TSTP Solution File: MSC006-1 by Z3---4.8.9.0

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Z3---4.8.9.0
% Problem  : MSC006-1 : TPTP v8.1.0. Released v1.0.0.
% Transfm  : none
% Format   : tptp
% Command  : z3_tptp -proof -model -t:%d -file:%s

% Computer : n018.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 : Sun Sep 18 05:34:00 EDT 2022

% Result   : Unsatisfiable 0.13s 0.42s
% Output   : Proof 0.21s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12  % Problem  : MSC006-1 : TPTP v8.1.0. Released v1.0.0.
% 0.12/0.13  % Command  : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34  % Computer : n018.cluster.edu
% 0.13/0.34  % Model    : x86_64 x86_64
% 0.13/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34  % Memory   : 8042.1875MB
% 0.13/0.34  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34  % CPULimit : 300
% 0.13/0.34  % WCLimit  : 300
% 0.13/0.34  % DateTime : Fri Sep  2 03:03:28 EDT 2022
% 0.13/0.34  % CPUTime  : 
% 0.13/0.35  Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.35  Usage: tptp [options] [-file:]file
% 0.13/0.35    -h, -?       prints this message.
% 0.13/0.35    -smt2        print SMT-LIB2 benchmark.
% 0.13/0.35    -m, -model   generate model.
% 0.13/0.35    -p, -proof   generate proof.
% 0.13/0.35    -c, -core    generate unsat core of named formulas.
% 0.13/0.35    -st, -statistics display statistics.
% 0.13/0.35    -t:timeout   set timeout (in second).
% 0.13/0.35    -smt2status  display status in smt2 format instead of SZS.
% 0.13/0.35    -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.35    -<param>:<value> configuration parameter and value.
% 0.13/0.35    -o:<output-file> file to place output in.
% 0.13/0.42  % SZS status Unsatisfiable
% 0.13/0.42  % SZS output start Proof
% 0.13/0.42  tff(q_type, type, (
% 0.13/0.42     q: ( $i * $i ) > $o)).
% 0.13/0.42  tff(d_type, type, (
% 0.13/0.42     d: $i)).
% 0.13/0.42  tff(a_type, type, (
% 0.13/0.42     a: $i)).
% 0.13/0.42  tff(p_type, type, (
% 0.13/0.42     p: ( $i * $i ) > $o)).
% 0.13/0.42  tff(b_type, type, (
% 0.13/0.42     b: $i)).
% 0.13/0.42  tff(c_type, type, (
% 0.13/0.42     c: $i)).
% 0.13/0.42  tff(1,assumption,(~q(a, c)), introduced(assumption)).
% 0.13/0.42  tff(2,assumption,(q(b, c)), introduced(assumption)).
% 0.13/0.42  tff(3,plain,
% 0.13/0.42      ((~p(a, b)) <=> (~p(a, b))),
% 0.13/0.42      inference(rewrite,[status(thm)],[])).
% 0.13/0.42  tff(4,axiom,(~p(a, b)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','p_is_not_total')).
% 0.13/0.42  tff(5,plain,
% 0.13/0.42      (~p(a, b)),
% 0.13/0.42      inference(modus_ponens,[status(thm)],[4, 3])).
% 0.13/0.42  tff(6,plain,
% 0.13/0.42      (^[Y: $i, X: $i] : refl((p(X, Y) | q(X, Y)) <=> (p(X, Y) | q(X, Y)))),
% 0.13/0.42      inference(bind,[status(th)],[])).
% 0.13/0.42  tff(7,plain,
% 0.13/0.42      (![Y: $i, X: $i] : (p(X, Y) | q(X, Y)) <=> ![Y: $i, X: $i] : (p(X, Y) | q(X, Y))),
% 0.13/0.42      inference(quant_intro,[status(thm)],[6])).
% 0.13/0.42  tff(8,plain,
% 0.13/0.42      (![Y: $i, X: $i] : (p(X, Y) | q(X, Y)) <=> ![Y: $i, X: $i] : (p(X, Y) | q(X, Y))),
% 0.13/0.42      inference(rewrite,[status(thm)],[])).
% 0.13/0.42  tff(9,axiom,(![Y: $i, X: $i] : (p(X, Y) | q(X, Y))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','all_related')).
% 0.13/0.42  tff(10,plain,
% 0.13/0.42      (![Y: $i, X: $i] : (p(X, Y) | q(X, Y))),
% 0.13/0.42      inference(modus_ponens,[status(thm)],[9, 8])).
