%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : CAT009-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.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 05:00:21 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.44s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : CAT009-1 : TPTP v9.0.0. Released v1.0.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n015.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Thu Jun 19 11:52:10 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.19/0.44 % SZS status Unsatisfiable
% 0.19/0.44 % SZS output start Proof
% 0.19/0.44 tff(domain_type, type, (
% 0.19/0.44 domain: $i > $i)).
% 0.19/0.44 tff(a_type, type, (
% 0.19/0.44 a: $i)).
% 0.19/0.44 tff(compose_type, type, (
% 0.19/0.44 compose: ( $i * $i ) > $i)).
% 0.19/0.44 tff(b_type, type, (
% 0.19/0.44 b: $i)).
% 0.19/0.44 tff(product_type, type, (
% 0.19/0.44 product: ( $i * $i * $i ) > $o)).
% 0.19/0.44 tff(defined_type, type, (
% 0.19/0.44 defined: ( $i * $i ) > $o)).
% 0.19/0.44 tff(identity_map_type, type, (
% 0.19/0.44 identity_map: $i > $o)).
% 0.19/0.44 tff(1,plain,
% 0.19/0.44 (^[X: $i] : refl(product(X, domain(X), X) <=> product(X, domain(X), X))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(2,plain,
% 0.19/0.44 (![X: $i] : product(X, domain(X), X) <=> ![X: $i] : product(X, domain(X), X)),
% 0.19/0.44 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.44 tff(3,plain,
% 0.19/0.44 (![X: $i] : product(X, domain(X), X) <=> ![X: $i] : product(X, domain(X), X)),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(4,axiom,(![X: $i] : product(X, domain(X), X)), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','product_on_domain')).
% 0.19/0.44 tff(5,plain,
% 0.19/0.44 (![X: $i] : product(X, domain(X), X)),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.44 tff(6,plain,(
% 0.19/0.44 ![X: $i] : product(X, domain(X), X)),
% 0.19/0.44 inference(skolemize,[status(sab)],[5])).
% 0.19/0.44 tff(7,plain,
% 0.19/0.44 (![X: $i] : product(X, domain(X), X)),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.19/0.44 tff(8,plain,
% 0.19/0.44 ((~![X: $i] : product(X, domain(X), X)) | product(domain(a), domain(domain(a)), domain(a))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(9,plain,
% 0.19/0.44 (product(domain(a), domain(domain(a)), domain(a))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.19/0.44 tff(10,plain,
% 0.19/0.44 (^[Z: $i, Y: $i, X: $i] : refl(((~product(X, Y, Z)) | defined(X, Y)) <=> ((~product(X, Y, Z)) | defined(X, Y)))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(11,plain,
% 0.19/0.44 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y)) <=> ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[10])).
% 0.19/0.44 tff(12,plain,
% 0.19/0.44 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y)) <=> ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(13,axiom,(![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','associative_property1')).
% 0.19/0.44 tff(14,plain,
% 0.19/0.44 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.19/0.44 tff(15,plain,(
% 0.19/0.44 ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))),
% 0.19/0.44 inference(skolemize,[status(sab)],[14])).
% 0.19/0.44 tff(16,plain,
% 0.19/0.44 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[15, 11])).
% 0.19/0.44 tff(17,plain,
% 0.19/0.44 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(domain(a), domain(domain(a)), domain(a))) | defined(domain(a), domain(domain(a))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(domain(a), domain(domain(a)), domain(a))) | defined(domain(a), domain(domain(a))))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(18,plain,
% 0.19/0.44 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(domain(a), domain(domain(a)), domain(a))) | defined(domain(a), domain(domain(a))))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(19,plain,
% 0.19/0.44 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(domain(a), domain(domain(a)), domain(a))) | defined(domain(a), domain(domain(a)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.19/0.44 tff(20,plain,
% 0.19/0.44 (defined(domain(a), domain(domain(a)))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[19, 16, 9])).
% 0.19/0.44 tff(21,plain,
% 0.19/0.44 (^[X: $i] : refl(identity_map(domain(X)) <=> identity_map(domain(X)))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(22,plain,
% 0.19/0.44 (![X: $i] : identity_map(domain(X)) <=> ![X: $i] : identity_map(domain(X))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[21])).
% 0.19/0.44 tff(23,plain,
% 0.19/0.44 (![X: $i] : identity_map(domain(X)) <=> ![X: $i] : identity_map(domain(X))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(24,axiom,(![X: $i] : identity_map(domain(X))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','domain_is_an_identity_map')).
% 0.19/0.44 tff(25,plain,
% 0.19/0.44 (![X: $i] : identity_map(domain(X))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[24, 23])).
% 0.19/0.44 tff(26,plain,(
% 0.19/0.44 ![X: $i] : identity_map(domain(X))),
% 0.19/0.44 inference(skolemize,[status(sab)],[25])).
% 0.19/0.44 tff(27,plain,
% 0.19/0.44 (![X: $i] : identity_map(domain(X))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[26, 22])).
% 0.19/0.44 tff(28,plain,
% 0.19/0.44 ((~![X: $i] : identity_map(domain(X))) | identity_map(domain(a))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(29,plain,
% 0.19/0.44 (identity_map(domain(a))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[28, 27])).
% 0.19/0.44 tff(30,plain,
% 0.19/0.44 (^[Y: $i, X: $i] : refl(((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X))) <=> ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X))))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(31,plain,
% 0.19/0.44 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X))) <=> ![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[30])).
% 0.19/0.44 tff(32,plain,
% 0.19/0.44 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X))) <=> ![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(33,plain,
% 0.19/0.44 (^[Y: $i, X: $i] : trans(monotonicity(rewrite(((~defined(X, Y)) | (~identity_map(X))) <=> ((~defined(X, Y)) | (~identity_map(X)))), ((((~defined(X, Y)) | (~identity_map(X))) | product(X, Y, Y)) <=> (((~defined(X, Y)) | (~identity_map(X))) | product(X, Y, Y)))), rewrite((((~defined(X, Y)) | (~identity_map(X))) | product(X, Y, Y)) <=> ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))), ((((~defined(X, Y)) | (~identity_map(X))) | product(X, Y, Y)) <=> ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(34,plain,
% 0.19/0.44 (![Y: $i, X: $i] : (((~defined(X, Y)) | (~identity_map(X))) | product(X, Y, Y)) <=> ![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[33])).
