%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO177+2 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.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 05:05:46 AM UTC 2025
% Result : Theorem 0.22s 0.41s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.13 % Problem : GEO177+2 : TPTP v9.0.0. Released v3.3.0.
% 0.04/0.13 % Command : run_E %s %d THM
% 0.13/0.35 % Computer : n003.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Thu Jun 19 14:39:54 EDT 2025
% 0.13/0.35 % CPUTime :
% 0.22/0.41 % SZS status Theorem
% 0.22/0.41 % SZS output start Proof
% 0.22/0.41 tff(distinct_points_type, type, (
% 0.22/0.41 distinct_points: ( $i * $i ) > $o)).
% 0.22/0.41 tff(tptp_fun_Y_2_type, type, (
% 0.22/0.41 tptp_fun_Y_2: $i)).
% 0.22/0.41 tff(tptp_fun_X_3_type, type, (
% 0.22/0.41 tptp_fun_X_3: $i)).
% 0.22/0.41 tff(apart_point_and_line_type, type, (
% 0.22/0.41 apart_point_and_line: ( $i * $i ) > $o)).
% 0.22/0.41 tff(line_connecting_type, type, (
% 0.22/0.41 line_connecting: ( $i * $i ) > $i)).
% 0.22/0.41 tff(distinct_lines_type, type, (
% 0.22/0.41 distinct_lines: ( $i * $i ) > $o)).
% 0.22/0.41 tff(tptp_fun_V_0_type, type, (
% 0.22/0.41 tptp_fun_V_0: $i)).
% 0.22/0.41 tff(tptp_fun_U_1_type, type, (
% 0.22/0.41 tptp_fun_U_1: $i)).
% 0.22/0.41 tff(1,plain,
% 0.22/0.41 (^[X: $i] : refl((~distinct_points(X, X)) <=> (~distinct_points(X, X)))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(2,plain,
% 0.22/0.41 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[1])).
% 0.22/0.41 tff(3,plain,
% 0.22/0.41 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.22/0.41 inference(rewrite,[status(thm)],[])).
% 0.22/0.41 tff(4,axiom,(![X: $i] : (~distinct_points(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','apart1')).
% 0.22/0.41 tff(5,plain,
% 0.22/0.41 (![X: $i] : (~distinct_points(X, X))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.22/0.41 tff(6,plain,(
% 0.22/0.41 ![X: $i] : (~distinct_points(X, X))),
% 0.22/0.41 inference(skolemize,[status(sab)],[5])).
% 0.22/0.41 tff(7,plain,
% 0.22/0.41 (![X: $i] : (~distinct_points(X, X))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.22/0.41 tff(8,plain,
% 0.22/0.41 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(X!3, X!3))),
% 0.22/0.41 inference(quant_inst,[status(thm)],[])).
% 0.22/0.41 tff(9,plain,
% 0.22/0.41 (~distinct_points(X!3, X!3)),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.22/0.41 tff(10,plain,
% 0.22/0.41 (((~distinct_points(X!3, X!3)) | (~distinct_points(X!3, Y!2))) | distinct_points(X!3, X!3)),
% 0.22/0.41 inference(tautology,[status(thm)],[])).
% 0.22/0.41 tff(11,plain,
% 0.22/0.41 ((~distinct_points(X!3, X!3)) | (~distinct_points(X!3, Y!2))),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[10, 9])).
% 0.22/0.41 tff(12,plain,
% 0.22/0.41 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(V!0, V!0))),
% 0.22/0.41 inference(quant_inst,[status(thm)],[])).
% 0.22/0.41 tff(13,plain,
% 0.22/0.41 (~distinct_points(V!0, V!0)),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[12, 7])).
% 0.22/0.41 tff(14,plain,
% 0.22/0.41 (((~distinct_points(V!0, U!1)) | (~distinct_points(V!0, V!0))) | distinct_points(V!0, V!0)),
% 0.22/0.41 inference(tautology,[status(thm)],[])).
% 0.22/0.41 tff(15,plain,
% 0.22/0.41 ((~distinct_points(V!0, U!1)) | (~distinct_points(V!0, V!0))),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[14, 13])).