% 0.13/0.42  tff(11,plain,(
% 0.13/0.42      ![Y: $i, X: $i] : (p(X, Y) | q(X, Y))),
% 0.13/0.42      inference(skolemize,[status(sab)],[10])).
% 0.13/0.42  tff(12,plain,
% 0.13/0.42      (![Y: $i, X: $i] : (p(X, Y) | q(X, Y))),
% 0.13/0.42      inference(modus_ponens,[status(thm)],[11, 7])).
% 0.13/0.42  tff(13,plain,
% 0.13/0.42      (((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | (p(a, b) | q(a, b))) <=> ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | p(a, b) | q(a, b))),
% 0.13/0.42      inference(rewrite,[status(thm)],[])).
% 0.13/0.42  tff(14,plain,
% 0.13/0.42      ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | (p(a, b) | q(a, b))),
% 0.13/0.42      inference(quant_inst,[status(thm)],[])).
% 0.13/0.42  tff(15,plain,
% 0.13/0.42      ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | p(a, b) | q(a, b)),
% 0.13/0.42      inference(modus_ponens,[status(thm)],[14, 13])).
% 0.13/0.42  tff(16,plain,
% 0.13/0.42      (q(a, b)),
% 0.13/0.42      inference(unit_resolution,[status(thm)],[15, 12, 5])).
% 0.13/0.42  tff(17,plain,
% 0.13/0.42      (![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y))) <=> ![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))),
% 0.13/0.42      inference(rewrite,[status(thm)],[])).
% 0.13/0.42  tff(18,plain,
% 0.13/0.42      (![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y))) <=> ![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))),
% 0.13/0.42      inference(rewrite,[status(thm)],[])).
% 0.13/0.42  tff(19,plain,
% 0.13/0.42      (^[Z: $i, Y: $i, X: $i] : rewrite((((~q(X, Y)) | (~q(Y, Z))) | q(X, Z)) <=> (q(X, Z) | (~q(Y, Z)) | (~q(X, Y))))),
% 0.13/0.42      inference(bind,[status(th)],[])).
% 0.13/0.42  tff(20,plain,
% 0.13/0.42      (![Z: $i, Y: $i, X: $i] : (((~q(X, Y)) | (~q(Y, Z))) | q(X, Z)) <=> ![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))),
% 0.13/0.42      inference(quant_intro,[status(thm)],[19])).
% 0.13/0.42  tff(21,axiom,(![Z: $i, Y: $i, X: $i] : (((~q(X, Y)) | (~q(Y, Z))) | q(X, Z))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','q_transitivity')).
% 0.13/0.42  tff(22,plain,
% 0.13/0.42      (![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))),
% 0.13/0.42      inference(modus_ponens,[status(thm)],[21, 20])).
% 0.13/0.42  tff(23,plain,
% 0.13/0.42      (![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))),
% 0.13/0.42      inference(modus_ponens,[status(thm)],[22, 18])).
% 0.13/0.42  tff(24,plain,(
% 0.13/0.42      ![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))),
% 0.13/0.42      inference(skolemize,[status(sab)],[23])).
% 0.13/0.42  tff(25,plain,
% 0.13/0.42      (![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))),
% 0.13/0.42      inference(modus_ponens,[status(thm)],[24, 17])).
% 0.13/0.42  tff(26,plain,
% 0.13/0.42      (((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | ((~q(a, b)) | q(a, c) | (~q(b, c)))) <=> ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (~q(a, b)) | q(a, c) | (~q(b, c)))),
% 0.13/0.42      inference(rewrite,[status(thm)],[])).
% 0.13/0.42  tff(27,plain,
% 0.13/0.42      ((q(a, c) | (~q(b, c)) | (~q(a, b))) <=> ((~q(a, b)) | q(a, c) | (~q(b, c)))),
% 0.13/0.42      inference(rewrite,[status(thm)],[])).
% 0.13/0.42  tff(28,plain,
% 0.13/0.42      (((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (q(a, c) | (~q(b, c)) | (~q(a, b)))) <=> ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | ((~q(a, b)) | q(a, c) | (~q(b, c))))),
% 0.21/0.42      inference(monotonicity,[status(thm)],[27])).
% 0.21/0.42  tff(29,plain,
% 0.21/0.42      (((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (q(a, c) | (~q(b, c)) | (~q(a, b)))) <=> ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (~q(a, b)) | q(a, c) | (~q(b, c)))),
% 0.21/0.42      inference(transitivity,[status(thm)],[28, 26])).