% 0.19/0.44 tff(35,axiom,(![Y: $i, X: $i] : (((~defined(X, Y)) | (~identity_map(X))) | product(X, Y, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','identity1')).
% 0.19/0.44 tff(36,plain,
% 0.19/0.44 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.19/0.44 tff(37,plain,
% 0.19/0.44 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[36, 32])).
% 0.19/0.44 tff(38,plain,(
% 0.19/0.44 ![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))),
% 0.19/0.44 inference(skolemize,[status(sab)],[37])).
% 0.19/0.44 tff(39,plain,
% 0.19/0.44 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[38, 31])).
% 0.19/0.44 tff(40,plain,
% 0.19/0.44 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | ((~identity_map(domain(a))) | product(domain(a), domain(domain(a)), domain(domain(a))) | (~defined(domain(a), domain(domain(a)))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | (~identity_map(domain(a))) | product(domain(a), domain(domain(a)), domain(domain(a))) | (~defined(domain(a), domain(domain(a)))))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(41,plain,
% 0.19/0.44 (((~defined(domain(a), domain(domain(a)))) | product(domain(a), domain(domain(a)), domain(domain(a))) | (~identity_map(domain(a)))) <=> ((~identity_map(domain(a))) | product(domain(a), domain(domain(a)), domain(domain(a))) | (~defined(domain(a), domain(domain(a)))))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(42,plain,
% 0.19/0.44 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | ((~defined(domain(a), domain(domain(a)))) | product(domain(a), domain(domain(a)), domain(domain(a))) | (~identity_map(domain(a))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | ((~identity_map(domain(a))) | product(domain(a), domain(domain(a)), domain(domain(a))) | (~defined(domain(a), domain(domain(a))))))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[41])).
% 0.19/0.45 tff(43,plain,
% 0.19/0.45 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | ((~defined(domain(a), domain(domain(a)))) | product(domain(a), domain(domain(a)), domain(domain(a))) | (~identity_map(domain(a))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | (~identity_map(domain(a))) | product(domain(a), domain(domain(a)), domain(domain(a))) | (~defined(domain(a), domain(domain(a)))))),
% 0.19/0.45 inference(transitivity,[status(thm)],[42, 40])).
% 0.19/0.45 tff(44,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | ((~defined(domain(a), domain(domain(a)))) | product(domain(a), domain(domain(a)), domain(domain(a))) | (~identity_map(domain(a))))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(45,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | (~identity_map(domain(a))) | product(domain(a), domain(domain(a)), domain(domain(a))) | (~defined(domain(a), domain(domain(a))))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.19/0.45 tff(46,plain,
% 0.19/0.45 (product(domain(a), domain(domain(a)), domain(domain(a)))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[45, 39, 29, 20])).
% 0.19/0.45 tff(47,plain,
% 0.19/0.45 (^[W: $i, Z: $i, Y: $i, X: $i] : refl(((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W)) <=> ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W)))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(48,plain,
% 0.19/0.45 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.19/0.45 inference(quant_intro,[status(thm)],[47])).
% 0.19/0.45 tff(49,plain,
% 0.19/0.45 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(50,plain,
% 0.19/0.45 (^[W: $i, Z: $i, Y: $i, X: $i] : rewrite((((~product(X, Y, Z)) | (~product(X, Y, W))) | (Z = W)) <=> ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W)))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(51,plain,
% 0.19/0.45 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, Z)) | (~product(X, Y, W))) | (Z = W)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.19/0.45 inference(quant_intro,[status(thm)],[50])).
% 0.19/0.45 tff(52,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, Z)) | (~product(X, Y, W))) | (Z = W))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','composition_is_well_defined')).
% 0.19/0.45 tff(53,plain,
% 0.19/0.45 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.19/0.45 tff(54,plain,
% 0.19/0.45 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[53, 49])).
% 0.19/0.45 tff(55,plain,(
% 0.19/0.45 ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.19/0.45 inference(skolemize,[status(sab)],[54])).
% 0.19/0.45 tff(56,plain,
% 0.19/0.45 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[55, 48])).
% 0.19/0.45 tff(57,plain,
% 0.19/0.45 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(domain(a), domain(domain(a)), domain(a))) | (~product(domain(a), domain(domain(a)), domain(domain(a)))) | (domain(a) = domain(domain(a))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(domain(a), domain(domain(a)), domain(a))) | (~product(domain(a), domain(domain(a)), domain(domain(a)))) | (domain(a) = domain(domain(a))))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(58,plain,
% 0.19/0.45 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(domain(a), domain(domain(a)), domain(a))) | (~product(domain(a), domain(domain(a)), domain(domain(a)))) | (domain(a) = domain(domain(a))))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(59,plain,
% 0.19/0.45 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(domain(a), domain(domain(a)), domain(a))) | (~product(domain(a), domain(domain(a)), domain(domain(a)))) | (domain(a) = domain(domain(a)))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.19/0.45 tff(60,plain,
% 0.19/0.45 (domain(a) = domain(domain(a))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[59, 56, 9, 46])).
% 0.19/0.45 tff(61,plain,
% 0.19/0.45 (domain(domain(a)) = domain(a)),
% 0.19/0.45 inference(symmetry,[status(thm)],[60])).
% 0.19/0.45 tff(62,plain,
% 0.19/0.45 ((~![X: $i] : product(X, domain(X), X)) | product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(63,plain,
% 0.19/0.45 (product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[62, 7])).
% 0.19/0.45 tff(64,plain,
% 0.19/0.45 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | defined(domain(compose(b, a)), domain(domain(compose(b, a)))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | defined(domain(compose(b, a)), domain(domain(compose(b, a)))))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(65,plain,
% 0.19/0.45 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | defined(domain(compose(b, a)), domain(domain(compose(b, a)))))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(66,plain,
% 0.19/0.45 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | defined(domain(compose(b, a)), domain(domain(compose(b, a))))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.19/0.45 tff(67,plain,
% 0.19/0.45 (defined(domain(compose(b, a)), domain(domain(compose(b, a))))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[66, 16, 63])).
