%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO190+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.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 05:05:50 AM UTC 2025
% Result : Theorem 0.20s 0.40s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : GEO190+1 : TPTP v9.0.0. Released v3.3.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n001.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Jun 19 14:44:33 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.20/0.40 % SZS status Theorem
% 0.20/0.40 % SZS output start Proof
% 0.20/0.40 tff(distinct_lines_type, type, (
% 0.20/0.40 distinct_lines: ( $i * $i ) > $o)).
% 0.20/0.40 tff(line_connecting_type, type, (
% 0.20/0.40 line_connecting: ( $i * $i ) > $i)).
% 0.20/0.40 tff(tptp_fun_Y_1_type, type, (
% 0.20/0.40 tptp_fun_Y_1: $i)).
% 0.20/0.40 tff(tptp_fun_X_2_type, type, (
% 0.20/0.40 tptp_fun_X_2: $i)).
% 0.20/0.40 tff(apart_point_and_line_type, type, (
% 0.20/0.40 apart_point_and_line: ( $i * $i ) > $o)).
% 0.20/0.40 tff(tptp_fun_Z_0_type, type, (
% 0.20/0.40 tptp_fun_Z_0: $i)).
% 0.20/0.40 tff(distinct_points_type, type, (
% 0.20/0.40 distinct_points: ( $i * $i ) > $o)).
% 0.20/0.40 tff(1,assumption,(~distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1))), introduced(assumption)).
% 0.20/0.40 tff(2,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(Y, X))) | (~(distinct_points(X, Y) & distinct_points(X, Z) & distinct_points(Y, Z) & (~apart_point_and_line(Z, line_connecting(X, Y))))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(Y, X))) | (~(distinct_points(X, Y) & distinct_points(X, Z) & distinct_points(Y, Z) & (~apart_point_and_line(Z, line_connecting(X, Y)))))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(3,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : ((((distinct_points(X, Y) & distinct_points(X, Z)) & distinct_points(Y, Z)) & (~apart_point_and_line(Z, line_connecting(X, Y)))) => (~apart_point_and_line(Z, line_connecting(Y, X))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(Y, X))) | (~(distinct_points(X, Y) & distinct_points(X, Z) & distinct_points(Y, Z) & (~apart_point_and_line(Z, line_connecting(X, Y)))))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(4,axiom,(~![X: $i, Y: $i, Z: $i] : ((((distinct_points(X, Y) & distinct_points(X, Z)) & distinct_points(Y, Z)) & (~apart_point_and_line(Z, line_connecting(X, Y)))) => (~apart_point_and_line(Z, line_connecting(Y, X))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.20/0.40 tff(5,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(Y, X))) | (~(distinct_points(X, Y) & distinct_points(X, Z) & distinct_points(Y, Z) & (~apart_point_and_line(Z, line_connecting(X, Y))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.20/0.40 tff(6,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(Y, X))) | (~(distinct_points(X, Y) & distinct_points(X, Z) & distinct_points(Y, Z) & (~apart_point_and_line(Z, line_connecting(X, Y))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[5, 2])).
% 0.20/0.40 tff(7,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(Y, X))) | (~(distinct_points(X, Y) & distinct_points(X, Z) & distinct_points(Y, Z) & (~apart_point_and_line(Z, line_connecting(X, Y))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.20/0.40 tff(8,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(Y, X))) | (~(distinct_points(X, Y) & distinct_points(X, Z) & distinct_points(Y, Z) & (~apart_point_and_line(Z, line_connecting(X, Y))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[7, 2])).
% 0.20/0.40 tff(9,plain,(
% 0.20/0.40 ~((~apart_point_and_line(Z!0, line_connecting(Y!1, X!2))) | (~(distinct_points(X!2, Y!1) & distinct_points(X!2, Z!0) & distinct_points(Y!1, Z!0) & (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))))),
% 0.20/0.40 inference(skolemize,[status(sab)],[8])).
% 0.20/0.40 tff(10,plain,
% 0.20/0.40 (distinct_points(X!2, Y!1) & distinct_points(X!2, Z!0) & distinct_points(Y!1, Z!0) & (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))),
% 0.20/0.40 inference(or_elim,[status(thm)],[9])).