% 0.21/0.42  tff(30,plain,
% 0.21/0.42      ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (q(a, c) | (~q(b, c)) | (~q(a, b)))),
% 0.21/0.42      inference(quant_inst,[status(thm)],[])).
% 0.21/0.42  tff(31,plain,
% 0.21/0.42      ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (~q(a, b)) | q(a, c) | (~q(b, c))),
% 0.21/0.42      inference(modus_ponens,[status(thm)],[30, 29])).
% 0.21/0.42  tff(32,plain,
% 0.21/0.42      ($false),
% 0.21/0.42      inference(unit_resolution,[status(thm)],[31, 25, 16, 1, 2])).
% 0.21/0.42  tff(33,plain,((~q(b, c)) | q(a, c)), inference(lemma,lemma(discharge,[]))).
% 0.21/0.42  tff(34,plain,
% 0.21/0.42      (~q(b, c)),
% 0.21/0.42      inference(unit_resolution,[status(thm)],[33, 1])).
% 0.21/0.42  tff(35,assumption,(~q(b, c)), introduced(assumption)).
% 0.21/0.42  tff(36,assumption,(q(c, b)), introduced(assumption)).
% 0.21/0.42  tff(37,plain,
% 0.21/0.42      (^[Y: $i, X: $i] : refl(((~q(X, Y)) | q(Y, X)) <=> ((~q(X, Y)) | q(Y, X)))),
% 0.21/0.42      inference(bind,[status(th)],[])).
% 0.21/0.42  tff(38,plain,
% 0.21/0.42      (![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X)) <=> ![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))),
% 0.21/0.42      inference(quant_intro,[status(thm)],[37])).
% 0.21/0.42  tff(39,plain,
% 0.21/0.42      (![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X)) <=> ![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))),
% 0.21/0.42      inference(rewrite,[status(thm)],[])).
% 0.21/0.42  tff(40,axiom,(![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','q_symmetry')).
% 0.21/0.42  tff(41,plain,
% 0.21/0.42      (![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))),
% 0.21/0.42      inference(modus_ponens,[status(thm)],[40, 39])).
% 0.21/0.42  tff(42,plain,(
% 0.21/0.42      ![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))),
% 0.21/0.42      inference(skolemize,[status(sab)],[41])).
% 0.21/0.42  tff(43,plain,
% 0.21/0.42      (![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))),
% 0.21/0.42      inference(modus_ponens,[status(thm)],[42, 38])).
% 0.21/0.42  tff(44,plain,
% 0.21/0.42      (((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | ((~q(c, b)) | q(b, c))) <=> ((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | (~q(c, b)) | q(b, c))),
% 0.21/0.42      inference(rewrite,[status(thm)],[])).
% 0.21/0.42  tff(45,plain,
% 0.21/0.42      ((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | ((~q(c, b)) | q(b, c))),
% 0.21/0.42      inference(quant_inst,[status(thm)],[])).
% 0.21/0.42  tff(46,plain,
% 0.21/0.42      ((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | (~q(c, b)) | q(b, c)),
% 0.21/0.42      inference(modus_ponens,[status(thm)],[45, 44])).
% 0.21/0.42  tff(47,plain,
% 0.21/0.42      ($false),
% 0.21/0.42      inference(unit_resolution,[status(thm)],[46, 43, 36, 35])).
% 0.21/0.42  tff(48,plain,((~q(c, b)) | q(b, c)), inference(lemma,lemma(discharge,[]))).
% 0.21/0.42  tff(49,plain,
% 0.21/0.42      (~q(c, b)),
% 0.21/0.42      inference(unit_resolution,[status(thm)],[48, 34])).
% 0.21/0.42  tff(50,plain,
% 0.21/0.42      (((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | (p(c, b) | q(c, b))) <=> ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | p(c, b) | q(c, b))),
% 0.21/0.42      inference(rewrite,[status(thm)],[])).
% 0.21/0.42  tff(51,plain,
% 0.21/0.42      ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | (p(c, b) | q(c, b))),
% 0.21/0.42      inference(quant_inst,[status(thm)],[])).
% 0.21/0.42  tff(52,plain,
% 0.21/0.42      ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | p(c, b) | q(c, b)),
% 0.21/0.42      inference(modus_ponens,[status(thm)],[51, 50])).
