%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET047-5 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% 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 : Sat Jun 21 05:24:59 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.37s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SET047-5 : TPTP v9.0.0. Released v1.0.0.
% 0.00/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n004.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 04:08:38 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.37 % SZS status Unsatisfiable
% 0.19/0.37 % SZS output start Proof
% 0.19/0.37 tff(element_type, type, (
% 0.19/0.37 element: ( $i * $i ) > $o)).
% 0.19/0.37 tff(a_type, type, (
% 0.19/0.37 a: $i)).
% 0.19/0.37 tff(f_type, type, (
% 0.19/0.37 f: ( $i * $i ) > $i)).
% 0.19/0.37 tff(b_type, type, (
% 0.19/0.37 b: $i)).
% 0.19/0.37 tff(set_equal_type, type, (
% 0.19/0.37 set_equal: ( $i * $i ) > $o)).
% 0.19/0.37 tff(1,assumption,(~element(f(b, a), a)), introduced(assumption)).
% 0.19/0.37 tff(2,assumption,(~set_equal(a, b)), introduced(assumption)).
% 0.19/0.37 tff(3,assumption,(~element(f(a, b), a)), introduced(assumption)).
% 0.19/0.37 tff(4,plain,
% 0.19/0.37 (^[Y: $i, X: $i] : refl((element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X)) <=> (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X)))),
% 0.19/0.37 inference(bind,[status(th)],[])).
% 0.19/0.37 tff(5,plain,
% 0.19/0.37 (![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X)) <=> ![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))),
% 0.19/0.37 inference(quant_intro,[status(thm)],[4])).
% 0.19/0.37 tff(6,plain,
% 0.19/0.37 (![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X)) <=> ![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))),
% 0.19/0.37 inference(rewrite,[status(thm)],[])).
% 0.19/0.37 tff(7,plain,
% 0.19/0.37 (^[Y: $i, X: $i] : rewrite(((element(f(X, Y), X) | element(f(X, Y), Y)) | set_equal(X, Y)) <=> (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X)))),
% 0.19/0.37 inference(bind,[status(th)],[])).
% 0.19/0.37 tff(8,plain,
% 0.19/0.37 (![Y: $i, X: $i] : ((element(f(X, Y), X) | element(f(X, Y), Y)) | set_equal(X, Y)) <=> ![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))),
% 0.19/0.37 inference(quant_intro,[status(thm)],[7])).
% 0.19/0.37 tff(9,axiom,(![Y: $i, X: $i] : ((element(f(X, Y), X) | element(f(X, Y), Y)) | set_equal(X, Y))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','clause_3')).
% 0.19/0.37 tff(10,plain,
% 0.19/0.37 (![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[9, 8])).
% 0.19/0.37 tff(11,plain,
% 0.19/0.37 (![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[10, 6])).
% 0.19/0.37 tff(12,plain,(
% 0.19/0.37 ![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))),
% 0.19/0.37 inference(skolemize,[status(sab)],[11])).
% 0.19/0.37 tff(13,plain,
% 0.19/0.37 (![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[12, 5])).
% 0.19/0.37 tff(14,plain,
% 0.19/0.37 (((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | (set_equal(a, b) | element(f(a, b), a) | element(f(a, b), b))) <=> ((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | set_equal(a, b) | element(f(a, b), a) | element(f(a, b), b))),
% 0.19/0.37 inference(rewrite,[status(thm)],[])).
% 0.19/0.37 tff(15,plain,
% 0.19/0.37 ((element(f(a, b), b) | set_equal(a, b) | element(f(a, b), a)) <=> (set_equal(a, b) | element(f(a, b), a) | element(f(a, b), b))),
% 0.19/0.37 inference(rewrite,[status(thm)],[])).
% 0.19/0.37 tff(16,plain,
% 0.19/0.37 (((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | (element(f(a, b), b) | set_equal(a, b) | element(f(a, b), a))) <=> ((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | (set_equal(a, b) | element(f(a, b), a) | element(f(a, b), b)))),
% 0.19/0.37 inference(monotonicity,[status(thm)],[15])).