% 0.19/0.45 tff(68,plain,
% 0.19/0.45 ((~![X: $i] : identity_map(domain(X))) | identity_map(domain(compose(b, a)))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(69,plain,
% 0.19/0.45 (identity_map(domain(compose(b, a)))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[68, 27])).
% 0.19/0.45 tff(70,plain,
% 0.19/0.45 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | ((~defined(domain(compose(b, a)), domain(domain(compose(b, a))))) | product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(domain(compose(b, a)))) | (~identity_map(domain(compose(b, a)))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | (~defined(domain(compose(b, a)), domain(domain(compose(b, a))))) | product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(domain(compose(b, a)))) | (~identity_map(domain(compose(b, a)))))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(71,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | ((~defined(domain(compose(b, a)), domain(domain(compose(b, a))))) | product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(domain(compose(b, a)))) | (~identity_map(domain(compose(b, a)))))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(72,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | (~defined(domain(compose(b, a)), domain(domain(compose(b, a))))) | product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(domain(compose(b, a)))) | (~identity_map(domain(compose(b, a))))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.19/0.45 tff(73,plain,
% 0.19/0.45 (product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(domain(compose(b, a))))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[72, 39, 69, 67])).
% 0.19/0.45 tff(74,plain,
% 0.19/0.45 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(domain(compose(b, a))))) | (domain(compose(b, a)) = domain(domain(compose(b, a)))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(domain(compose(b, a))))) | (domain(compose(b, a)) = domain(domain(compose(b, a)))))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(75,plain,
% 0.19/0.45 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(domain(compose(b, a))))) | (domain(compose(b, a)) = domain(domain(compose(b, a)))))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(76,plain,
% 0.19/0.45 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(domain(compose(b, a))))) | (domain(compose(b, a)) = domain(domain(compose(b, a))))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.19/0.45 tff(77,plain,
% 0.19/0.45 (domain(compose(b, a)) = domain(domain(compose(b, a)))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[76, 56, 63, 73])).
% 0.19/0.45 tff(78,plain,
% 0.19/0.45 (domain(domain(compose(b, a))) = domain(compose(b, a))),
% 0.19/0.45 inference(symmetry,[status(thm)],[77])).
% 0.19/0.45 tff(79,plain,
% 0.19/0.45 (product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(compose(b, a)))) <=> product(domain(a), domain(compose(b, a)), domain(compose(b, a)))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[61, 78, 78])).
% 0.19/0.45 tff(80,plain,
% 0.19/0.45 (product(domain(a), domain(compose(b, a)), domain(compose(b, a))) <=> product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(compose(b, a))))),
% 0.19/0.45 inference(symmetry,[status(thm)],[79])).
% 0.19/0.45 tff(81,plain,
% 0.19/0.45 (defined(domain(domain(a)), domain(domain(compose(b, a)))) <=> defined(domain(a), domain(compose(b, a)))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[61, 78])).
% 0.19/0.45 tff(82,plain,
% 0.19/0.45 (defined(domain(a), domain(compose(b, a))) <=> defined(domain(domain(a)), domain(domain(compose(b, a))))),
% 0.19/0.45 inference(symmetry,[status(thm)],[81])).
% 0.19/0.45 tff(83,plain,
% 0.19/0.45 ((~defined(domain(a), domain(compose(b, a)))) <=> (~defined(domain(domain(a)), domain(domain(compose(b, a)))))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[82])).
% 0.19/0.45 tff(84,assumption,(~defined(domain(a), domain(compose(b, a)))), introduced(assumption)).
% 0.19/0.45 tff(85,plain,
% 0.19/0.45 (~defined(domain(domain(a)), domain(domain(compose(b, a))))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.19/0.45 tff(86,plain,
% 0.19/0.45 ((~![X: $i] : identity_map(domain(X))) | identity_map(domain(domain(a)))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(87,plain,
% 0.19/0.45 (identity_map(domain(domain(a)))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[86, 27])).
% 0.19/0.45 tff(88,plain,
% 0.19/0.45 (^[X: $i] : refl(defined(X, domain(X)) <=> defined(X, domain(X)))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(89,plain,
% 0.19/0.45 (![X: $i] : defined(X, domain(X)) <=> ![X: $i] : defined(X, domain(X))),
% 0.19/0.45 inference(quant_intro,[status(thm)],[88])).
% 0.19/0.45 tff(90,plain,
% 0.19/0.45 (![X: $i] : defined(X, domain(X)) <=> ![X: $i] : defined(X, domain(X))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(91,axiom,(![X: $i] : defined(X, domain(X))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','mapping_from_x_to_its_domain')).
% 0.19/0.45 tff(92,plain,
% 0.19/0.45 (![X: $i] : defined(X, domain(X))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.19/0.45 tff(93,plain,(
% 0.19/0.45 ![X: $i] : defined(X, domain(X))),
% 0.19/0.45 inference(skolemize,[status(sab)],[92])).
% 0.19/0.45 tff(94,plain,
% 0.19/0.45 (![X: $i] : defined(X, domain(X))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[93, 89])).
% 0.19/0.45 tff(95,plain,
% 0.19/0.45 ((~![X: $i] : defined(X, domain(X))) | defined(compose(b, a), domain(compose(b, a)))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(96,plain,
% 0.19/0.46 (defined(compose(b, a), domain(compose(b, a)))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[95, 94])).
% 0.19/0.46 tff(97,plain,
% 0.19/0.46 (^[Y: $i, X: $i] : refl(((~defined(X, Y)) | product(X, Y, compose(X, Y))) <=> ((~defined(X, Y)) | product(X, Y, compose(X, Y))))),
% 0.19/0.46 inference(bind,[status(th)],[])).
% 0.19/0.46 tff(98,plain,
% 0.19/0.46 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y))) <=> ![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))),
% 0.19/0.46 inference(quant_intro,[status(thm)],[97])).
% 0.19/0.46 tff(99,plain,
% 0.19/0.46 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y))) <=> ![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(100,axiom,(![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','closure_of_composition')).