% 0.21/0.42  tff(53,plain,
% 0.21/0.42      (p(c, b) | q(c, b)),
% 0.21/0.42      inference(unit_resolution,[status(thm)],[52, 12])).
% 0.21/0.42  tff(54,plain,
% 0.21/0.42      (p(c, b)),
% 0.21/0.42      inference(unit_resolution,[status(thm)],[53, 49])).
% 0.21/0.42  tff(55,plain,
% 0.21/0.42      (((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | (p(a, c) | q(a, c))) <=> ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | p(a, c) | q(a, c))),
% 0.21/0.42      inference(rewrite,[status(thm)],[])).
% 0.21/0.42  tff(56,plain,
% 0.21/0.42      ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | (p(a, c) | q(a, c))),
% 0.21/0.42      inference(quant_inst,[status(thm)],[])).
% 0.21/0.42  tff(57,plain,
% 0.21/0.42      ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | p(a, c) | q(a, c)),
% 0.21/0.42      inference(modus_ponens,[status(thm)],[56, 55])).
% 0.21/0.42  tff(58,plain,
% 0.21/0.42      (p(a, c) | q(a, c)),
% 0.21/0.42      inference(unit_resolution,[status(thm)],[57, 12])).
% 0.21/0.42  tff(59,plain,
% 0.21/0.42      (p(a, c)),
% 0.21/0.42      inference(unit_resolution,[status(thm)],[58, 1])).
% 0.21/0.42  tff(60,plain,
% 0.21/0.42      ((~q(c, d)) <=> (~q(c, d))),
% 0.21/0.42      inference(rewrite,[status(thm)],[])).
% 0.21/0.43  tff(61,axiom,(~q(c, d)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_q_is_total')).
% 0.21/0.43  tff(62,plain,
% 0.21/0.43      (~q(c, d)),
% 0.21/0.43      inference(modus_ponens,[status(thm)],[61, 60])).
% 0.21/0.43  tff(63,plain,
% 0.21/0.43      (((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | (p(c, d) | q(c, d))) <=> ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | p(c, d) | q(c, d))),
% 0.21/0.43      inference(rewrite,[status(thm)],[])).
% 0.21/0.43  tff(64,plain,
% 0.21/0.43      ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | (p(c, d) | q(c, d))),
% 0.21/0.43      inference(quant_inst,[status(thm)],[])).
% 0.21/0.43  tff(65,plain,
% 0.21/0.43      ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | p(c, d) | q(c, d)),
% 0.21/0.43      inference(modus_ponens,[status(thm)],[64, 63])).
% 0.21/0.43  tff(66,plain,
% 0.21/0.43      (p(c, d)),
% 0.21/0.43      inference(unit_resolution,[status(thm)],[65, 12, 62])).
% 0.21/0.43  tff(67,plain,
% 0.21/0.43      (![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y))) <=> ![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))),
% 0.21/0.43      inference(rewrite,[status(thm)],[])).
% 0.21/0.43  tff(68,plain,
% 0.21/0.43      (![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y))) <=> ![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))),
% 0.21/0.43      inference(rewrite,[status(thm)],[])).
% 0.21/0.43  tff(69,plain,
% 0.21/0.43      (^[Z: $i, Y: $i, X: $i] : rewrite((((~p(X, Y)) | (~p(Y, Z))) | p(X, Z)) <=> (p(X, Z) | (~p(Y, Z)) | (~p(X, Y))))),
% 0.21/0.43      inference(bind,[status(th)],[])).
% 0.21/0.43  tff(70,plain,
% 0.21/0.43      (![Z: $i, Y: $i, X: $i] : (((~p(X, Y)) | (~p(Y, Z))) | p(X, Z)) <=> ![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))),
% 0.21/0.43      inference(quant_intro,[status(thm)],[69])).
% 0.21/0.43  tff(71,axiom,(![Z: $i, Y: $i, X: $i] : (((~p(X, Y)) | (~p(Y, Z))) | p(X, Z))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','p_transitivity')).
% 0.21/0.43  tff(72,plain,
% 0.21/0.43      (![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))),
% 0.21/0.43      inference(modus_ponens,[status(thm)],[71, 70])).
% 0.21/0.43  tff(73,plain,
% 0.21/0.43      (![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))),
% 0.21/0.43      inference(modus_ponens,[status(thm)],[72, 68])).
