TSTP Solution File: GEO179+1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : GEO179+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n004.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 0.41s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GEO179+1 : TPTP v8.1.0. Released v3.3.0.
% 0.03/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.34 % Computer : n004.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 06:43:05 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.
% 0.20/0.41 % SZS status Theorem
% 0.20/0.41 % SZS output start Proof
% 0.20/0.41 tff(apart_point_and_line_type, type, (
% 0.20/0.41 apart_point_and_line: ( $i * $i ) > $o)).
% 0.20/0.41 tff(line_connecting_type, type, (
% 0.20/0.41 line_connecting: ( $i * $i ) > $i)).
% 0.20/0.41 tff(tptp_fun_X_2_type, type, (
% 0.20/0.41 tptp_fun_X_2: $i)).
% 0.20/0.41 tff(tptp_fun_Z_0_type, type, (
% 0.20/0.41 tptp_fun_Z_0: $i)).
% 0.20/0.41 tff(distinct_points_type, type, (
% 0.20/0.41 distinct_points: ( $i * $i ) > $o)).
% 0.20/0.41 tff(tptp_fun_Y_1_type, type, (
% 0.20/0.41 tptp_fun_Y_1: $i)).
% 0.20/0.41 tff(distinct_lines_type, type, (
% 0.20/0.41 distinct_lines: ( $i * $i ) > $o)).
% 0.20/0.41 tff(1,assumption,(~distinct_points(Z!0, X!2)), introduced(assumption)).
% 0.20/0.41 tff(2,plain,
% 0.20/0.41 ((~![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)))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(3,plain,
% 0.20/0.41 ((~![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)))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(4,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/sandbox2/benchmark/theBenchmark.p','con')).
% 0.20/0.41 tff(5,plain,
% 0.20/0.41 (~![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))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.20/0.41 tff(6,plain,
% 0.20/0.41 (~![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))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[5, 2])).
% 0.20/0.41 tff(7,plain,
% 0.20/0.41 (~![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))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.20/0.41 tff(8,plain,
% 0.20/0.41 (~![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))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[7, 2])).
% 0.20/0.41 tff(9,plain,
% 0.20/0.41 (~![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))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[8, 2])).
% 0.20/0.41 tff(10,plain,
% 0.20/0.41 (~![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))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[9, 2])).
% 0.20/0.41 tff(11,plain,
% 0.20/0.41 (~![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))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[10, 2])).
% 0.20/0.41 tff(12,plain,(
% 0.20/0.42 ~((~(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))))),
% 0.20/0.42 inference(skolemize,[status(sab)],[11])).
% 0.20/0.42 tff(13,plain,
% 0.20/0.42 (distinct_points(X!2, Y!1) & apart_point_and_line(Z!0, line_connecting(X!2, Y!1))),
% 0.20/0.42 inference(or_elim,[status(thm)],[12])).
% 0.20/0.42 tff(14,plain,
% 0.20/0.42 (distinct_points(X!2, Y!1)),
% 0.20/0.42 inference(and_elim,[status(thm)],[13])).
% 0.20/0.42 tff(15,plain,
% 0.20/0.42 (^[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)))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(16,plain,
% 0.20/0.42 (![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))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[15])).
% 0.20/0.42 tff(17,plain,
% 0.20/0.42 (![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))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(18,plain,
% 0.20/0.42 (^[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)))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(19,plain,
% 0.20/0.42 (![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))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[18])).
% 0.20/0.42 tff(20,axiom,(![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(X, line_connecting(X, Y))))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO006+0.ax','ci1')).
% 0.20/0.42 tff(21,plain,
% 0.20/0.42 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.20/0.42 tff(22,plain,
% 0.20/0.42 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[21, 17])).
% 0.20/0.42 tff(23,plain,(
% 0.20/0.42 ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.42 inference(skolemize,[status(sab)],[22])).
% 0.20/0.42 tff(24,plain,
% 0.20/0.42 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[23, 16])).
% 0.20/0.42 tff(25,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(26,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(27,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)],[26, 25])).
% 0.20/0.42 tff(28,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)],[27, 24, 14])).
% 0.20/0.42 tff(29,plain,
% 0.20/0.42 (apart_point_and_line(Z!0, line_connecting(X!2, Y!1))),
% 0.20/0.42 inference(and_elim,[status(thm)],[13])).
% 0.20/0.42 tff(30,plain,
% 0.20/0.42 (^[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))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(31,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))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[30])).
% 0.20/0.42 tff(32,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))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(33,plain,
% 0.20/0.42 (^[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)))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(34,plain,
% 0.20/0.42 (![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)))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[33])).
% 0.20/0.42 tff(35,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/sandbox2/benchmark/Axioms/GEO006+0.ax','ceq1')).
% 0.20/0.42 tff(36,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)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.20/0.42 tff(37,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)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[36, 32])).
% 0.20/0.42 tff(38,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)))),
% 0.20/0.42 inference(skolemize,[status(sab)],[37])).
% 0.20/0.42 tff(39,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)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[38, 31])).
% 0.20/0.42 tff(40,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(41,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(42,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)],[41])).
% 0.20/0.42 tff(43,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)],[42, 40])).
% 0.20/0.42 tff(44,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(45,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, X!2)),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.20/0.42 tff(46,plain,
% 0.20/0.42 ($false),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[45, 39, 29, 28, 1])).
% 0.20/0.42 tff(47,plain,(distinct_points(Z!0, X!2)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.42 tff(48,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(49,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(50,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)],[49, 48])).
% 0.20/0.42 tff(51,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)],[50, 24])).
% 0.20/0.42 tff(52,plain,
% 0.20/0.42 (~apart_point_and_line(Z!0, line_connecting(Z!0, X!2))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[51, 47])).
