TSTP Solution File: GEO179+3 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : GEO179+3 : TPTP v8.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n021.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 : Fri Sep 16 20:35:19 EDT 2022
% Result : Theorem 0.20s 0.40s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GEO179+3 : TPTP v8.1.0. Released v4.0.0.
% 0.07/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.34 % Computer : n021.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Wed Aug 31 06:43:11 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.35 Usage: tptp [options] [-file:]file
% 0.12/0.35 -h, -? prints this message.
% 0.12/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.35 -m, -model generate model.
% 0.12/0.35 -p, -proof generate proof.
% 0.12/0.35 -c, -core generate unsat core of named formulas.
% 0.12/0.35 -st, -statistics display statistics.
% 0.12/0.35 -t:timeout set timeout (in second).
% 0.12/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.35 -<param>:<value> configuration parameter and value.
% 0.12/0.35 -o:<output-file> file to place output in.
% 0.20/0.40 % SZS status Theorem
% 0.20/0.40 % SZS output start Proof
% 0.20/0.40 tff(distinct_points_type, type, (
% 0.20/0.40 distinct_points: ( $i * $i ) > $o)).
% 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(tptp_fun_Z_0_type, type, (
% 0.20/0.40 tptp_fun_Z_0: $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(line_connecting_type, type, (
% 0.20/0.40 line_connecting: ( $i * $i ) > $i)).
% 0.20/0.40 tff(distinct_lines_type, type, (
% 0.20/0.40 distinct_lines: ( $i * $i ) > $o)).
% 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(1,assumption,(~apart_point_and_line(Z!0, line_connecting(Z!0, X!2))), introduced(assumption)).
% 0.20/0.40 tff(2,assumption,(~distinct_points(Z!0, Y!1)), introduced(assumption)).
% 0.20/0.40 tff(3,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y)))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(4,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : ((distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y))) => (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y)))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(5,axiom,(~![X: $i, Y: $i, Z: $i] : ((distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y))) => (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.20/0.40 tff(6,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.20/0.40 tff(7,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[6, 3])).
% 0.20/0.40 tff(8,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.20/0.40 tff(9,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[8, 3])).
% 0.20/0.40 tff(10,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[9, 3])).
% 0.20/0.40 tff(11,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[10, 3])).
% 0.20/0.40 tff(12,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[11, 3])).
% 0.20/0.40 tff(13,plain,(
% 0.20/0.40 ~((~(distinct_points(X!2, Y!1) & apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) & distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))))),
% 0.20/0.40 inference(skolemize,[status(sab)],[12])).
% 0.20/0.40 tff(14,plain,
% 0.20/0.40 (distinct_points(X!2, Y!1) & apart_point_and_line(Z!0, line_connecting(X!2, Y!1))),
% 0.20/0.40 inference(or_elim,[status(thm)],[13])).
% 0.20/0.40 tff(15,plain,
% 0.20/0.40 (distinct_points(X!2, Y!1)),
% 0.20/0.40 inference(and_elim,[status(thm)],[14])).
% 0.20/0.40 tff(16,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(17,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)],[16])).
% 0.20/0.40 tff(18,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(19,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(20,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)],[19])).
% 0.20/0.40 tff(21,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(22,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)],[21, 20])).
% 0.20/0.40 tff(23,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)],[22, 18])).
% 0.20/0.40 tff(24,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)],[23])).
% 0.20/0.40 tff(25,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)],[24, 17])).
% 0.20/0.40 tff(26,plain,
% 0.20/0.40 (((~![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))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(27,plain,
% 0.20/0.40 ((~![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))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(28,plain,
% 0.20/0.40 ((~![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)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.20/0.40 tff(29,plain,
% 0.20/0.40 (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[28, 25, 15])).
% 0.20/0.40 tff(30,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)],[14])).
% 0.20/0.40 tff(31,plain,
% 0.20/0.40 (^[X: $i, Y: $i, Z: $i] : refl((distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(32,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[31])).
% 0.20/0.41 tff(33,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(34,plain,
% 0.20/0.41 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_points(X, Z) | apart_point_and_line(Z, Y)) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y))), ((apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> (apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))))), rewrite((apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))), ((apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(35,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[34])).
% 0.20/0.41 tff(36,axiom,(![X: $i, Y: $i, Z: $i] : (apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ceq1')).
% 0.20/0.41 tff(37,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.20/0.41 tff(38,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[37, 33])).
% 0.20/0.41 tff(39,plain,(
% 0.20/0.41 ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(skolemize,[status(sab)],[38])).
% 0.20/0.41 tff(40,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[39, 32])).
% 0.20/0.41 tff(41,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, Y!1))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, Y!1))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(42,plain,
% 0.20/0.41 ((distinct_points(Z!0, Y!1) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))) <=> (apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, Y!1))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(43,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(Z!0, Y!1) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, Y!1)))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[42])).
% 0.20/0.41 tff(44,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(Z!0, Y!1) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, Y!1))),
% 0.20/0.41 inference(transitivity,[status(thm)],[43, 41])).
% 0.20/0.41 tff(45,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(Z!0, Y!1) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(46,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, Y!1)),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.20/0.41 tff(47,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[46, 40, 30, 29, 2])).
% 0.20/0.41 tff(48,plain,(distinct_points(Z!0, Y!1)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.41 tff(49,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(50,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)],[49])).