% 0.21/0.43  tff(74,plain,(
% 0.21/0.43      ![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))),
% 0.21/0.43      inference(skolemize,[status(sab)],[73])).
% 0.21/0.43  tff(75,plain,
% 0.21/0.43      (![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))),
% 0.21/0.43      inference(modus_ponens,[status(thm)],[74, 67])).
% 0.21/0.43  tff(76,plain,
% 0.21/0.43      (((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | ((~p(c, d)) | p(a, d) | (~p(a, c)))) <=> ((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | (~p(c, d)) | p(a, d) | (~p(a, c)))),
% 0.21/0.43      inference(rewrite,[status(thm)],[])).
% 0.21/0.43  tff(77,plain,
% 0.21/0.43      ((p(a, d) | (~p(c, d)) | (~p(a, c))) <=> ((~p(c, d)) | p(a, d) | (~p(a, c)))),
% 0.21/0.43      inference(rewrite,[status(thm)],[])).
% 0.21/0.43  tff(78,plain,
% 0.21/0.43      (((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | (p(a, d) | (~p(c, d)) | (~p(a, c)))) <=> ((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | ((~p(c, d)) | p(a, d) | (~p(a, c))))),
% 0.21/0.43      inference(monotonicity,[status(thm)],[77])).
% 0.21/0.43  tff(79,plain,
% 0.21/0.43      (((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | (p(a, d) | (~p(c, d)) | (~p(a, c)))) <=> ((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | (~p(c, d)) | p(a, d) | (~p(a, c)))),
% 0.21/0.43      inference(transitivity,[status(thm)],[78, 76])).
% 0.21/0.43  tff(80,plain,
% 0.21/0.43      ((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | (p(a, d) | (~p(c, d)) | (~p(a, c)))),
% 0.21/0.43      inference(quant_inst,[status(thm)],[])).
% 0.21/0.43  tff(81,plain,
% 0.21/0.43      ((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | (~p(c, d)) | p(a, d) | (~p(a, c))),
% 0.21/0.43      inference(modus_ponens,[status(thm)],[80, 79])).
% 0.21/0.43  tff(82,plain,
% 0.21/0.43      (p(a, d)),
% 0.21/0.43      inference(unit_resolution,[status(thm)],[81, 75, 66, 59])).
% 0.21/0.43  tff(83,assumption,(p(a, d)), introduced(assumption)).
% 0.21/0.43  tff(84,assumption,(p(d, b)), introduced(assumption)).
% 0.21/0.43  tff(85,plain,
% 0.21/0.43      (((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | (p(a, b) | (~p(d, b)) | (~p(a, d)))) <=> ((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | p(a, b) | (~p(d, b)) | (~p(a, d)))),
% 0.21/0.43      inference(rewrite,[status(thm)],[])).
% 0.21/0.43  tff(86,plain,
% 0.21/0.43      ((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | (p(a, b) | (~p(d, b)) | (~p(a, d)))),
% 0.21/0.43      inference(quant_inst,[status(thm)],[])).
% 0.21/0.43  tff(87,plain,
% 0.21/0.43      ((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | p(a, b) | (~p(d, b)) | (~p(a, d))),
% 0.21/0.43      inference(modus_ponens,[status(thm)],[86, 85])).
% 0.21/0.43  tff(88,plain,
% 0.21/0.43      ($false),
% 0.21/0.43      inference(unit_resolution,[status(thm)],[87, 75, 5, 84, 83])).
% 0.21/0.43  tff(89,plain,((~p(d, b)) | (~p(a, d))), inference(lemma,lemma(discharge,[]))).
% 0.21/0.43  tff(90,plain,
% 0.21/0.43      (~p(d, b)),
% 0.21/0.43      inference(unit_resolution,[status(thm)],[89, 82])).
% 0.21/0.43  tff(91,plain,
% 0.21/0.43      (((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | ((~q(d, c)) | q(c, d))) <=> ((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | (~q(d, c)) | q(c, d))),
% 0.21/0.43      inference(rewrite,[status(thm)],[])).
% 0.21/0.43  tff(92,plain,
% 0.21/0.43      ((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | ((~q(d, c)) | q(c, d))),
% 0.21/0.43      inference(quant_inst,[status(thm)],[])).
% 0.21/0.43  tff(93,plain,
% 0.21/0.43      ((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | (~q(d, c)) | q(c, d)),
% 0.21/0.43      inference(modus_ponens,[status(thm)],[92, 91])).