% 0.19/0.46 tff(101,plain,
% 0.19/0.46 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[100, 99])).
% 0.19/0.46 tff(102,plain,(
% 0.19/0.46 ![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))),
% 0.19/0.46 inference(skolemize,[status(sab)],[101])).
% 0.19/0.46 tff(103,plain,
% 0.19/0.46 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[102, 98])).
% 0.19/0.46 tff(104,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(compose(b, a), domain(compose(b, a)))) | product(compose(b, a), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a)))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(compose(b, a), domain(compose(b, a)))) | product(compose(b, a), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a)))))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(105,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(compose(b, a), domain(compose(b, a)))) | product(compose(b, a), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a)))))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(106,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(compose(b, a), domain(compose(b, a)))) | product(compose(b, a), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[105, 104])).
% 0.19/0.46 tff(107,plain,
% 0.19/0.46 (product(compose(b, a), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[106, 103, 96])).
% 0.19/0.46 tff(108,plain,
% 0.19/0.46 ((~![X: $i] : product(X, domain(X), X)) | product(compose(b, a), domain(compose(b, a)), compose(b, a))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(109,plain,
% 0.19/0.46 (product(compose(b, a), domain(compose(b, a)), compose(b, a))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[108, 7])).
% 0.19/0.46 tff(110,plain,
% 0.19/0.46 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(compose(b, a), domain(compose(b, a)), compose(b, a))) | (~product(compose(b, a), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))) | (compose(b, a) = compose(compose(b, a), domain(compose(b, a)))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(compose(b, a), domain(compose(b, a)), compose(b, a))) | (~product(compose(b, a), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))) | (compose(b, a) = compose(compose(b, a), domain(compose(b, a)))))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(111,plain,
% 0.19/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(compose(b, a), domain(compose(b, a)), compose(b, a))) | (~product(compose(b, a), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))) | (compose(b, a) = compose(compose(b, a), domain(compose(b, a)))))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(112,plain,
% 0.19/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(compose(b, a), domain(compose(b, a)), compose(b, a))) | (~product(compose(b, a), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))) | (compose(b, a) = compose(compose(b, a), domain(compose(b, a))))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[111, 110])).
% 0.19/0.46 tff(113,plain,
% 0.19/0.46 (compose(b, a) = compose(compose(b, a), domain(compose(b, a)))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[112, 56, 109, 107])).
% 0.19/0.46 tff(114,plain,
% 0.19/0.46 (compose(compose(b, a), domain(compose(b, a))) = compose(b, a)),
% 0.19/0.46 inference(symmetry,[status(thm)],[113])).
% 0.19/0.46 tff(115,plain,
% 0.19/0.46 (defined(compose(compose(b, a), domain(compose(b, a))), domain(a)) <=> defined(compose(b, a), domain(a))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[114])).
% 0.19/0.46 tff(116,plain,
% 0.19/0.46 (defined(compose(b, a), domain(a)) <=> defined(compose(compose(b, a), domain(compose(b, a))), domain(a))),
% 0.19/0.46 inference(symmetry,[status(thm)],[115])).
% 0.19/0.46 tff(117,plain,
% 0.19/0.46 ((~![X: $i] : product(X, domain(X), X)) | product(a, domain(a), a)),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(118,plain,
% 0.19/0.46 (product(a, domain(a), a)),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[117, 7])).
% 0.19/0.46 tff(119,plain,
% 0.19/0.46 (defined(b, a) <=> defined(b, a)),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(120,axiom,(defined(b, a)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','ba_defined')).
% 0.19/0.46 tff(121,plain,
% 0.19/0.46 (defined(b, a)),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[120, 119])).
% 0.19/0.46 tff(122,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(b, a)) | product(b, a, compose(b, a)))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(b, a)) | product(b, a, compose(b, a)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(123,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(b, a)) | product(b, a, compose(b, a)))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(124,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(b, a)) | product(b, a, compose(b, a))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[123, 122])).
% 0.19/0.46 tff(125,plain,
% 0.19/0.46 (product(b, a, compose(b, a))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[124, 103, 121])).
% 0.19/0.46 tff(126,plain,
% 0.19/0.46 (^[Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : refl(((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy))) <=> ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy))))),
% 0.19/0.46 inference(bind,[status(th)],[])).
% 0.19/0.46 tff(127,plain,
% 0.19/0.46 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy))) <=> ![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.19/0.46 inference(quant_intro,[status(thm)],[126])).
% 0.19/0.46 tff(128,plain,
% 0.19/0.46 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy))) <=> ![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(129,plain,
% 0.19/0.46 (^[Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~product(Y, Z, Yz)) | (~product(X, Y, Xy))) <=> ((~product(Y, Z, Yz)) | (~product(X, Y, Xy)))), ((((~product(Y, Z, Yz)) | (~product(X, Y, Xy))) | (~defined(X, Yz))) <=> (((~product(Y, Z, Yz)) | (~product(X, Y, Xy))) | (~defined(X, Yz))))), rewrite((((~product(Y, Z, Yz)) | (~product(X, Y, Xy))) | (~defined(X, Yz))) <=> ((~defined(X, Yz)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))), ((((~product(Y, Z, Yz)) | (~product(X, Y, Xy))) | (~defined(X, Yz))) <=> ((~defined(X, Yz)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy))))), (((((~product(Y, Z, Yz)) | (~product(X, Y, Xy))) | (~defined(X, Yz))) | defined(Xy, Z)) <=> (((~defined(X, Yz)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy))) | defined(Xy, Z)))), rewrite((((~defined(X, Yz)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy))) | defined(Xy, Z)) <=> ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))), (((((~product(Y, Z, Yz)) | (~product(X, Y, Xy))) | (~defined(X, Yz))) | defined(Xy, Z)) <=> ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))))),
% 0.19/0.46 inference(bind,[status(th)],[])).
% 0.19/0.46 tff(130,plain,
% 0.19/0.46 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((((~product(Y, Z, Yz)) | (~product(X, Y, Xy))) | (~defined(X, Yz))) | defined(Xy, Z)) <=> ![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.19/0.46 inference(quant_intro,[status(thm)],[129])).