% 0.19/0.37 tff(17,plain,
% 0.19/0.37 (((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | (element(f(a, b), b) | set_equal(a, b) | element(f(a, b), a))) <=> ((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | set_equal(a, b) | element(f(a, b), a) | element(f(a, b), b))),
% 0.19/0.37 inference(transitivity,[status(thm)],[16, 14])).
% 0.19/0.37 tff(18,plain,
% 0.19/0.37 ((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | (element(f(a, b), b) | set_equal(a, b) | element(f(a, b), a))),
% 0.19/0.37 inference(quant_inst,[status(thm)],[])).
% 0.19/0.37 tff(19,plain,
% 0.19/0.37 ((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | set_equal(a, b) | element(f(a, b), a) | element(f(a, b), b)),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.19/0.37 tff(20,plain,
% 0.19/0.37 (element(f(a, b), a) | element(f(a, b), b)),
% 0.19/0.37 inference(unit_resolution,[status(thm)],[19, 13, 2])).
% 0.19/0.37 tff(21,plain,
% 0.19/0.37 (element(f(a, b), b)),
% 0.19/0.37 inference(unit_resolution,[status(thm)],[20, 3])).
% 0.19/0.37 tff(22,plain,
% 0.19/0.37 ((set_equal(a, b) | set_equal(b, a)) <=> (set_equal(a, b) | set_equal(b, a))),
% 0.19/0.37 inference(rewrite,[status(thm)],[])).
% 0.19/0.37 tff(23,axiom,(set_equal(a, b) | set_equal(b, a)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_symmetry1')).
% 0.19/0.37 tff(24,plain,
% 0.19/0.37 (set_equal(a, b) | set_equal(b, a)),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.19/0.37 tff(25,plain,
% 0.19/0.37 (set_equal(b, a)),
% 0.19/0.37 inference(unit_resolution,[status(thm)],[24, 2])).
% 0.19/0.37 tff(26,plain,
% 0.19/0.37 (^[Z: $i, Y: $i, X: $i] : refl((element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y))) <=> (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y))))),
% 0.19/0.37 inference(bind,[status(th)],[])).
% 0.19/0.37 tff(27,plain,
% 0.19/0.37 (![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y))) <=> ![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))),
% 0.19/0.37 inference(quant_intro,[status(thm)],[26])).
% 0.19/0.37 tff(28,plain,
% 0.19/0.37 (![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y))) <=> ![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))),
% 0.19/0.37 inference(rewrite,[status(thm)],[])).
% 0.19/0.37 tff(29,plain,
% 0.19/0.37 (^[Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~set_equal(X, Y)) | (~element(Z, X))) <=> ((~element(Z, X)) | (~set_equal(X, Y)))), ((((~set_equal(X, Y)) | (~element(Z, X))) | element(Z, Y)) <=> (((~element(Z, X)) | (~set_equal(X, Y))) | element(Z, Y)))), rewrite((((~element(Z, X)) | (~set_equal(X, Y))) | element(Z, Y)) <=> (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))), ((((~set_equal(X, Y)) | (~element(Z, X))) | element(Z, Y)) <=> (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))))),
% 0.19/0.37 inference(bind,[status(th)],[])).
% 0.19/0.37 tff(30,plain,
% 0.19/0.37 (![Z: $i, Y: $i, X: $i] : (((~set_equal(X, Y)) | (~element(Z, X))) | element(Z, Y)) <=> ![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[29])).
% 0.19/0.38 tff(31,axiom,(![Z: $i, Y: $i, X: $i] : (((~set_equal(X, Y)) | (~element(Z, X))) | element(Z, Y))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_substitution1')).
% 0.19/0.38 tff(32,plain,
% 0.19/0.38 (![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[31, 30])).
% 0.19/0.38 tff(33,plain,
% 0.19/0.38 (![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[32, 28])).
% 0.19/0.38 tff(34,plain,(
% 0.19/0.38 ![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[33])).
