0.19/0.40 (~![X: $i, Y: $i] : ((~(point(X) & line(Y))) | ?[Z: $i] : (point(Z) & incident_point_and_line(Z, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[84, 80])).
% 0.19/0.40 tff(86,plain,
% 0.19/0.40 (~![X: $i, Y: $i] : ((~(point(X) & line(Y))) | ?[Z: $i] : (point(Z) & incident_point_and_line(Z, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[85, 80])).
% 0.19/0.40 tff(87,plain,
% 0.19/0.40 (point(X!1) & line(Y!0) & ![Z: $i] : (~(point(Z) & incident_point_and_line(Z, Y!0)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[86, 79])).
% 0.19/0.40 tff(88,plain,
% 0.19/0.40 (![Z: $i] : (~(point(Z) & incident_point_and_line(Z, Y!0)))),
% 0.19/0.40 inference(and_elim,[status(thm)],[87])).
% 0.19/0.40 tff(89,plain,
% 0.19/0.40 (![Z: $i] : ((~point(Z)) | (~incident_point_and_line(Z, Y!0)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[88, 78])).
% 0.19/0.40 tff(90,plain,
% 0.19/0.40 (![Z: $i] : ((~point(Z)) | (~incident_point_and_line(Z, Y!0)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[89, 76])).
% 0.19/0.40 tff(91,plain,
% 0.19/0.40 (point(X!1)),
% 0.19/0.40 inference(and_elim,[status(thm)],[87])).
% 0.19/0.40 tff(92,plain,
% 0.19/0.40 (((~![Z: $i] : ((~point(Z)) | (~incident_point_and_line(Z, Y!0)))) | ((~point(X!1)) | (~incident_point_and_line(X!1, Y!0)))) <=> ((~![Z: $i] : ((~point(Z)) | (~incident_point_and_line(Z, Y!0)))) | (~point(X!1)) | (~incident_point_and_line(X!1, Y!0)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(93,plain,
% 0.19/0.40 ((~![Z: $i] : ((~point(Z)) | (~incident_point_and_line(Z, Y!0)))) | ((~point(X!1)) | (~incident_point_and_line(X!1, Y!0)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(94,plain,
% 0.19/0.40 ((~![Z: $i] : ((~point(Z)) | (~incident_point_and_line(Z, Y!0)))) | (~point(X!1)) | (~incident_point_and_line(X!1, Y!0))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.19/0.40 tff(95,plain,
% 0.19/0.40 (~incident_point_and_line(X!1, Y!0)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[94, 91, 90])).
% 0.19/0.40 tff(96,plain,
% 0.19/0.40 ((apart_point_and_line(X!1, Y!0) <=> incident_point_and_line(X!1, Y!0)) | apart_point_and_line(X!1, Y!0) | incident_point_and_line(X!1, Y!0)),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(97,plain,
% 0.19/0.40 ((apart_point_and_line(X!1, Y!0) <=> incident_point_and_line(X!1, Y!0)) | apart_point_and_line(X!1, Y!0)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[96, 95])).
% 0.19/0.40 tff(98,plain,
% 0.19/0.40 (apart_point_and_line(X!1, Y!0)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[97, 74])).
% 0.19/0.40 tff(99,plain,
% 0.19/0.40 (^[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))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(100,plain,
% 0.19/0.40 (![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)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[99])).
% 0.19/0.40 tff(101,plain,
% 0.19/0.40 (![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)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(102,plain,
% 0.19/0.40 (^[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)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(103,plain,
% 0.19/0.40 (![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)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[102])).
% 0.19/0.40 tff(104,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')).
% 0.19/0.40 tff(105,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[104, 103])).
% 0.19/0.40 tff(106,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[105, 101])).
% 0.19/0.40 tff(107,plain,(
% 0.19/0.40 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.40 inference(skolemize,[status(sab)],[106])).
% 0.19/0.40 tff(108,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[107, 100])).
% 0.19/0.40 tff(109,plain,
% 0.19/0.40 (((~![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(X!1, Y!0)) | apart_point_and_line(X!1, parallel_through_point(Y!0, X!1)) | distinct_lines(Y!0, parallel_through_point(Y!0, X!1)))) <=> ((~![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(X!1, Y!0)) | apart_point_and_line(X!1, parallel_through_point(Y!0, X!1)) | distinct_lines(Y!0, parallel_through_point(Y!0, X!1)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(110,plain,
% 0.19/0.40 ((distinct_lines(Y!0, parallel_through_point(Y!0, X!1)) | apart_point_and_line(X!1, parallel_through_point(Y!0, X!1)) | (~apart_point_and_line(X!1, Y!0))) <=> ((~apart_point_and_line(X!1, Y!0)) | apart_point_and_line(X!1, parallel_through_point(Y!0, X!1)) | distinct_lines(Y!0, parallel_through_point(Y!0, X!1)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(111,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(Y!0, parallel_through_point(Y!0, X!1)) | apart_point_and_line(X!1, parallel_through_point(Y!0, X!1)) | (~apart_point_and_line(X!1, Y!0)))) <=> ((~![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(X!1, Y!0)) | apart_point_and_line(X!1, parallel_through_point(Y!0, X!1)) | distinct_lines(Y!0, parallel_through_point(Y!0, X!1))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[110])).
% 0.19/0.40 tff(112,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(Y!0, parallel_through_point(Y!0, X!1)) | apart_point_and_line(X!1, parallel_through_point(Y!0, X!1)) | (~apart_point_and_line(X!1, Y!0)))) <=> ((~![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(X!1, Y!0)) | apart_point_and_line(X!1, parallel_through_point(Y!0, X!1)) | distinct_lines(Y!0, parallel_through_point(Y!0, X!1)))),
% 0.19/0.40 inference(transitivity,[status(thm)],[111, 109])).
% 0.19/0.40 tff(113,plain,
% 0.19/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(Y!0, parallel_through_point(Y!0, X!1)) | apart_point_and_line(X!1, parallel_through_point(Y!0, X!1)) | (~apart_point_and_line(X!1, Y!0)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(114,plain,
% 0.19/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(X!1, Y!0)) | apart_point_and_line(X!1, parallel_through_point(Y!0, X!1)) | distinct_lines(Y!0, parallel_through_point(Y!0, X!1))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[113, 112])).
% 0.19/0.40 tff(115,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[114, 108, 98, 62, 53])).
% 0.19/0.40 % SZS output end Proof
% 0.19/0.40 % E exiting
%------------------------------------------------------------------------------