% 0.19/0.46 tff(131,axiom,(![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((((~product(Y, Z, Yz)) | (~product(X, Y, Xy))) | (~defined(X, Yz))) | defined(Xy, Z))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','category_theory_axiom4')).
% 0.19/0.46 tff(132,plain,
% 0.19/0.46 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[131, 130])).
% 0.19/0.46 tff(133,plain,
% 0.19/0.46 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[132, 128])).
% 0.19/0.46 tff(134,plain,(
% 0.19/0.46 ![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.19/0.46 inference(skolemize,[status(sab)],[133])).
% 0.19/0.46 tff(135,plain,
% 0.19/0.46 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[134, 127])).
% 0.19/0.46 tff(136,plain,
% 0.19/0.46 (((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~defined(b, a)) | defined(compose(b, a), domain(a)) | (~product(a, domain(a), a)) | (~product(b, a, compose(b, a))))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | (~defined(b, a)) | defined(compose(b, a), domain(a)) | (~product(a, domain(a), a)) | (~product(b, a, compose(b, a))))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(137,plain,
% 0.19/0.46 ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~defined(b, a)) | defined(compose(b, a), domain(a)) | (~product(a, domain(a), a)) | (~product(b, a, compose(b, a))))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(138,plain,
% 0.19/0.46 ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | (~defined(b, a)) | defined(compose(b, a), domain(a)) | (~product(a, domain(a), a)) | (~product(b, a, compose(b, a)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[137, 136])).
% 0.19/0.46 tff(139,plain,
% 0.19/0.46 (defined(compose(b, a), domain(a))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[138, 135, 121, 125, 118])).
% 0.19/0.46 tff(140,plain,
% 0.19/0.46 (defined(compose(compose(b, a), domain(compose(b, a))), domain(a))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[139, 116])).
% 0.19/0.46 tff(141,plain,
% 0.19/0.46 (product(compose(compose(b, a), domain(compose(b, a))), domain(a), compose(compose(b, a), domain(compose(b, a)))) <=> product(compose(b, a), domain(a), compose(b, a))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[114, 114])).
% 0.19/0.46 tff(142,plain,
% 0.19/0.46 (product(compose(b, a), domain(a), compose(b, a)) <=> product(compose(compose(b, a), domain(compose(b, a))), domain(a), compose(compose(b, a), domain(compose(b, a))))),
% 0.19/0.46 inference(symmetry,[status(thm)],[141])).
% 0.19/0.46 tff(143,plain,
% 0.19/0.46 (^[Y: $i, X: $i] : refl(((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X)) <=> ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X)))),
% 0.19/0.46 inference(bind,[status(th)],[])).
% 0.19/0.46 tff(144,plain,
% 0.19/0.46 (![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X)) <=> ![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))),
% 0.19/0.46 inference(quant_intro,[status(thm)],[143])).
% 0.19/0.47 tff(145,plain,
% 0.19/0.47 (![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X)) <=> ![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(146,plain,
% 0.19/0.47 (^[Y: $i, X: $i] : trans(monotonicity(rewrite(((~defined(X, Y)) | (~identity_map(Y))) <=> ((~defined(X, Y)) | (~identity_map(Y)))), ((((~defined(X, Y)) | (~identity_map(Y))) | product(X, Y, X)) <=> (((~defined(X, Y)) | (~identity_map(Y))) | product(X, Y, X)))), rewrite((((~defined(X, Y)) | (~identity_map(Y))) | product(X, Y, X)) <=> ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))), ((((~defined(X, Y)) | (~identity_map(Y))) | product(X, Y, X)) <=> ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))))),
% 0.19/0.47 inference(bind,[status(th)],[])).
% 0.19/0.47 tff(147,plain,
% 0.19/0.47 (![Y: $i, X: $i] : (((~defined(X, Y)) | (~identity_map(Y))) | product(X, Y, X)) <=> ![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))),
% 0.19/0.47 inference(quant_intro,[status(thm)],[146])).
% 0.19/0.47 tff(148,axiom,(![Y: $i, X: $i] : (((~defined(X, Y)) | (~identity_map(Y))) | product(X, Y, X))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','identity2')).
% 0.19/0.47 tff(149,plain,
% 0.19/0.47 (![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[148, 147])).
% 0.19/0.47 tff(150,plain,
% 0.19/0.47 (![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[149, 145])).
% 0.19/0.47 tff(151,plain,(
% 0.19/0.47 ![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))),
% 0.19/0.47 inference(skolemize,[status(sab)],[150])).
% 0.19/0.47 tff(152,plain,
% 0.19/0.47 (![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[151, 144])).
% 0.19/0.47 tff(153,plain,
% 0.19/0.47 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | ((~defined(compose(b, a), domain(a))) | (~identity_map(domain(a))) | product(compose(b, a), domain(a), compose(b, a)))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | (~defined(compose(b, a), domain(a))) | (~identity_map(domain(a))) | product(compose(b, a), domain(a), compose(b, a)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(154,plain,
% 0.19/0.47 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | ((~defined(compose(b, a), domain(a))) | (~identity_map(domain(a))) | product(compose(b, a), domain(a), compose(b, a)))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(155,plain,
% 0.19/0.47 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | (~defined(compose(b, a), domain(a))) | (~identity_map(domain(a))) | product(compose(b, a), domain(a), compose(b, a))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[154, 153])).
% 0.19/0.47 tff(156,plain,
% 0.19/0.47 (product(compose(b, a), domain(a), compose(b, a))),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[155, 152, 29, 139])).
% 0.19/0.47 tff(157,plain,
% 0.19/0.47 (product(compose(compose(b, a), domain(compose(b, a))), domain(a), compose(compose(b, a), domain(compose(b, a))))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[156, 142])).