% 0.22/0.41 tff(16,assumption,(apart_point_and_line(V!0, line_connecting(U!1, V!0))), introduced(assumption)).
% 0.22/0.41 tff(17,plain,
% 0.22/0.41 ((~(apart_point_and_line(V!0, line_connecting(X!3, Y!2)) | apart_point_and_line(U!1, line_connecting(X!3, Y!2)) | (~(apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | apart_point_and_line(X!3, line_connecting(U!1, V!0)))) | (~(distinct_points(X!3, Y!2) & distinct_points(U!1, V!0))))) <=> (~(apart_point_and_line(V!0, line_connecting(X!3, Y!2)) | apart_point_and_line(U!1, line_connecting(X!3, Y!2)) | (~(apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | apart_point_and_line(X!3, line_connecting(U!1, V!0)))) | (~(distinct_points(X!3, Y!2) & distinct_points(U!1, V!0)))))),
% 0.22/0.41 inference(rewrite,[status(thm)],[])).
% 0.22/0.41 tff(18,plain,
% 0.22/0.41 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(V, line_connecting(X, Y)) | apart_point_and_line(U, line_connecting(X, Y)) | (~(apart_point_and_line(Y, line_connecting(U, V)) | apart_point_and_line(X, line_connecting(U, V)))) | (~(distinct_points(X, Y) & distinct_points(U, V))))) <=> (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(V, line_connecting(X, Y)) | apart_point_and_line(U, line_connecting(X, Y)) | (~(apart_point_and_line(Y, line_connecting(U, V)) | apart_point_and_line(X, line_connecting(U, V)))) | (~(distinct_points(X, Y) & distinct_points(U, V)))))),
% 0.22/0.41 inference(rewrite,[status(thm)],[])).
% 0.22/0.41 tff(19,plain,
% 0.22/0.41 ((~![X: $i, Y: $i, U: $i, V: $i] : ((distinct_points(X, Y) & distinct_points(U, V)) => ((apart_point_and_line(X, line_connecting(U, V)) | apart_point_and_line(Y, line_connecting(U, V))) => (apart_point_and_line(U, line_connecting(X, Y)) | apart_point_and_line(V, line_connecting(X, Y)))))) <=> (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(V, line_connecting(X, Y)) | apart_point_and_line(U, line_connecting(X, Y)) | (~(apart_point_and_line(Y, line_connecting(U, V)) | apart_point_and_line(X, line_connecting(U, V)))) | (~(distinct_points(X, Y) & distinct_points(U, V)))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(20,axiom,(~![X: $i, Y: $i, U: $i, V: $i] : ((distinct_points(X, Y) & distinct_points(U, V)) => ((apart_point_and_line(X, line_connecting(U, V)) | apart_point_and_line(Y, line_connecting(U, V))) => (apart_point_and_line(U, line_connecting(X, Y)) | apart_point_and_line(V, line_connecting(X, Y)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.22/0.42 tff(21,plain,
% 0.22/0.42 (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(V, line_connecting(X, Y)) | apart_point_and_line(U, line_connecting(X, Y)) | (~(apart_point_and_line(Y, line_connecting(U, V)) | apart_point_and_line(X, line_connecting(U, V)))) | (~(distinct_points(X, Y) & distinct_points(U, V))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.22/0.42 tff(22,plain,
% 0.22/0.42 (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(V, line_connecting(X, Y)) | apart_point_and_line(U, line_connecting(X, Y)) | (~(apart_point_and_line(Y, line_connecting(U, V)) | apart_point_and_line(X, line_connecting(U, V)))) | (~(distinct_points(X, Y) & distinct_points(U, V))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[21, 18])).
% 0.22/0.42 tff(23,plain,
% 0.22/0.42 (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(V, line_connecting(X, Y)) | apart_point_and_line(U, line_connecting(X, Y)) | (~(apart_point_and_line(Y, line_connecting(U, V)) | apart_point_and_line(X, line_connecting(U, V)))) | (~(distinct_points(X, Y) & distinct_points(U, V))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[22, 18])).