% 0.20/0.41 tff(51,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(52,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(53,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)],[52])).
% 0.20/0.41 tff(54,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(55,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)],[54, 53])).
% 0.20/0.41 tff(56,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)],[55, 51])).
% 0.20/0.41 tff(57,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)],[56])).
% 0.20/0.41 tff(58,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)],[57, 50])).
% 0.20/0.41 tff(59,plain,
% 0.20/0.41 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(60,plain,
% 0.20/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(61,plain,
% 0.20/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[60, 59])).
% 0.20/0.41 tff(62,plain,
% 0.20/0.41 (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[61, 58, 48])).
% 0.20/0.41 tff(63,assumption,(~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))), introduced(assumption)).
% 0.20/0.41 tff(64,assumption,(~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))), introduced(assumption)).
% 0.20/0.41 tff(65,plain,
% 0.20/0.41 (^[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))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(66,plain,
% 0.20/0.41 (![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)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[65])).
% 0.20/0.41 tff(67,plain,
% 0.20/0.41 (![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)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(68,plain,
% 0.20/0.41 (^[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)))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(69,plain,
% 0.20/0.41 (![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)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[68])).
% 0.20/0.41 tff(70,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')).
% 0.20/0.41 tff(71,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[70, 69])).
% 0.20/0.41 tff(72,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[71, 67])).
% 0.20/0.41 tff(73,plain,(
% 0.20/0.41 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(skolemize,[status(sab)],[72])).
% 0.20/0.41 tff(74,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[73, 66])).
% 0.20/0.41 tff(75,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(76,plain,
% 0.20/0.41 ((distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))) <=> (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(77,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, 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)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[76])).
% 0.20/0.41 tff(78,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, 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)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)))),
% 0.20/0.41 inference(transitivity,[status(thm)],[77, 75])).
% 0.20/0.41 tff(79,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(80,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[79, 78])).
% 0.20/0.41 tff(81,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[80, 74, 30, 64, 63])).
% 0.20/0.41 tff(82,plain,(apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.41 tff(83,plain,
% 0.20/0.41 (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[82, 62])).
% 0.20/0.41 tff(84,plain,
% 0.20/0.41 ((~(~((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)))))) <=> ((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(85,plain,
% 0.20/0.41 ((distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) & distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))) <=> (~((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(86,plain,
% 0.20/0.41 ((~(distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) & distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)))) <=> (~(~((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[85])).
% 0.20/0.42 tff(87,plain,
% 0.20/0.42 ((~(distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) & distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)))) <=> ((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))))),
% 0.20/0.42 inference(transitivity,[status(thm)],[86, 84])).
% 0.20/0.42 tff(88,plain,
% 0.20/0.42 (~(distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) & distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)))),
% 0.20/0.42 inference(or_elim,[status(thm)],[13])).
% 0.20/0.42 tff(89,plain,
% 0.20/0.42 ((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.20/0.42 tff(90,plain,
% 0.20/0.42 (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[89, 83])).
% 0.20/0.42 tff(91,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, X!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(X!2, Y!1), line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(92,plain,
% 0.20/0.42 ((distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))) <=> (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(93,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, 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)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[92])).
% 0.20/0.42 tff(94,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, 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)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2)))),
% 0.20/0.42 inference(transitivity,[status(thm)],[93, 91])).
% 0.20/0.42 tff(95,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(96,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[95, 94])).
% 0.20/0.42 tff(97,plain,
% 0.20/0.42 ($false),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[96, 74, 30, 90, 1])).
% 0.20/0.42 tff(98,plain,(apart_point_and_line(Z!0, line_connecting(Z!0, X!2))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.42 tff(99,plain,
% 0.20/0.42 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, X!2))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, X!2))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(100,plain,
% 0.20/0.42 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, X!2))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(101,plain,
% 0.20/0.42 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, X!2)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[100, 99])).
% 0.20/0.42 tff(102,plain,
% 0.20/0.42 ((~distinct_points(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, X!2)))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[101, 58])).
% 0.20/0.42 tff(103,plain,
% 0.20/0.42 (~distinct_points(Z!0, X!2)),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[102, 98])).
% 0.20/0.42 tff(104,plain,
% 0.20/0.42 (((~![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))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(105,plain,
% 0.20/0.42 ((~![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))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(106,plain,
% 0.20/0.42 ((~![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)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[105, 104])).
% 0.20/0.42 tff(107,plain,
% 0.20/0.42 (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[106, 58, 15])).
% 0.20/0.42 tff(108,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, X!2))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, X!2))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(109,plain,
% 0.20/0.42 ((distinct_points(Z!0, X!2) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))) <=> (apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, X!2))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(110,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(Z!0, X!2) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, X!2)))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[109])).
% 0.20/0.42 tff(111,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(Z!0, X!2) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, X!2))),
% 0.20/0.42 inference(transitivity,[status(thm)],[110, 108])).
% 0.20/0.42 tff(112,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(Z!0, X!2) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(113,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, X!2)),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[112, 111])).
% 0.20/0.42 tff(114,plain,
% 0.20/0.42 ($false),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[113, 40, 30, 107, 103])).
% 0.20/0.42 % SZS output end Proof
%------------------------------------------------------------------------------