% 0.19/0.47 tff(158,plain,
% 0.19/0.47 (((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~product(compose(compose(b, a), domain(compose(b, a))), domain(a), compose(compose(b, a), domain(compose(b, a))))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(a))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a))) | (~product(domain(a), domain(domain(a)), domain(a))))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(a), compose(compose(b, a), domain(compose(b, a))))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(a))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a))) | (~product(domain(a), domain(domain(a)), domain(a))))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(159,plain,
% 0.19/0.47 (((~defined(compose(compose(b, a), domain(compose(b, a))), domain(a))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a))) | (~product(domain(a), domain(domain(a)), domain(a))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(a), compose(compose(b, a), domain(compose(b, a)))))) <=> ((~product(compose(compose(b, a), domain(compose(b, a))), domain(a), compose(compose(b, a), domain(compose(b, a))))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(a))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a))) | (~product(domain(a), domain(domain(a)), domain(a))))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(160,plain,
% 0.19/0.47 (((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(a))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a))) | (~product(domain(a), domain(domain(a)), domain(a))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(a), compose(compose(b, a), domain(compose(b, a))))))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~product(compose(compose(b, a), domain(compose(b, a))), domain(a), compose(compose(b, a), domain(compose(b, a))))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(a))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a))) | (~product(domain(a), domain(domain(a)), domain(a)))))),
% 0.19/0.47 inference(monotonicity,[status(thm)],[159])).
% 0.19/0.47 tff(161,plain,
% 0.19/0.47 (((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(a))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a))) | (~product(domain(a), domain(domain(a)), domain(a))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(a), compose(compose(b, a), domain(compose(b, a))))))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(a), compose(compose(b, a), domain(compose(b, a))))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(a))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a))) | (~product(domain(a), domain(domain(a)), domain(a))))),
% 0.19/0.47 inference(transitivity,[status(thm)],[160, 158])).
% 0.19/0.47 tff(162,plain,
% 0.19/0.47 ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(a))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a))) | (~product(domain(a), domain(domain(a)), domain(a))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(a), compose(compose(b, a), domain(compose(b, a))))))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(163,plain,
% 0.19/0.47 ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(a), compose(compose(b, a), domain(compose(b, a))))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(a))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a))) | (~product(domain(a), domain(domain(a)), domain(a)))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[162, 161])).
% 0.19/0.47 tff(164,plain,
% 0.19/0.47 ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(a))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)))),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[163, 135, 157, 9])).
% 0.19/0.47 tff(165,plain,
% 0.19/0.47 (defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)))),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[164, 140])).
% 0.19/0.47 tff(166,plain,
% 0.19/0.47 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)))) | product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a)))) | (~identity_map(domain(domain(a)))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)))) | product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a)))) | (~identity_map(domain(domain(a)))))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(167,plain,
% 0.19/0.47 (((~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)))) | (~identity_map(domain(domain(a)))) | product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a))))) <=> ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)))) | product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a)))) | (~identity_map(domain(domain(a)))))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(168,plain,
% 0.19/0.47 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)))) | (~identity_map(domain(domain(a)))) | product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a)))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)))) | product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a)))) | (~identity_map(domain(domain(a))))))),
% 0.19/0.47 inference(monotonicity,[status(thm)],[167])).
% 0.19/0.47 tff(169,plain,
% 0.19/0.47 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)))) | (~identity_map(domain(domain(a)))) | product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a)))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)))) | product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a)))) | (~identity_map(domain(domain(a)))))),
% 0.19/0.47 inference(transitivity,[status(thm)],[168, 166])).
% 0.19/0.47 tff(170,plain,
% 0.19/0.47 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)))) | (~identity_map(domain(domain(a)))) | product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a)))))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(171,plain,
% 0.19/0.47 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)))) | product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a)))) | (~identity_map(domain(domain(a))))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[170, 169])).
% 0.19/0.47 tff(172,plain,
% 0.19/0.47 (product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a))))),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[171, 152, 165, 87])).
% 0.19/0.47 tff(173,plain,
% 0.19/0.47 (defined(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a))) <=> defined(compose(b, a), domain(compose(b, a)))),
% 0.19/0.47 inference(monotonicity,[status(thm)],[114])).
% 0.19/0.47 tff(174,plain,
% 0.19/0.47 (defined(compose(b, a), domain(compose(b, a))) <=> defined(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)))),
% 0.19/0.47 inference(symmetry,[status(thm)],[173])).
% 0.19/0.48 tff(175,plain,
% 0.19/0.48 (defined(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)))),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[96, 174])).
% 0.19/0.48 tff(176,plain,
% 0.19/0.48 (product(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a)))) <=> product(compose(b, a), domain(compose(b, a)), compose(b, a))),
% 0.19/0.48 inference(monotonicity,[status(thm)],[114, 114])).
% 0.19/0.48 tff(177,plain,
% 0.19/0.48 (product(compose(b, a), domain(compose(b, a)), compose(b, a)) <=> product(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))),
% 0.19/0.48 inference(symmetry,[status(thm)],[176])).
% 0.19/0.48 tff(178,plain,
% 0.19/0.48 (product(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[109, 177])).
% 0.19/0.48 tff(179,plain,
% 0.19/0.48 (((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~product(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a)))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)))))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a)))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)))))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(180,plain,
% 0.19/0.48 (((~defined(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a)))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a)))))) <=> ((~product(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a)))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)))))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(181,plain,
% 0.19/0.48 (((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a)))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~product(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a)))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a))))))),
% 0.19/0.48 inference(monotonicity,[status(thm)],[180])).
% 0.19/0.48 tff(182,plain,
% 0.19/0.48 (((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a)))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a)))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)))))),
% 0.19/0.48 inference(transitivity,[status(thm)],[181, 179])).
% 0.19/0.48 tff(183,plain,
% 0.19/0.48 ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a)))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))))),
% 0.19/0.48 inference(quant_inst,[status(thm)],[])).
% 0.19/0.48 tff(184,plain,
% 0.19/0.48 ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a)), compose(compose(b, a), domain(compose(b, a))))) | defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a)))) | (~product(domain(compose(b, a)), domain(domain(compose(b, a))), domain(compose(b, a)))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a))))),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[183, 182])).
% 0.19/0.48 tff(185,plain,
% 0.19/0.48 (defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a)))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(compose(b, a))))),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[184, 135, 178, 63])).
% 0.19/0.48 tff(186,plain,
% 0.19/0.48 (defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a))))),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[185, 175])).
% 0.19/0.48 tff(187,plain,
% 0.19/0.48 (^[Xy: $i, Z: $i, Y: $i, X: $i] : refl((defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy))) <=> (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy))))),
% 0.19/0.48 inference(bind,[status(th)],[])).