% 0.19/0.38 tff(35,plain,
% 0.19/0.38 (![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[34, 27])).
% 0.19/0.38 tff(36,plain,
% 0.19/0.38 (((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | ((~set_equal(b, a)) | element(f(a, b), a) | (~element(f(a, b), b)))) <=> ((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | (~set_equal(b, a)) | element(f(a, b), a) | (~element(f(a, b), b)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(37,plain,
% 0.19/0.38 ((element(f(a, b), a) | (~element(f(a, b), b)) | (~set_equal(b, a))) <=> ((~set_equal(b, a)) | element(f(a, b), a) | (~element(f(a, b), b)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(38,plain,
% 0.19/0.38 (((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | (element(f(a, b), a) | (~element(f(a, b), b)) | (~set_equal(b, a)))) <=> ((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | ((~set_equal(b, a)) | element(f(a, b), a) | (~element(f(a, b), b))))),
% 0.19/0.38 inference(monotonicity,[status(thm)],[37])).
% 0.19/0.38 tff(39,plain,
% 0.19/0.38 (((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | (element(f(a, b), a) | (~element(f(a, b), b)) | (~set_equal(b, a)))) <=> ((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | (~set_equal(b, a)) | element(f(a, b), a) | (~element(f(a, b), b)))),
% 0.19/0.38 inference(transitivity,[status(thm)],[38, 36])).
% 0.19/0.38 tff(40,plain,
% 0.19/0.38 ((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | (element(f(a, b), a) | (~element(f(a, b), b)) | (~set_equal(b, a)))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(41,plain,
% 0.19/0.38 ((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | (~set_equal(b, a)) | element(f(a, b), a) | (~element(f(a, b), b))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.19/0.38 tff(42,plain,
% 0.19/0.38 ($false),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[41, 35, 25, 3, 21])).
% 0.19/0.38 tff(43,plain,(element(f(a, b), a) | set_equal(a, b)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.38 tff(44,plain,
% 0.19/0.38 (element(f(a, b), a)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[43, 2])).
% 0.19/0.38 tff(45,plain,
% 0.19/0.38 (^[Y: $i, X: $i] : refl((set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y))) <=> (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(46,plain,
% 0.19/0.38 (![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y))) <=> ![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[45])).
% 0.19/0.38 tff(47,plain,
% 0.19/0.38 (![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y))) <=> ![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(48,plain,
% 0.19/0.38 (^[Y: $i, X: $i] : trans(monotonicity(rewrite(((~element(f(X, Y), Y)) | (~element(f(X, Y), X))) <=> ((~element(f(X, Y), X)) | (~element(f(X, Y), Y)))), ((((~element(f(X, Y), Y)) | (~element(f(X, Y), X))) | set_equal(X, Y)) <=> (((~element(f(X, Y), X)) | (~element(f(X, Y), Y))) | set_equal(X, Y)))), rewrite((((~element(f(X, Y), X)) | (~element(f(X, Y), Y))) | set_equal(X, Y)) <=> (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))), ((((~element(f(X, Y), Y)) | (~element(f(X, Y), X))) | set_equal(X, Y)) <=> (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(49,plain,
% 0.19/0.38 (![Y: $i, X: $i] : (((~element(f(X, Y), Y)) | (~element(f(X, Y), X))) | set_equal(X, Y)) <=> ![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[48])).
% 0.19/0.38 tff(50,axiom,(![Y: $i, X: $i] : (((~element(f(X, Y), Y)) | (~element(f(X, Y), X))) | set_equal(X, Y))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','clause_4')).
% 0.19/0.38 tff(51,plain,
% 0.19/0.38 (![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.19/0.38 tff(52,plain,
% 0.19/0.38 (![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[51, 47])).
% 0.19/0.38 tff(53,plain,(
% 0.19/0.38 ![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[52])).
% 0.19/0.38 tff(54,plain,
% 0.19/0.38 (![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[53, 46])).