% 0.20/0.42 tff(53,assumption,(~distinct_points(Z!0, Y!1)), introduced(assumption)).
% 0.20/0.42 tff(54,plain,
% 0.20/0.42 (^[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)))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(55,plain,
% 0.20/0.42 (![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))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[54])).
% 0.20/0.42 tff(56,plain,
% 0.20/0.42 (![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))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(57,plain,
% 0.20/0.42 (^[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)))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(58,plain,
% 0.20/0.42 (![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))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[57])).
% 0.20/0.42 tff(59,axiom,(![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(Y, line_connecting(X, Y))))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO006+0.ax','ci2')).
% 0.20/0.42 tff(60,plain,
% 0.20/0.42 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.20/0.42 tff(61,plain,
% 0.20/0.42 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[60, 56])).
% 0.20/0.42 tff(62,plain,(
% 0.20/0.42 ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.42 inference(skolemize,[status(sab)],[61])).
% 0.20/0.42 tff(63,plain,
% 0.20/0.42 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[62, 55])).
% 0.20/0.42 tff(64,plain,
% 0.20/0.42 (((~![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))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(65,plain,
% 0.20/0.42 ((~![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))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(66,plain,
% 0.20/0.42 ((~![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)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.20/0.42 tff(67,plain,
% 0.20/0.42 (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[66, 63, 14])).
% 0.20/0.42 tff(68,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(Y!1, line_connecting(X!2, Y!1)) | distinct_points(Z!0, 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)) | distinct_points(Z!0, Y!1) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(69,plain,
% 0.20/0.42 ((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)) | distinct_points(Z!0, Y!1) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(70,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, 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)) | distinct_points(Z!0, Y!1) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[69])).
% 0.20/0.42 tff(71,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, 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)) | distinct_points(Z!0, Y!1) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.42 inference(transitivity,[status(thm)],[70, 68])).
% 0.20/0.42 tff(72,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, 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))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(73,plain,
% 0.20/0.43 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | distinct_points(Z!0, Y!1) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[72, 71])).
% 0.20/0.43 tff(74,plain,
% 0.20/0.43 ($false),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[73, 39, 29, 67, 53])).
% 0.20/0.43 tff(75,plain,(distinct_points(Z!0, Y!1)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.43 tff(76,plain,
% 0.20/0.43 (((~![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))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(77,plain,
% 0.20/0.43 ((~![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))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(78,plain,
% 0.20/0.43 ((~![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)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.20/0.43 tff(79,plain,
% 0.20/0.43 (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[78, 24, 75])).
% 0.20/0.43 tff(80,assumption,(~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))), introduced(assumption)).
% 0.20/0.43 tff(81,assumption,(~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))), introduced(assumption)).
% 0.20/0.43 tff(82,plain,
% 0.20/0.43 (^[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))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(83,plain,
% 0.20/0.43 (![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)))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[82])).
% 0.20/0.43 tff(84,plain,
% 0.20/0.43 (![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)))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(85,plain,
% 0.20/0.43 (^[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)))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(86,plain,
% 0.20/0.43 (![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)))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[85])).
% 0.20/0.43 tff(87,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/sandbox2/benchmark/Axioms/GEO006+0.ax','ceq2')).
% 0.20/0.43 tff(88,plain,
% 0.20/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[87, 86])).
% 0.20/0.43 tff(89,plain,
% 0.20/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[88, 84])).
% 0.20/0.43 tff(90,plain,(
% 0.20/0.43 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.43 inference(skolemize,[status(sab)],[89])).
% 0.20/0.43 tff(91,plain,
% 0.20/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[90, 83])).
% 0.20/0.43 tff(92,plain,
% 0.20/0.43 (((~![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(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(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(93,plain,
% 0.20/0.43 ((~![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(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))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(94,plain,
% 0.20/0.43 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.20/0.43 tff(95,plain,
% 0.20/0.43 ($false),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[94, 91, 29, 81, 80])).
% 0.20/0.43 tff(96,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,[]))).
% 0.20/0.43 tff(97,plain,
% 0.20/0.43 (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[96, 79])).
% 0.20/0.43 tff(98,plain,
% 0.20/0.43 ((~(~((~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))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(99,plain,
% 0.20/0.43 ((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)))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(100,plain,
% 0.20/0.43 ((~(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))))))),
% 0.20/0.43 inference(monotonicity,[status(thm)],[99])).
% 0.20/0.43 tff(101,plain,
% 0.20/0.43 ((~(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))))),
% 0.20/0.43 inference(transitivity,[status(thm)],[100, 98])).
% 0.20/0.43 tff(102,plain,
% 0.20/0.43 (~(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)))),
% 0.20/0.43 inference(or_elim,[status(thm)],[12])).
% 0.20/0.43 tff(103,plain,
% 0.20/0.43 ((~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)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.20/0.43 tff(104,plain,
% 0.20/0.43 (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[103, 97])).
% 0.20/0.43 tff(105,plain,
% 0.20/0.43 (((~![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(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)))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(106,plain,
% 0.20/0.43 ((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)))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(107,plain,
% 0.20/0.43 (((~![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(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))))),
% 0.20/0.43 inference(monotonicity,[status(thm)],[106])).
% 0.20/0.43 tff(108,plain,
% 0.20/0.43 (((~![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(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)))),
% 0.20/0.43 inference(transitivity,[status(thm)],[107, 105])).
% 0.20/0.43 tff(109,plain,
% 0.20/0.43 ((~![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(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))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(110,plain,
% 0.20/0.43 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[109, 108])).
% 0.20/0.43 tff(111,plain,
% 0.20/0.43 ($false),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[110, 91, 29, 104, 52])).
% 0.20/0.43 % SZS output end Proof
%------------------------------------------------------------------------------