% 0.22/0.42 tff(24,plain,
% 0.22/0.42 (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(V, line_connecting(X, Y)) | apart_point_and_line(U, line_connecting(X, Y)) | (~(apart_point_and_line(Y, line_connecting(U, V)) | apart_point_and_line(X, line_connecting(U, V)))) | (~(distinct_points(X, Y) & distinct_points(U, V))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[23, 18])).
% 0.22/0.42 tff(25,plain,(
% 0.22/0.42 ~(apart_point_and_line(V!0, line_connecting(X!3, Y!2)) | apart_point_and_line(U!1, line_connecting(X!3, Y!2)) | (~(apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | apart_point_and_line(X!3, line_connecting(U!1, V!0)))) | (~(distinct_points(X!3, Y!2) & distinct_points(U!1, V!0))))),
% 0.22/0.42 inference(skolemize,[status(sab)],[24])).
% 0.22/0.42 tff(26,plain,
% 0.22/0.42 (~(apart_point_and_line(V!0, line_connecting(X!3, Y!2)) | apart_point_and_line(U!1, line_connecting(X!3, Y!2)) | (~(apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | apart_point_and_line(X!3, line_connecting(U!1, V!0)))) | (~(distinct_points(X!3, Y!2) & distinct_points(U!1, V!0))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[25, 17])).
% 0.22/0.42 tff(27,plain,
% 0.22/0.42 (distinct_points(X!3, Y!2) & distinct_points(U!1, V!0)),
% 0.22/0.42 inference(or_elim,[status(thm)],[26])).
% 0.22/0.42 tff(28,plain,
% 0.22/0.42 (distinct_points(U!1, V!0)),
% 0.22/0.42 inference(and_elim,[status(thm)],[27])).
% 0.22/0.42 tff(29,plain,
% 0.22/0.42 (^[X: $i, Y: $i, Z: $i] : refl(((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(30,plain,
% 0.22/0.42 (![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))) <=> ![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[29])).
% 0.22/0.42 tff(31,plain,
% 0.22/0.42 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_points(Z, X) & distinct_points(Z, Y)) <=> (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))), (((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ((~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))))), rewrite(((~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))), (((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(32,plain,
% 0.22/0.42 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[31])).
% 0.22/0.42 tff(33,plain,
% 0.22/0.42 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(34,plain,
% 0.22/0.42 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y))) <=> ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))), ((distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> (distinct_points(X, Y) => ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))))), rewrite((distinct_points(X, Y) => ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))), ((distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(35,plain,
% 0.22/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[34])).
% 0.22/0.42 tff(36,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y))))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','con1')).
% 0.22/0.42 tff(37,plain,
% 0.22/0.42 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.22/0.42 tff(38,plain,
% 0.22/0.42 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[37, 33])).
% 0.22/0.42 tff(39,plain,(
% 0.22/0.42 ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.22/0.42 inference(skolemize,[status(sab)],[38])).
% 0.22/0.42 tff(40,plain,
% 0.22/0.42 (![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[39, 32])).
% 0.22/0.42 tff(41,plain,
% 0.22/0.42 (![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[40, 30])).
% 0.22/0.42 tff(42,plain,
% 0.22/0.42 (((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(U!1, V!0))) | (~((~distinct_points(V!0, U!1)) | (~distinct_points(V!0, V!0)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(U!1, V!0))) | (~((~distinct_points(V!0, U!1)) | (~distinct_points(V!0, V!0)))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(43,plain,
% 0.22/0.42 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(U!1, V!0))) | (~((~distinct_points(V!0, U!1)) | (~distinct_points(V!0, V!0)))))),
% 0.22/0.42 inference(quant_inst,[status(thm)],[])).
% 0.22/0.42 tff(44,plain,
% 0.22/0.42 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(U!1, V!0)) | (~apart_point_and_line(V!0, line_connecting(U!1, V!0))) | (~((~distinct_points(V!0, U!1)) | (~distinct_points(V!0, V!0))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.22/0.42 tff(45,plain,
% 0.22/0.42 ($false),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[44, 41, 28, 16, 15])).