% 0.19/0.48 tff(188,plain,
% 0.19/0.48 (![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy))) <=> ![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))),
% 0.19/0.48 inference(quant_intro,[status(thm)],[187])).
% 0.19/0.48 tff(189,plain,
% 0.19/0.48 (![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy))) <=> ![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(190,plain,
% 0.19/0.48 (^[Xy: $i, Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~product(X, Y, Xy)) | (~defined(Xy, Z))) <=> ((~defined(Xy, Z)) | (~product(X, Y, Xy)))), ((((~product(X, Y, Xy)) | (~defined(Xy, Z))) | defined(Y, Z)) <=> (((~defined(Xy, Z)) | (~product(X, Y, Xy))) | defined(Y, Z)))), rewrite((((~defined(Xy, Z)) | (~product(X, Y, Xy))) | defined(Y, Z)) <=> (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))), ((((~product(X, Y, Xy)) | (~defined(Xy, Z))) | defined(Y, Z)) <=> (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))))),
% 0.19/0.48 inference(bind,[status(th)],[])).
% 0.19/0.48 tff(191,plain,
% 0.19/0.48 (![Xy: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, Xy)) | (~defined(Xy, Z))) | defined(Y, Z)) <=> ![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))),
% 0.19/0.49 inference(quant_intro,[status(thm)],[190])).
% 0.19/0.49 tff(192,axiom,(![Xy: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, Xy)) | (~defined(Xy, Z))) | defined(Y, Z))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','associative_property2')).
% 0.19/0.49 tff(193,plain,
% 0.19/0.49 (![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[192, 191])).
% 0.19/0.49 tff(194,plain,
% 0.19/0.49 (![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[193, 189])).
% 0.19/0.49 tff(195,plain,(
% 0.19/0.49 ![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))),
% 0.19/0.49 inference(skolemize,[status(sab)],[194])).
% 0.19/0.49 tff(196,plain,
% 0.19/0.49 (![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[195, 188])).
% 0.19/0.49 tff(197,plain,
% 0.19/0.49 (((~![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))) | ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a))))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a))))) | defined(domain(domain(a)), domain(domain(compose(b, a)))))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a))))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a))))) | defined(domain(domain(a)), domain(domain(compose(b, a)))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(198,plain,
% 0.19/0.49 ((defined(domain(domain(a)), domain(domain(compose(b, a)))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a))))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a)))))) <=> ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a))))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a))))) | defined(domain(domain(a)), domain(domain(compose(b, a)))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(199,plain,
% 0.19/0.49 (((~![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))) | (defined(domain(domain(a)), domain(domain(compose(b, a)))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a))))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a))))))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))) | ((~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a))))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a))))) | defined(domain(domain(a)), domain(domain(compose(b, a))))))),
% 0.19/0.49 inference(monotonicity,[status(thm)],[198])).
% 0.19/0.49 tff(200,plain,
% 0.19/0.49 (((~![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))) | (defined(domain(domain(a)), domain(domain(compose(b, a)))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a))))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a))))))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a))))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a))))) | defined(domain(domain(a)), domain(domain(compose(b, a)))))),
% 0.19/0.49 inference(transitivity,[status(thm)],[199, 197])).
% 0.19/0.49 tff(201,plain,
% 0.19/0.49 ((~![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))) | (defined(domain(domain(a)), domain(domain(compose(b, a)))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a))))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a))))))),
% 0.19/0.49 inference(quant_inst,[status(thm)],[])).
% 0.19/0.49 tff(202,plain,
% 0.19/0.49 ((~![Xy: $i, Z: $i, Y: $i, X: $i] : (defined(Y, Z) | (~defined(Xy, Z)) | (~product(X, Y, Xy)))) | (~defined(compose(compose(b, a), domain(compose(b, a))), domain(domain(compose(b, a))))) | (~product(compose(compose(b, a), domain(compose(b, a))), domain(domain(a)), compose(compose(b, a), domain(compose(b, a))))) | defined(domain(domain(a)), domain(domain(compose(b, a))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[201, 200])).
% 0.19/0.49 tff(203,plain,
% 0.19/0.49 (defined(domain(domain(a)), domain(domain(compose(b, a))))),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[202, 196, 186, 172])).
% 0.19/0.49 tff(204,plain,
% 0.19/0.49 ($false),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[203, 85])).
% 0.19/0.49 tff(205,plain,(defined(domain(a), domain(compose(b, a)))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.49 tff(206,plain,
% 0.19/0.49 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | ((~identity_map(domain(a))) | product(domain(a), domain(compose(b, a)), domain(compose(b, a))) | (~defined(domain(a), domain(compose(b, a)))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | (~identity_map(domain(a))) | product(domain(a), domain(compose(b, a)), domain(compose(b, a))) | (~defined(domain(a), domain(compose(b, a)))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(207,plain,
% 0.19/0.49 (((~defined(domain(a), domain(compose(b, a)))) | product(domain(a), domain(compose(b, a)), domain(compose(b, a))) | (~identity_map(domain(a)))) <=> ((~identity_map(domain(a))) | product(domain(a), domain(compose(b, a)), domain(compose(b, a))) | (~defined(domain(a), domain(compose(b, a)))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(208,plain,
% 0.19/0.49 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | ((~defined(domain(a), domain(compose(b, a)))) | product(domain(a), domain(compose(b, a)), domain(compose(b, a))) | (~identity_map(domain(a))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | ((~identity_map(domain(a))) | product(domain(a), domain(compose(b, a)), domain(compose(b, a))) | (~defined(domain(a), domain(compose(b, a))))))),
% 0.19/0.49 inference(monotonicity,[status(thm)],[207])).
% 0.19/0.49 tff(209,plain,
% 0.19/0.49 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | ((~defined(domain(a), domain(compose(b, a)))) | product(domain(a), domain(compose(b, a)), domain(compose(b, a))) | (~identity_map(domain(a))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | (~identity_map(domain(a))) | product(domain(a), domain(compose(b, a)), domain(compose(b, a))) | (~defined(domain(a), domain(compose(b, a)))))),
% 0.19/0.49 inference(transitivity,[status(thm)],[208, 206])).