% 0.19/0.38 tff(55,plain,
% 0.19/0.38 (((~![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))) | (set_equal(a, b) | (~element(f(a, b), a)) | (~element(f(a, b), b)))) <=> ((~![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))) | set_equal(a, b) | (~element(f(a, b), a)) | (~element(f(a, b), b)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(56,plain,
% 0.19/0.38 ((~![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))) | (set_equal(a, b) | (~element(f(a, b), a)) | (~element(f(a, b), b)))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(57,plain,
% 0.19/0.38 ((~![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))) | set_equal(a, b) | (~element(f(a, b), a)) | (~element(f(a, b), b))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.19/0.38 tff(58,plain,
% 0.19/0.38 ((~element(f(a, b), a)) | (~element(f(a, b), b))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[57, 54, 2])).
% 0.19/0.38 tff(59,plain,
% 0.19/0.38 (~element(f(a, b), b)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[58, 44])).
% 0.19/0.38 tff(60,plain,
% 0.19/0.38 (^[Z: $i, Y: $i, X: $i] : refl((element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y))) <=> (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(61,plain,
% 0.19/0.38 (![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y))) <=> ![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[60])).
% 0.19/0.38 tff(62,plain,
% 0.19/0.38 (![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y))) <=> ![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(63,plain,
% 0.19/0.38 (^[Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~set_equal(X, Y)) | (~element(Z, Y))) <=> ((~set_equal(X, Y)) | (~element(Z, Y)))), ((((~set_equal(X, Y)) | (~element(Z, Y))) | element(Z, X)) <=> (((~set_equal(X, Y)) | (~element(Z, Y))) | element(Z, X)))), rewrite((((~set_equal(X, Y)) | (~element(Z, Y))) | element(Z, X)) <=> (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))), ((((~set_equal(X, Y)) | (~element(Z, Y))) | element(Z, X)) <=> (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(64,plain,
% 0.19/0.38 (![Z: $i, Y: $i, X: $i] : (((~set_equal(X, Y)) | (~element(Z, Y))) | element(Z, X)) <=> ![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[63])).
% 0.19/0.38 tff(65,axiom,(![Z: $i, Y: $i, X: $i] : (((~set_equal(X, Y)) | (~element(Z, Y))) | element(Z, X))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_substitution2')).
% 0.19/0.38 tff(66,plain,
% 0.19/0.38 (![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.19/0.38 tff(67,plain,
% 0.19/0.38 (![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[66, 62])).
% 0.19/0.38 tff(68,plain,(
% 0.19/0.38 ![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[67])).
% 0.19/0.38 tff(69,plain,
% 0.19/0.38 (![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[68, 61])).
% 0.19/0.38 tff(70,plain,
% 0.19/0.38 (((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | ((~set_equal(b, a)) | (~element(f(a, b), a)) | element(f(a, b), b))) <=> ((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | (~set_equal(b, a)) | (~element(f(a, b), a)) | element(f(a, b), b))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(71,plain,
% 0.19/0.38 ((element(f(a, b), b) | (~set_equal(b, a)) | (~element(f(a, b), a))) <=> ((~set_equal(b, a)) | (~element(f(a, b), a)) | element(f(a, b), b))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(72,plain,
% 0.19/0.38 (((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | (element(f(a, b), b) | (~set_equal(b, a)) | (~element(f(a, b), a)))) <=> ((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | ((~set_equal(b, a)) | (~element(f(a, b), a)) | element(f(a, b), b)))),
% 0.19/0.38 inference(monotonicity,[status(thm)],[71])).
% 0.19/0.38 tff(73,plain,
% 0.19/0.38 (((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | (element(f(a, b), b) | (~set_equal(b, a)) | (~element(f(a, b), a)))) <=> ((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | (~set_equal(b, a)) | (~element(f(a, b), a)) | element(f(a, b), b))),
% 0.19/0.38 inference(transitivity,[status(thm)],[72, 70])).