% 0.20/0.40 tff(11,plain,
% 0.20/0.40 (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))),
% 0.20/0.40 inference(and_elim,[status(thm)],[10])).
% 0.20/0.40 tff(12,plain,
% 0.20/0.40 (apart_point_and_line(Z!0, line_connecting(Y!1, X!2))),
% 0.20/0.40 inference(or_elim,[status(thm)],[9])).
% 0.20/0.40 tff(13,plain,
% 0.20/0.40 (^[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))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(14,plain,
% 0.20/0.40 (![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)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[13])).
% 0.20/0.40 tff(15,plain,
% 0.20/0.40 (![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)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(16,plain,
% 0.20/0.40 (^[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)))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(17,plain,
% 0.20/0.40 (![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)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[16])).
% 0.20/0.40 tff(18,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')).
% 0.20/0.40 tff(19,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.20/0.40 tff(20,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[19, 15])).
% 0.20/0.40 tff(21,plain,(
% 0.20/0.40 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.40 inference(skolemize,[status(sab)],[20])).
% 0.20/0.40 tff(22,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[21, 14])).
% 0.20/0.40 tff(23,plain,
% 0.20/0.40 (((~![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(Z!0, line_connecting(Y!1, X!2))) | apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!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(Z!0, line_connecting(Y!1, X!2))) | apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(24,plain,
% 0.20/0.40 ((distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1)) | apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(Y!1, X!2)))) <=> ((~apart_point_and_line(Z!0, line_connecting(Y!1, X!2))) | apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(25,plain,
% 0.20/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1)) | apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(Y!1, X!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(Z!0, line_connecting(Y!1, X!2))) | apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1))))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[24])).
% 0.20/0.40 tff(26,plain,
% 0.20/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1)) | apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(Y!1, X!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(Z!0, line_connecting(Y!1, X!2))) | apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1)))),
% 0.20/0.40 inference(transitivity,[status(thm)],[25, 23])).
% 0.20/0.40 tff(27,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1)) | apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(Y!1, X!2))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(28,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(Z!0, line_connecting(Y!1, X!2))) | apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.20/0.40 tff(29,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[28, 22, 12, 11, 1])).
% 0.20/0.40 tff(30,plain,(distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.40 tff(31,plain,
% 0.20/0.40 (^[X: $i] : refl((~distinct_points(X, X)) <=> (~distinct_points(X, X)))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(32,plain,
% 0.20/0.40 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[31])).
% 0.20/0.40 tff(33,plain,
% 0.20/0.40 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(34,axiom,(![X: $i] : (~distinct_points(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','apart1')).
% 0.20/0.40 tff(35,plain,
% 0.20/0.40 (![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.20/0.40 tff(36,plain,(
% 0.20/0.40 ![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(skolemize,[status(sab)],[35])).
% 0.20/0.40 tff(37,plain,
% 0.20/0.40 (![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[36, 32])).
% 0.20/0.40 tff(38,plain,
% 0.20/0.40 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(X!2, X!2))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(39,plain,
% 0.20/0.40 (~distinct_points(X!2, X!2)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[38, 37])).
% 0.20/0.40 tff(40,assumption,(~distinct_points(Y!1, X!2)), introduced(assumption)).
% 0.20/0.40 tff(41,plain,
% 0.20/0.40 (distinct_points(X!2, Y!1)),
% 0.20/0.40 inference(and_elim,[status(thm)],[10])).
% 0.20/0.40 tff(42,plain,
% 0.20/0.40 (^[X: $i, Y: $i, Z: $i] : refl((distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(43,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[42])).
% 0.20/0.40 tff(44,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(45,plain,
% 0.20/0.40 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_points(X, Z) | distinct_points(Y, Z)) <=> (distinct_points(Y, Z) | distinct_points(X, Z))), ((distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> (distinct_points(X, Y) => (distinct_points(Y, Z) | distinct_points(X, Z))))), rewrite((distinct_points(X, Y) => (distinct_points(Y, Z) | distinct_points(X, Z))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))), ((distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(46,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[45])).
% 0.20/0.40 tff(47,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','apart4')).
% 0.20/0.40 tff(48,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.20/0.40 tff(49,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[48, 44])).