% 0.21/0.43  tff(94,plain,
% 0.21/0.43      (~q(d, c)),
% 0.21/0.43      inference(unit_resolution,[status(thm)],[93, 43, 62])).
% 0.21/0.43  tff(95,plain,
% 0.21/0.43      (((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | (p(d, c) | q(d, c))) <=> ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | p(d, c) | q(d, c))),
% 0.21/0.43      inference(rewrite,[status(thm)],[])).
% 0.21/0.43  tff(96,plain,
% 0.21/0.43      ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | (p(d, c) | q(d, c))),
% 0.21/0.43      inference(quant_inst,[status(thm)],[])).
% 0.21/0.43  tff(97,plain,
% 0.21/0.43      ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | p(d, c) | q(d, c)),
% 0.21/0.43      inference(modus_ponens,[status(thm)],[96, 95])).
% 0.21/0.43  tff(98,plain,
% 0.21/0.43      (p(d, c)),
% 0.21/0.43      inference(unit_resolution,[status(thm)],[97, 12, 94])).
% 0.21/0.43  tff(99,plain,
% 0.21/0.43      (((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | ((~p(d, c)) | p(d, b) | (~p(c, b)))) <=> ((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | (~p(d, c)) | p(d, b) | (~p(c, b)))),
% 0.21/0.43      inference(rewrite,[status(thm)],[])).
% 0.21/0.43  tff(100,plain,
% 0.21/0.43      ((p(d, b) | (~p(c, b)) | (~p(d, c))) <=> ((~p(d, c)) | p(d, b) | (~p(c, b)))),
% 0.21/0.43      inference(rewrite,[status(thm)],[])).
% 0.21/0.43  tff(101,plain,
% 0.21/0.43      (((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | (p(d, b) | (~p(c, b)) | (~p(d, c)))) <=> ((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | ((~p(d, c)) | p(d, b) | (~p(c, b))))),
% 0.21/0.43      inference(monotonicity,[status(thm)],[100])).
% 0.21/0.43  tff(102,plain,
% 0.21/0.43      (((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | (p(d, b) | (~p(c, b)) | (~p(d, c)))) <=> ((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | (~p(d, c)) | p(d, b) | (~p(c, b)))),
% 0.21/0.43      inference(transitivity,[status(thm)],[101, 99])).
% 0.21/0.43  tff(103,plain,
% 0.21/0.43      ((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | (p(d, b) | (~p(c, b)) | (~p(d, c)))),
% 0.21/0.43      inference(quant_inst,[status(thm)],[])).
% 0.21/0.43  tff(104,plain,
% 0.21/0.43      ((~![Z: $i, Y: $i, X: $i] : (p(X, Z) | (~p(Y, Z)) | (~p(X, Y)))) | (~p(d, c)) | p(d, b) | (~p(c, b))),
% 0.21/0.43      inference(modus_ponens,[status(thm)],[103, 102])).
% 0.21/0.43  tff(105,plain,
% 0.21/0.43      ($false),
% 0.21/0.43      inference(unit_resolution,[status(thm)],[104, 75, 98, 90, 54])).
% 0.21/0.43  tff(106,plain,(q(a, c)), inference(lemma,lemma(discharge,[]))).
% 0.21/0.43  tff(107,plain,
% 0.21/0.43      (((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | ((~q(a, b)) | q(b, a))) <=> ((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | (~q(a, b)) | q(b, a))),
% 0.21/0.43      inference(rewrite,[status(thm)],[])).
% 0.21/0.43  tff(108,plain,
% 0.21/0.43      ((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | ((~q(a, b)) | q(b, a))),
% 0.21/0.43      inference(quant_inst,[status(thm)],[])).
% 0.21/0.43  tff(109,plain,
% 0.21/0.43      ((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | (~q(a, b)) | q(b, a)),
% 0.21/0.43      inference(modus_ponens,[status(thm)],[108, 107])).
% 0.21/0.43  tff(110,plain,
% 0.21/0.43      (q(b, a)),
% 0.21/0.43      inference(unit_resolution,[status(thm)],[109, 43, 16])).
% 0.21/0.43  tff(111,plain,
% 0.21/0.43      (((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | ((~q(b, a)) | q(b, c) | (~q(a, c)))) <=> ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (~q(b, a)) | q(b, c) | (~q(a, c)))),
% 0.21/0.43      inference(rewrite,[status(thm)],[])).