% 0.22/0.42 tff(46,plain,(~apart_point_and_line(V!0, line_connecting(U!1, V!0))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.42 tff(47,plain,
% 0.22/0.42 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(U!1, U!1))),
% 0.22/0.42 inference(quant_inst,[status(thm)],[])).
% 0.22/0.42 tff(48,plain,
% 0.22/0.42 (~distinct_points(U!1, U!1)),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[47, 7])).
% 0.22/0.42 tff(49,plain,
% 0.22/0.42 (((~distinct_points(U!1, U!1)) | (~distinct_points(U!1, V!0))) | distinct_points(U!1, U!1)),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(50,plain,
% 0.22/0.42 ((~distinct_points(U!1, U!1)) | (~distinct_points(U!1, V!0))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[49, 48])).
% 0.22/0.42 tff(51,assumption,(apart_point_and_line(U!1, line_connecting(U!1, V!0))), introduced(assumption)).
% 0.22/0.42 tff(52,plain,
% 0.22/0.42 (((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(U!1, V!0))) | (~((~distinct_points(U!1, U!1)) | (~distinct_points(U!1, V!0)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(U!1, V!0))) | (~((~distinct_points(U!1, U!1)) | (~distinct_points(U!1, V!0)))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(53,plain,
% 0.22/0.42 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(U!1, V!0))) | (~((~distinct_points(U!1, U!1)) | (~distinct_points(U!1, V!0)))))),
% 0.22/0.42 inference(quant_inst,[status(thm)],[])).
% 0.22/0.42 tff(54,plain,
% 0.22/0.42 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(U!1, V!0)) | (~apart_point_and_line(U!1, line_connecting(U!1, V!0))) | (~((~distinct_points(U!1, U!1)) | (~distinct_points(U!1, V!0))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.22/0.42 tff(55,plain,
% 0.22/0.42 ($false),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[54, 41, 28, 51, 50])).
% 0.22/0.42 tff(56,plain,(~apart_point_and_line(U!1, line_connecting(U!1, V!0))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.42 tff(57,plain,
% 0.22/0.42 (~apart_point_and_line(U!1, line_connecting(X!3, Y!2))),
% 0.22/0.42 inference(or_elim,[status(thm)],[26])).
% 0.22/0.42 tff(58,plain,
% 0.22/0.42 (~apart_point_and_line(V!0, line_connecting(X!3, Y!2))),
% 0.22/0.42 inference(or_elim,[status(thm)],[26])).
% 0.22/0.42 tff(59,plain,
% 0.22/0.42 (^[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))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(60,plain,
% 0.22/0.42 (![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))) <=> ![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)))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[59])).
% 0.22/0.42 tff(61,plain,
% 0.22/0.42 (^[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)))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(62,plain,
% 0.22/0.42 (![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)))) <=> ![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)))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[61])).
% 0.22/0.42 tff(63,plain,
% 0.22/0.42 (![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)))) <=> ![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))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(64,plain,
% 0.22/0.42 (^[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))))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(65,plain,
% 0.22/0.42 (![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))))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[64])).
% 0.22/0.42 tff(66,axiom,(![X: $i, Y: $i, U: $i, V: $i] : ((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','cu1')).
% 0.22/0.42 tff(67,plain,
% 0.22/0.42 (![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))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.22/0.42 tff(68,plain,
% 0.22/0.42 (![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))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[67, 63])).
% 0.22/0.42 tff(69,plain,(
% 0.22/0.42 ![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))))),
% 0.22/0.43 inference(skolemize,[status(sab)],[68])).
% 0.22/0.43 tff(70,plain,
% 0.22/0.43 (![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)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[69, 62])).
% 0.22/0.43 tff(71,plain,
% 0.22/0.43 (![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)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[70, 60])).