% 0.19/0.38 tff(74,plain,
% 0.19/0.38 ((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | (element(f(a, b), b) | (~set_equal(b, a)) | (~element(f(a, b), a)))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(75,plain,
% 0.19/0.38 ((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | (~set_equal(b, a)) | (~element(f(a, b), a)) | element(f(a, b), b)),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[74, 73])).
% 0.19/0.38 tff(76,plain,
% 0.19/0.38 ($false),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[75, 69, 25, 44, 59])).
% 0.19/0.38 tff(77,plain,(set_equal(a, b)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.38 tff(78,plain,
% 0.19/0.38 (((~set_equal(a, b)) | (~set_equal(b, a))) <=> ((~set_equal(a, b)) | (~set_equal(b, a)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(79,plain,
% 0.19/0.38 (((~set_equal(b, a)) | (~set_equal(a, b))) <=> ((~set_equal(a, b)) | (~set_equal(b, a)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(80,axiom,((~set_equal(b, a)) | (~set_equal(a, b))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_symmetry2')).
% 0.19/0.38 tff(81,plain,
% 0.19/0.38 ((~set_equal(a, b)) | (~set_equal(b, a))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[80, 79])).
% 0.19/0.38 tff(82,plain,
% 0.19/0.38 ((~set_equal(a, b)) | (~set_equal(b, a))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[81, 78])).
% 0.19/0.38 tff(83,plain,
% 0.19/0.38 (~set_equal(b, a)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[82, 77])).
% 0.19/0.38 tff(84,plain,
% 0.19/0.38 (((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | (set_equal(b, a) | element(f(b, a), b) | element(f(b, a), a))) <=> ((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | set_equal(b, a) | element(f(b, a), b) | element(f(b, a), a))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(85,plain,
% 0.19/0.38 ((element(f(b, a), a) | set_equal(b, a) | element(f(b, a), b)) <=> (set_equal(b, a) | element(f(b, a), b) | element(f(b, a), a))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(86,plain,
% 0.19/0.38 (((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | (element(f(b, a), a) | set_equal(b, a) | element(f(b, a), b))) <=> ((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | (set_equal(b, a) | element(f(b, a), b) | element(f(b, a), a)))),
% 0.19/0.38 inference(monotonicity,[status(thm)],[85])).
% 0.19/0.38 tff(87,plain,
% 0.19/0.38 (((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | (element(f(b, a), a) | set_equal(b, a) | element(f(b, a), b))) <=> ((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | set_equal(b, a) | element(f(b, a), b) | element(f(b, a), a))),
% 0.19/0.38 inference(transitivity,[status(thm)],[86, 84])).
% 0.19/0.38 tff(88,plain,
% 0.19/0.38 ((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | (element(f(b, a), a) | set_equal(b, a) | element(f(b, a), b))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(89,plain,
% 0.19/0.38 ((~![Y: $i, X: $i] : (element(f(X, Y), Y) | set_equal(X, Y) | element(f(X, Y), X))) | set_equal(b, a) | element(f(b, a), b) | element(f(b, a), a)),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.19/0.38 tff(90,plain,
% 0.19/0.38 (element(f(b, a), b) | element(f(b, a), a)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[89, 13, 83])).
% 0.19/0.38 tff(91,plain,
% 0.19/0.38 (element(f(b, a), b)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[90, 1])).
% 0.19/0.38 tff(92,plain,
% 0.19/0.38 (((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | ((~set_equal(a, b)) | (~element(f(b, a), b)) | element(f(b, a), a))) <=> ((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | (~set_equal(a, b)) | (~element(f(b, a), b)) | element(f(b, a), a))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(93,plain,
% 0.19/0.38 ((element(f(b, a), a) | (~set_equal(a, b)) | (~element(f(b, a), b))) <=> ((~set_equal(a, b)) | (~element(f(b, a), b)) | element(f(b, a), a))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(94,plain,
% 0.19/0.38 (((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | (element(f(b, a), a) | (~set_equal(a, b)) | (~element(f(b, a), b)))) <=> ((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | ((~set_equal(a, b)) | (~element(f(b, a), b)) | element(f(b, a), a)))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[93])).