% 0.20/0.40 tff(50,plain,(
% 0.20/0.40 ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(skolemize,[status(sab)],[49])).
% 0.20/0.40 tff(51,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[50, 43])).
% 0.20/0.40 tff(52,plain,
% 0.20/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | ((~distinct_points(X!2, Y!1)) | distinct_points(Y!1, X!2) | distinct_points(X!2, X!2))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (~distinct_points(X!2, Y!1)) | distinct_points(Y!1, X!2) | distinct_points(X!2, X!2))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(53,plain,
% 0.20/0.40 ((distinct_points(Y!1, X!2) | distinct_points(X!2, X!2) | (~distinct_points(X!2, Y!1))) <=> ((~distinct_points(X!2, Y!1)) | distinct_points(Y!1, X!2) | distinct_points(X!2, X!2))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(54,plain,
% 0.20/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(Y!1, X!2) | distinct_points(X!2, X!2) | (~distinct_points(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | ((~distinct_points(X!2, Y!1)) | distinct_points(Y!1, X!2) | distinct_points(X!2, X!2)))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[53])).
% 0.20/0.40 tff(55,plain,
% 0.20/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(Y!1, X!2) | distinct_points(X!2, X!2) | (~distinct_points(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (~distinct_points(X!2, Y!1)) | distinct_points(Y!1, X!2) | distinct_points(X!2, X!2))),
% 0.20/0.40 inference(transitivity,[status(thm)],[54, 52])).
% 0.20/0.40 tff(56,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(Y!1, X!2) | distinct_points(X!2, X!2) | (~distinct_points(X!2, Y!1)))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(57,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (~distinct_points(X!2, Y!1)) | distinct_points(Y!1, X!2) | distinct_points(X!2, X!2)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.20/0.40 tff(58,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[57, 51, 41, 40, 39])).
% 0.20/0.40 tff(59,plain,(distinct_points(Y!1, X!2)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.40 tff(60,plain,
% 0.20/0.40 (^[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)))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(61,plain,
% 0.20/0.40 (![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))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[60])).
% 0.20/0.40 tff(62,plain,
% 0.20/0.40 (![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))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(63,plain,
% 0.20/0.40 (^[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)))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(64,plain,
% 0.20/0.40 (![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))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[63])).
% 0.20/0.40 tff(65,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')).
% 0.20/0.40 tff(66,plain,
% 0.20/0.40 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.20/0.40 tff(67,plain,
% 0.20/0.40 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[66, 62])).
% 0.20/0.40 tff(68,plain,(
% 0.20/0.40 ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.40 inference(skolemize,[status(sab)],[67])).
% 0.20/0.40 tff(69,plain,
% 0.20/0.40 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[68, 61])).
% 0.20/0.40 tff(70,plain,
% 0.20/0.40 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(Y!1, X!2)) | (~apart_point_and_line(X!2, line_connecting(Y!1, X!2))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(Y!1, X!2)) | (~apart_point_and_line(X!2, line_connecting(Y!1, X!2))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(71,plain,
% 0.20/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(Y!1, X!2)) | (~apart_point_and_line(X!2, line_connecting(Y!1, X!2))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(72,plain,
% 0.20/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(Y!1, X!2)) | (~apart_point_and_line(X!2, line_connecting(Y!1, X!2)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.20/0.41 tff(73,plain,
% 0.20/0.41 ((~distinct_points(Y!1, X!2)) | (~apart_point_and_line(X!2, line_connecting(Y!1, X!2)))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[72, 69])).
% 0.20/0.41 tff(74,plain,
% 0.20/0.41 (~apart_point_and_line(X!2, line_connecting(Y!1, X!2))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[73, 59])).
% 0.20/0.41 tff(75,plain,
% 0.20/0.41 (^[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)))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(76,plain,
% 0.20/0.41 (![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))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[75])).
% 0.20/0.41 tff(77,plain,
% 0.20/0.41 (![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))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(78,plain,
% 0.20/0.41 (^[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)))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(79,plain,
% 0.20/0.41 (![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))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[78])).
% 0.20/0.41 tff(80,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')).
% 0.20/0.41 tff(81,plain,
% 0.20/0.41 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[80, 79])).
% 0.20/0.41 tff(82,plain,
% 0.20/0.41 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[81, 77])).