% 0.22/0.43 tff(72,plain,
% 0.22/0.43 (((~![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(V!0, line_connecting(X!3, Y!2)) | apart_point_and_line(U!1, line_connecting(X!3, Y!2)) | (~distinct_points(U!1, V!0)) | (~distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2))) | apart_point_and_line(U!1, line_connecting(U!1, V!0)) | apart_point_and_line(V!0, 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_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(V!0, line_connecting(X!3, Y!2)) | apart_point_and_line(U!1, line_connecting(X!3, Y!2)) | (~distinct_points(U!1, V!0)) | (~distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2))) | apart_point_and_line(U!1, line_connecting(U!1, V!0)) | apart_point_and_line(V!0, line_connecting(U!1, V!0)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(73,plain,
% 0.22/0.43 ((apart_point_and_line(V!0, line_connecting(X!3, Y!2)) | apart_point_and_line(V!0, 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_points(U!1, V!0)) | (~distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2)))) <=> (apart_point_and_line(V!0, line_connecting(X!3, Y!2)) | apart_point_and_line(U!1, line_connecting(X!3, Y!2)) | (~distinct_points(U!1, V!0)) | (~distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2))) | apart_point_and_line(U!1, line_connecting(U!1, V!0)) | apart_point_and_line(V!0, line_connecting(U!1, V!0)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(74,plain,
% 0.22/0.43 (((~![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(V!0, line_connecting(X!3, Y!2)) | apart_point_and_line(V!0, 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_points(U!1, V!0)) | (~distinct_lines(line_connecting(U!1, V!0), line_connecting(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_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(V!0, line_connecting(X!3, Y!2)) | apart_point_and_line(U!1, line_connecting(X!3, Y!2)) | (~distinct_points(U!1, V!0)) | (~distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2))) | apart_point_and_line(U!1, line_connecting(U!1, V!0)) | apart_point_and_line(V!0, line_connecting(U!1, V!0))))),
% 0.22/0.43 inference(monotonicity,[status(thm)],[73])).
% 0.22/0.43 tff(75,plain,
% 0.22/0.43 (((~![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(V!0, line_connecting(X!3, Y!2)) | apart_point_and_line(V!0, 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_points(U!1, V!0)) | (~distinct_lines(line_connecting(U!1, V!0), line_connecting(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_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(V!0, line_connecting(X!3, Y!2)) | apart_point_and_line(U!1, line_connecting(X!3, Y!2)) | (~distinct_points(U!1, V!0)) | (~distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2))) | apart_point_and_line(U!1, line_connecting(U!1, V!0)) | apart_point_and_line(V!0, line_connecting(U!1, V!0)))),
% 0.22/0.43 inference(transitivity,[status(thm)],[74, 72])).
% 0.22/0.43 tff(76,plain,
% 0.22/0.43 ((~![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(V!0, line_connecting(X!3, Y!2)) | apart_point_and_line(V!0, 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_points(U!1, V!0)) | (~distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2))))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(77,plain,
% 0.22/0.43 ((~![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(V!0, line_connecting(X!3, Y!2)) | apart_point_and_line(U!1, line_connecting(X!3, Y!2)) | (~distinct_points(U!1, V!0)) | (~distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2))) | apart_point_and_line(U!1, line_connecting(U!1, V!0)) | apart_point_and_line(V!0, line_connecting(U!1, V!0))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.22/0.43 tff(78,plain,
% 0.22/0.43 ((~distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2))) | apart_point_and_line(U!1, line_connecting(U!1, V!0)) | apart_point_and_line(V!0, line_connecting(U!1, V!0))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[77, 71, 58, 57, 28])).
% 0.22/0.43 tff(79,plain,
% 0.22/0.43 (~distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[78, 56, 46])).
% 0.22/0.43 tff(80,plain,
% 0.22/0.43 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(Y!2, Y!2))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(81,plain,
% 0.22/0.43 (~distinct_points(Y!2, Y!2)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[80, 7])).
% 0.22/0.43 tff(82,plain,
% 0.22/0.43 (((~distinct_points(Y!2, X!3)) | (~distinct_points(Y!2, Y!2))) | distinct_points(Y!2, Y!2)),
% 0.22/0.43 inference(tautology,[status(thm)],[])).
% 0.22/0.43 tff(83,plain,
% 0.22/0.43 ((~distinct_points(Y!2, X!3)) | (~distinct_points(Y!2, Y!2))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[82, 81])).