% 0.21/0.43  tff(112,plain,
% 0.21/0.43      ((q(b, c) | (~q(a, c)) | (~q(b, a))) <=> ((~q(b, a)) | q(b, c) | (~q(a, c)))),
% 0.21/0.43      inference(rewrite,[status(thm)],[])).
% 0.21/0.43  tff(113,plain,
% 0.21/0.43      (((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (q(b, c) | (~q(a, c)) | (~q(b, a)))) <=> ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | ((~q(b, a)) | q(b, c) | (~q(a, c))))),
% 0.21/0.44      inference(monotonicity,[status(thm)],[112])).
% 0.21/0.44  tff(114,plain,
% 0.21/0.44      (((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (q(b, c) | (~q(a, c)) | (~q(b, a)))) <=> ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (~q(b, a)) | q(b, c) | (~q(a, c)))),
% 0.21/0.44      inference(transitivity,[status(thm)],[113, 111])).
% 0.21/0.44  tff(115,plain,
% 0.21/0.44      ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (q(b, c) | (~q(a, c)) | (~q(b, a)))),
% 0.21/0.44      inference(quant_inst,[status(thm)],[])).
% 0.21/0.44  tff(116,plain,
% 0.21/0.44      ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (~q(b, a)) | q(b, c) | (~q(a, c))),
% 0.21/0.44      inference(modus_ponens,[status(thm)],[115, 114])).
% 0.21/0.44  tff(117,plain,
% 0.21/0.44      (q(b, c)),
% 0.21/0.44      inference(unit_resolution,[status(thm)],[116, 25, 110, 106])).
% 0.21/0.44  tff(118,assumption,(q(d, b)), introduced(assumption)).
% 0.21/0.44  tff(119,plain,
% 0.21/0.44      (((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (q(d, c) | (~q(d, b)) | (~q(b, c)))) <=> ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | q(d, c) | (~q(d, b)) | (~q(b, c)))),
% 0.21/0.44      inference(rewrite,[status(thm)],[])).
% 0.21/0.44  tff(120,plain,
% 0.21/0.44      ((q(d, c) | (~q(b, c)) | (~q(d, b))) <=> (q(d, c) | (~q(d, b)) | (~q(b, c)))),
% 0.21/0.44      inference(rewrite,[status(thm)],[])).
% 0.21/0.44  tff(121,plain,
% 0.21/0.44      (((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (q(d, c) | (~q(b, c)) | (~q(d, b)))) <=> ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (q(d, c) | (~q(d, b)) | (~q(b, c))))),
% 0.21/0.44      inference(monotonicity,[status(thm)],[120])).
% 0.21/0.44  tff(122,plain,
% 0.21/0.44      (((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (q(d, c) | (~q(b, c)) | (~q(d, b)))) <=> ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | q(d, c) | (~q(d, b)) | (~q(b, c)))),
% 0.21/0.44      inference(transitivity,[status(thm)],[121, 119])).
% 0.21/0.44  tff(123,plain,
% 0.21/0.44      ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (q(d, c) | (~q(b, c)) | (~q(d, b)))),
% 0.21/0.44      inference(quant_inst,[status(thm)],[])).
% 0.21/0.44  tff(124,plain,
% 0.21/0.44      ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | q(d, c) | (~q(d, b)) | (~q(b, c))),
% 0.21/0.44      inference(modus_ponens,[status(thm)],[123, 122])).
% 0.21/0.44  tff(125,plain,
% 0.21/0.44      ($false),
% 0.21/0.44      inference(unit_resolution,[status(thm)],[124, 25, 94, 118, 117])).
% 0.21/0.44  tff(126,plain,(~q(d, b)), inference(lemma,lemma(discharge,[]))).
% 0.21/0.44  tff(127,plain,
% 0.21/0.44      (((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | (p(d, b) | q(d, b))) <=> ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | p(d, b) | q(d, b))),
% 0.21/0.44      inference(rewrite,[status(thm)],[])).
% 0.21/0.44  tff(128,plain,
% 0.21/0.44      ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | (p(d, b) | q(d, b))),
% 0.21/0.44      inference(quant_inst,[status(thm)],[])).
% 0.21/0.44  tff(129,plain,
% 0.21/0.44      ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | p(d, b) | q(d, b)),
% 0.21/0.44      inference(modus_ponens,[status(thm)],[128, 127])).
% 0.21/0.44  tff(130,plain,
% 0.21/0.44      (p(d, b) | q(d, b)),
% 0.21/0.44      inference(unit_resolution,[status(thm)],[129, 12])).