% 0.19/0.39 tff(95,plain,
% 0.19/0.39 (((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | (element(f(b, a), a) | (~set_equal(a, b)) | (~element(f(b, a), b)))) <=> ((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | (~set_equal(a, b)) | (~element(f(b, a), b)) | element(f(b, a), a))),
% 0.19/0.39 inference(transitivity,[status(thm)],[94, 92])).
% 0.19/0.39 tff(96,plain,
% 0.19/0.39 ((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | (element(f(b, a), a) | (~set_equal(a, b)) | (~element(f(b, a), b)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(97,plain,
% 0.19/0.39 ((~![Z: $i, Y: $i, X: $i] : (element(Z, X) | (~set_equal(X, Y)) | (~element(Z, Y)))) | (~set_equal(a, b)) | (~element(f(b, a), b)) | element(f(b, a), a)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[96, 95])).
% 0.19/0.39 tff(98,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[97, 69, 77, 91, 1])).
% 0.19/0.39 tff(99,plain,(element(f(b, a), a)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.39 tff(100,plain,
% 0.19/0.39 (((~![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))) | (set_equal(b, a) | (~element(f(b, a), b)) | (~element(f(b, a), a)))) <=> ((~![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))) | set_equal(b, a) | (~element(f(b, a), b)) | (~element(f(b, a), a)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(101,plain,
% 0.19/0.39 ((~![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))) | (set_equal(b, a) | (~element(f(b, a), b)) | (~element(f(b, a), a)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(102,plain,
% 0.19/0.39 ((~![Y: $i, X: $i] : (set_equal(X, Y) | (~element(f(X, Y), X)) | (~element(f(X, Y), Y)))) | set_equal(b, a) | (~element(f(b, a), b)) | (~element(f(b, a), a))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.19/0.39 tff(103,plain,
% 0.19/0.39 ((~element(f(b, a), b)) | (~element(f(b, a), a))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[102, 54, 83])).
% 0.19/0.39 tff(104,plain,
% 0.19/0.39 (~element(f(b, a), b)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[103, 99])).
% 0.19/0.39 tff(105,plain,
% 0.19/0.39 (((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | ((~set_equal(a, b)) | element(f(b, a), b) | (~element(f(b, a), a)))) <=> ((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | (~set_equal(a, b)) | element(f(b, a), b) | (~element(f(b, a), a)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(106,plain,
% 0.19/0.39 ((element(f(b, a), b) | (~element(f(b, a), a)) | (~set_equal(a, b))) <=> ((~set_equal(a, b)) | element(f(b, a), b) | (~element(f(b, a), a)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(107,plain,
% 0.19/0.39 (((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | (element(f(b, a), b) | (~element(f(b, a), a)) | (~set_equal(a, b)))) <=> ((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | ((~set_equal(a, b)) | element(f(b, a), b) | (~element(f(b, a), a))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[106])).
% 0.19/0.39 tff(108,plain,
% 0.19/0.39 (((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | (element(f(b, a), b) | (~element(f(b, a), a)) | (~set_equal(a, b)))) <=> ((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | (~set_equal(a, b)) | element(f(b, a), b) | (~element(f(b, a), a)))),
% 0.19/0.39 inference(transitivity,[status(thm)],[107, 105])).
% 0.19/0.39 tff(109,plain,
% 0.19/0.39 ((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | (element(f(b, a), b) | (~element(f(b, a), a)) | (~set_equal(a, b)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(110,plain,
% 0.19/0.39 ((~![Z: $i, Y: $i, X: $i] : (element(Z, Y) | (~element(Z, X)) | (~set_equal(X, Y)))) | (~set_equal(a, b)) | element(f(b, a), b) | (~element(f(b, a), a))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[109, 108])).
% 0.19/0.39 tff(111,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[110, 35, 77, 104, 99])).
% 0.19/0.39 % SZS output end Proof
% 0.19/0.39 % E exiting
%------------------------------------------------------------------------------