% 0.22/0.43 tff(84,assumption,(apart_point_and_line(Y!2, line_connecting(X!3, Y!2))), introduced(assumption)).
% 0.22/0.43 tff(85,plain,
% 0.22/0.43 (distinct_points(X!3, Y!2)),
% 0.22/0.43 inference(and_elim,[status(thm)],[27])).
% 0.22/0.43 tff(86,plain,
% 0.22/0.43 (((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(X!3, Y!2)) | (~apart_point_and_line(Y!2, line_connecting(X!3, Y!2))) | (~((~distinct_points(Y!2, X!3)) | (~distinct_points(Y!2, Y!2)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!3, Y!2)) | (~apart_point_and_line(Y!2, line_connecting(X!3, Y!2))) | (~((~distinct_points(Y!2, X!3)) | (~distinct_points(Y!2, Y!2)))))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(87,plain,
% 0.22/0.43 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(X!3, Y!2)) | (~apart_point_and_line(Y!2, line_connecting(X!3, Y!2))) | (~((~distinct_points(Y!2, X!3)) | (~distinct_points(Y!2, Y!2)))))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(88,plain,
% 0.22/0.43 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!3, Y!2)) | (~apart_point_and_line(Y!2, line_connecting(X!3, Y!2))) | (~((~distinct_points(Y!2, X!3)) | (~distinct_points(Y!2, Y!2))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[87, 86])).
% 0.22/0.43 tff(89,plain,
% 0.22/0.43 ($false),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[88, 41, 85, 84, 83])).
% 0.22/0.43 tff(90,plain,(~apart_point_and_line(Y!2, line_connecting(X!3, Y!2))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.43 tff(91,plain,
% 0.22/0.43 (^[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))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(92,plain,
% 0.22/0.43 (![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)))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[91])).
% 0.22/0.43 tff(93,plain,
% 0.22/0.43 (![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)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(94,plain,
% 0.22/0.43 (^[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)))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(95,plain,
% 0.22/0.43 (![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)))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[94])).
% 0.22/0.43 tff(96,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/GEO008+0.ax','ceq2')).
% 0.22/0.43 tff(97,plain,
% 0.22/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[96, 95])).
% 0.22/0.43 tff(98,plain,
% 0.22/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[97, 93])).
% 0.22/0.43 tff(99,plain,(
% 0.22/0.43 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.22/0.43 inference(skolemize,[status(sab)],[98])).
% 0.22/0.43 tff(100,plain,
% 0.22/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[99, 92])).
% 0.22/0.43 tff(101,plain,
% 0.22/0.43 (((~![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(Y!2, line_connecting(U!1, V!0))) | distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2)) | apart_point_and_line(Y!2, 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(Y!2, line_connecting(U!1, V!0))) | distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(102,plain,
% 0.22/0.43 ((distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)) | (~apart_point_and_line(Y!2, line_connecting(U!1, V!0)))) <=> ((~apart_point_and_line(Y!2, line_connecting(U!1, V!0))) | distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(103,plain,
% 0.22/0.43 (((~![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(U!1, V!0), line_connecting(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)) | (~apart_point_and_line(Y!2, line_connecting(U!1, V!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(Y!2, line_connecting(U!1, V!0))) | distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2))))),
% 0.22/0.43 inference(monotonicity,[status(thm)],[102])).
% 0.22/0.43 tff(104,plain,
% 0.22/0.43 (((~![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(U!1, V!0), line_connecting(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)) | (~apart_point_and_line(Y!2, line_connecting(U!1, V!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(Y!2, line_connecting(U!1, V!0))) | distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)))),
% 0.22/0.43 inference(transitivity,[status(thm)],[103, 101])).
% 0.22/0.43 tff(105,plain,
% 0.22/0.43 ((~![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(U!1, V!0), line_connecting(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2)) | (~apart_point_and_line(Y!2, line_connecting(U!1, V!0))))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(106,plain,
% 0.22/0.43 ((~![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(Y!2, line_connecting(U!1, V!0))) | distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[105, 104])).