% 0.20/0.41 tff(83,plain,(
% 0.20/0.41 ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.41 inference(skolemize,[status(sab)],[82])).
% 0.20/0.41 tff(84,plain,
% 0.20/0.41 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[83, 76])).
% 0.20/0.41 tff(85,plain,
% 0.20/0.41 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(Y!1, X!2)) | (~apart_point_and_line(Y!1, line_connecting(Y!1, X!2))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(Y!1, X!2)) | (~apart_point_and_line(Y!1, line_connecting(Y!1, X!2))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(86,plain,
% 0.20/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(Y!1, X!2)) | (~apart_point_and_line(Y!1, line_connecting(Y!1, X!2))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(87,plain,
% 0.20/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(Y!1, X!2)) | (~apart_point_and_line(Y!1, line_connecting(Y!1, X!2)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[86, 85])).
% 0.20/0.41 tff(88,plain,
% 0.20/0.41 ((~distinct_points(Y!1, X!2)) | (~apart_point_and_line(Y!1, line_connecting(Y!1, X!2)))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[87, 84])).
% 0.20/0.41 tff(89,plain,
% 0.20/0.41 (~apart_point_and_line(Y!1, line_connecting(Y!1, X!2))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[88, 59])).
% 0.20/0.41 tff(90,plain,
% 0.20/0.41 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(91,plain,
% 0.20/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(92,plain,
% 0.20/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.20/0.41 tff(93,plain,
% 0.20/0.41 (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[92, 84, 41])).
% 0.20/0.41 tff(94,plain,
% 0.20/0.41 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(95,plain,
% 0.20/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(96,plain,
% 0.20/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[95, 94])).
% 0.20/0.41 tff(97,plain,
% 0.20/0.41 (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[96, 69, 41])).
% 0.20/0.41 tff(98,plain,
% 0.20/0.41 (^[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))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(99,plain,
% 0.20/0.41 (![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))) <=> ![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)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[98])).
% 0.20/0.41 tff(100,plain,
% 0.20/0.41 (^[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)))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(101,plain,
% 0.20/0.41 (![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)))) <=> ![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)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[100])).
% 0.20/0.41 tff(102,plain,
% 0.20/0.41 (![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)))) <=> ![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))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(103,plain,
% 0.20/0.41 (^[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))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(104,plain,
% 0.20/0.41 (![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))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[103])).
% 0.20/0.41 tff(105,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')).
% 0.20/0.41 tff(106,plain,
% 0.20/0.41 (![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))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[105, 104])).
% 0.20/0.41 tff(107,plain,
% 0.20/0.41 (![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))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[106, 102])).
% 0.20/0.41 tff(108,plain,(
% 0.20/0.41 ![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))))),
% 0.20/0.41 inference(skolemize,[status(sab)],[107])).
% 0.20/0.41 tff(109,plain,
% 0.20/0.41 (![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)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[108, 101])).
% 0.20/0.41 tff(110,plain,
% 0.20/0.41 (![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)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[109, 99])).
% 0.20/0.41 tff(111,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | ((~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | (~distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | (~distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(112,plain,
% 0.20/0.41 ((apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | (~distinct_points(X!2, Y!1)) | (~distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1)))) <=> ((~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | (~distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(113,plain,
% 0.20/0.41 (((~![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(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | (~distinct_points(X!2, Y!1)) | (~distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | ((~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | (~distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1)))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[112])).
% 0.20/0.41 tff(114,plain,
% 0.20/0.41 (((~![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(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | (~distinct_points(X!2, Y!1)) | (~distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | (~distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1))))),
% 0.20/0.42 inference(transitivity,[status(thm)],[113, 111])).
% 0.20/0.42 tff(115,plain,
% 0.20/0.42 ((~![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(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | (~distinct_points(X!2, Y!1)) | (~distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(116,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | (~distinct_lines(line_connecting(Y!1, X!2), line_connecting(X!2, Y!1)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.20/0.42 tff(117,plain,
% 0.20/0.42 ($false),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[116, 110, 41, 97, 93, 89, 74, 30])).
% 0.20/0.42 % SZS output end Proof
% 0.20/0.42 % E exiting
%------------------------------------------------------------------------------