% 0.19/0.49 tff(210,plain,
% 0.19/0.49 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | ((~defined(domain(a), domain(compose(b, a)))) | product(domain(a), domain(compose(b, a)), domain(compose(b, a))) | (~identity_map(domain(a))))),
% 0.19/0.49 inference(quant_inst,[status(thm)],[])).
% 0.19/0.49 tff(211,plain,
% 0.19/0.49 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, Y) | (~identity_map(X)))) | (~identity_map(domain(a))) | product(domain(a), domain(compose(b, a)), domain(compose(b, a))) | (~defined(domain(a), domain(compose(b, a))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[210, 209])).
% 0.19/0.49 tff(212,plain,
% 0.19/0.49 (product(domain(a), domain(compose(b, a)), domain(compose(b, a))) | (~defined(domain(a), domain(compose(b, a))))),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[211, 39, 29])).
% 0.19/0.49 tff(213,plain,
% 0.19/0.49 (product(domain(a), domain(compose(b, a)), domain(compose(b, a)))),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[212, 205])).
% 0.19/0.49 tff(214,plain,
% 0.19/0.49 (product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(compose(b, a))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[213, 80])).
% 0.19/0.49 tff(215,plain,
% 0.19/0.49 (defined(domain(domain(a)), domain(domain(compose(b, a))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[205, 82])).
% 0.19/0.49 tff(216,plain,
% 0.19/0.49 ((~![X: $i] : identity_map(domain(X))) | identity_map(domain(domain(compose(b, a))))),
% 0.19/0.49 inference(quant_inst,[status(thm)],[])).
% 0.19/0.49 tff(217,plain,
% 0.19/0.49 (identity_map(domain(domain(compose(b, a))))),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[216, 27])).
% 0.19/0.49 tff(218,plain,
% 0.19/0.49 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | ((~identity_map(domain(domain(compose(b, a))))) | product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a))) | (~defined(domain(domain(a)), domain(domain(compose(b, a))))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | (~identity_map(domain(domain(compose(b, a))))) | product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a))) | (~defined(domain(domain(a)), domain(domain(compose(b, a))))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(219,plain,
% 0.19/0.49 (((~defined(domain(domain(a)), domain(domain(compose(b, a))))) | (~identity_map(domain(domain(compose(b, a))))) | product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a)))) <=> ((~identity_map(domain(domain(compose(b, a))))) | product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a))) | (~defined(domain(domain(a)), domain(domain(compose(b, a))))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(220,plain,
% 0.19/0.49 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | ((~defined(domain(domain(a)), domain(domain(compose(b, a))))) | (~identity_map(domain(domain(compose(b, a))))) | product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | ((~identity_map(domain(domain(compose(b, a))))) | product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a))) | (~defined(domain(domain(a)), domain(domain(compose(b, a)))))))),
% 0.19/0.49 inference(monotonicity,[status(thm)],[219])).
% 0.19/0.49 tff(221,plain,
% 0.19/0.49 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | ((~defined(domain(domain(a)), domain(domain(compose(b, a))))) | (~identity_map(domain(domain(compose(b, a))))) | product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | (~identity_map(domain(domain(compose(b, a))))) | product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a))) | (~defined(domain(domain(a)), domain(domain(compose(b, a))))))),
% 0.19/0.49 inference(transitivity,[status(thm)],[220, 218])).
% 0.19/0.49 tff(222,plain,
% 0.19/0.49 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | ((~defined(domain(domain(a)), domain(domain(compose(b, a))))) | (~identity_map(domain(domain(compose(b, a))))) | product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a))))),
% 0.19/0.49 inference(quant_inst,[status(thm)],[])).
% 0.19/0.49 tff(223,plain,
% 0.19/0.49 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | (~identity_map(Y)) | product(X, Y, X))) | (~identity_map(domain(domain(compose(b, a))))) | product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a))) | (~defined(domain(domain(a)), domain(domain(compose(b, a)))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[222, 221])).
% 0.19/0.49 tff(224,plain,
% 0.19/0.49 (product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a))) | (~defined(domain(domain(a)), domain(domain(compose(b, a)))))),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[223, 152, 217])).
% 0.19/0.49 tff(225,plain,
% 0.19/0.49 (product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a)))),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[224, 215])).
% 0.19/0.49 tff(226,plain,
% 0.19/0.49 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(compose(b, a))))) | (~product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a)))) | (domain(domain(compose(b, a))) = domain(domain(a))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(compose(b, a))))) | (~product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a)))) | (domain(domain(compose(b, a))) = domain(domain(a))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(227,plain,
% 0.19/0.49 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(compose(b, a))))) | (~product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a)))) | (domain(domain(compose(b, a))) = domain(domain(a))))),
% 0.19/0.49 inference(quant_inst,[status(thm)],[])).
% 0.19/0.49 tff(228,plain,
% 0.19/0.49 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(compose(b, a))))) | (~product(domain(domain(a)), domain(domain(compose(b, a))), domain(domain(a)))) | (domain(domain(compose(b, a))) = domain(domain(a)))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[227, 226])).
% 0.19/0.49 tff(229,plain,
% 0.19/0.49 (domain(domain(compose(b, a))) = domain(domain(a))),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[228, 56, 225, 214])).
% 0.19/0.49 tff(230,plain,
% 0.19/0.49 (domain(compose(b, a)) = domain(a)),
% 0.19/0.49 inference(transitivity,[status(thm)],[77, 229, 61])).
% 0.19/0.49 tff(231,plain,
% 0.19/0.49 ((~(domain(compose(b, a)) = domain(a))) <=> (~(domain(compose(b, a)) = domain(a)))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(232,axiom,(~(domain(compose(b, a)) = domain(a))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_domain_of_ba_equals_domain_of_a')).
% 0.19/0.49 tff(233,plain,
% 0.19/0.49 (~(domain(compose(b, a)) = domain(a))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[232, 231])).
% 0.19/0.49 tff(234,plain,
% 0.19/0.49 ($false),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[233, 230])).
% 0.19/0.49 % SZS output end Proof
% 0.19/0.49 % E exiting
%------------------------------------------------------------------------------