% 0.22/0.43 tff(107,plain,
% 0.22/0.43 ((~apart_point_and_line(Y!2, line_connecting(U!1, V!0))) | distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2)) | apart_point_and_line(Y!2, line_connecting(X!3, Y!2))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[106, 100])).
% 0.22/0.43 tff(108,plain,
% 0.22/0.43 ((~apart_point_and_line(Y!2, line_connecting(U!1, V!0))) | distinct_lines(line_connecting(U!1, V!0), line_connecting(X!3, Y!2))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[107, 90])).
% 0.22/0.43 tff(109,plain,
% 0.22/0.43 (~apart_point_and_line(Y!2, line_connecting(U!1, V!0))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[108, 79])).
% 0.22/0.43 tff(110,plain,
% 0.22/0.43 (apart_point_and_line(Y!2, line_connecting(U!1, V!0)) | apart_point_and_line(X!3, line_connecting(U!1, V!0))),
% 0.22/0.43 inference(or_elim,[status(thm)],[26])).
% 0.22/0.43 tff(111,plain,
% 0.22/0.43 (apart_point_and_line(X!3, line_connecting(U!1, V!0))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[110, 109])).
% 0.22/0.43 tff(112,plain,
% 0.22/0.43 (((~![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(U!1, V!0), 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)))) <=> ((~![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(U!1, V!0), 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)))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(113,plain,
% 0.22/0.44 ((distinct_lines(line_connecting(U!1, V!0), 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)))) <=> (distinct_lines(line_connecting(U!1, V!0), 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)))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(114,plain,
% 0.22/0.44 (((~![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(U!1, V!0), 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))))) <=> ((~![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(U!1, V!0), 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))))),
% 0.22/0.44 inference(monotonicity,[status(thm)],[113])).
% 0.22/0.44 tff(115,plain,
% 0.22/0.44 (((~![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(U!1, V!0), 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))))) <=> ((~![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(U!1, V!0), 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)))),
% 0.22/0.44 inference(transitivity,[status(thm)],[114, 112])).
% 0.22/0.44 tff(116,plain,
% 0.22/0.44 ((~![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(U!1, V!0), 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))))),
% 0.22/0.44 inference(quant_inst,[status(thm)],[])).
% 0.22/0.44 tff(117,plain,
% 0.22/0.44 ((~![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(U!1, V!0), 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))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[116, 115])).
% 0.22/0.44 tff(118,plain,
% 0.22/0.44 (distinct_lines(line_connecting(U!1, V!0), 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))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[117, 100])).
% 0.22/0.44 tff(119,plain,
% 0.22/0.44 (apart_point_and_line(X!3, line_connecting(X!3, Y!2))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[118, 111, 79])).
% 0.22/0.44 tff(120,plain,
% 0.22/0.44 (((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(X!3, Y!2)) | (~apart_point_and_line(X!3, line_connecting(X!3, Y!2))) | (~((~distinct_points(X!3, X!3)) | (~distinct_points(X!3, Y!2)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!3, Y!2)) | (~apart_point_and_line(X!3, line_connecting(X!3, Y!2))) | (~((~distinct_points(X!3, X!3)) | (~distinct_points(X!3, Y!2)))))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(121,plain,
% 0.22/0.44 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(X!3, Y!2)) | (~apart_point_and_line(X!3, line_connecting(X!3, Y!2))) | (~((~distinct_points(X!3, X!3)) | (~distinct_points(X!3, Y!2)))))),
% 0.22/0.44 inference(quant_inst,[status(thm)],[])).
% 0.22/0.44 tff(122,plain,
% 0.22/0.44 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!3, Y!2)) | (~apart_point_and_line(X!3, line_connecting(X!3, Y!2))) | (~((~distinct_points(X!3, X!3)) | (~distinct_points(X!3, Y!2))))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[121, 120])).
% 0.22/0.44 tff(123,plain,
% 0.22/0.44 ($false),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[122, 41, 85, 119, 11])).
% 0.22/0.44 % SZS output end Proof
% 0.22/0.44 % E exiting
%------------------------------------------------------------------------------