% 0.21/0.44  tff(131,plain,
% 0.21/0.44      (p(d, b)),
% 0.21/0.44      inference(unit_resolution,[status(thm)],[130, 126])).
% 0.21/0.44  tff(132,plain,
% 0.21/0.44      (~p(a, d)),
% 0.21/0.44      inference(unit_resolution,[status(thm)],[89, 131])).
% 0.21/0.44  tff(133,plain,
% 0.21/0.44      (((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | (p(a, d) | q(a, d))) <=> ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | p(a, d) | q(a, d))),
% 0.21/0.44      inference(rewrite,[status(thm)],[])).
% 0.21/0.44  tff(134,plain,
% 0.21/0.44      ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | (p(a, d) | q(a, d))),
% 0.21/0.44      inference(quant_inst,[status(thm)],[])).
% 0.21/0.44  tff(135,plain,
% 0.21/0.44      ((~![Y: $i, X: $i] : (p(X, Y) | q(X, Y))) | p(a, d) | q(a, d)),
% 0.21/0.44      inference(modus_ponens,[status(thm)],[134, 133])).
% 0.21/0.44  tff(136,plain,
% 0.21/0.44      (q(a, d)),
% 0.21/0.44      inference(unit_resolution,[status(thm)],[135, 12, 132])).
% 0.21/0.44  tff(137,plain,
% 0.21/0.44      (((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | ((~q(b, d)) | q(d, b))) <=> ((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | (~q(b, d)) | q(d, b))),
% 0.21/0.44      inference(rewrite,[status(thm)],[])).
% 0.21/0.44  tff(138,plain,
% 0.21/0.44      ((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | ((~q(b, d)) | q(d, b))),
% 0.21/0.44      inference(quant_inst,[status(thm)],[])).
% 0.21/0.44  tff(139,plain,
% 0.21/0.44      ((~![Y: $i, X: $i] : ((~q(X, Y)) | q(Y, X))) | (~q(b, d)) | q(d, b)),
% 0.21/0.44      inference(modus_ponens,[status(thm)],[138, 137])).
% 0.21/0.44  tff(140,plain,
% 0.21/0.44      (~q(b, d)),
% 0.21/0.44      inference(unit_resolution,[status(thm)],[139, 43, 126])).
% 0.21/0.44  tff(141,plain,
% 0.21/0.44      (((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | ((~q(b, a)) | (~q(a, d)) | q(b, d))) <=> ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (~q(b, a)) | (~q(a, d)) | q(b, d))),
% 0.21/0.44      inference(rewrite,[status(thm)],[])).
% 0.21/0.44  tff(142,plain,
% 0.21/0.44      ((q(b, d) | (~q(a, d)) | (~q(b, a))) <=> ((~q(b, a)) | (~q(a, d)) | q(b, d))),
% 0.21/0.44      inference(rewrite,[status(thm)],[])).
% 0.21/0.44  tff(143,plain,
% 0.21/0.44      (((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (q(b, d) | (~q(a, d)) | (~q(b, a)))) <=> ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | ((~q(b, a)) | (~q(a, d)) | q(b, d)))),
% 0.21/0.44      inference(monotonicity,[status(thm)],[142])).
% 0.21/0.44  tff(144,plain,
% 0.21/0.44      (((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (q(b, d) | (~q(a, d)) | (~q(b, a)))) <=> ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (~q(b, a)) | (~q(a, d)) | q(b, d))),
% 0.21/0.44      inference(transitivity,[status(thm)],[143, 141])).
% 0.21/0.44  tff(145,plain,
% 0.21/0.44      ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (q(b, d) | (~q(a, d)) | (~q(b, a)))),
% 0.21/0.44      inference(quant_inst,[status(thm)],[])).
% 0.21/0.44  tff(146,plain,
% 0.21/0.44      ((~![Z: $i, Y: $i, X: $i] : (q(X, Z) | (~q(Y, Z)) | (~q(X, Y)))) | (~q(b, a)) | (~q(a, d)) | q(b, d)),
% 0.21/0.44      inference(modus_ponens,[status(thm)],[145, 144])).
% 0.21/0.44  tff(147,plain,
% 0.21/0.44      ($false),
% 0.21/0.44      inference(unit_resolution,[status(thm)],[146, 25, 110, 140, 136])).
% 0.21/0.44  % SZS output end Proof
%------------------------------------------------------------------------------