%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : CAT002-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.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 05:00:18 AM UTC 2025
% Result : Unsatisfiable 0.79s 1.00s
% Output : Proof 0.85s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : CAT002-1 : TPTP v9.0.0. Released v1.0.0.
% 0.12/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n001.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 : Thu Jun 19 11:55:33 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.79/1.00 % SZS status Unsatisfiable
% 0.79/1.00 % SZS output start Proof
% 0.79/1.00 tff(compose_type, type, (
% 0.79/1.00 compose: ( $i * $i ) > $i)).
% 0.79/1.00 tff(domain_type, type, (
% 0.79/1.00 domain: $i > $i)).
% 0.79/1.00 tff(h_type, type, (
% 0.79/1.00 h: $i)).
% 0.79/1.00 tff(g_type, type, (
% 0.79/1.00 g: $i)).
% 0.79/1.00 tff(product_type, type, (
% 0.79/1.00 product: ( $i * $i * $i ) > $o)).
% 0.79/1.00 tff(defined_type, type, (
% 0.79/1.00 defined: ( $i * $i ) > $o)).
% 0.79/1.00 tff(b_type, type, (
% 0.79/1.00 b: $i)).
% 0.79/1.00 tff(c_type, type, (
% 0.79/1.00 c: $i)).
% 0.79/1.00 tff(a_type, type, (
% 0.79/1.00 a: $i)).
% 0.79/1.00 tff(codomain_type, type, (
% 0.79/1.00 codomain: $i > $i)).
% 0.79/1.00 tff(d_type, type, (
% 0.79/1.00 d: $i)).
% 0.79/1.00 tff(identity_map_type, type, (
% 0.79/1.00 identity_map: $i > $o)).
% 0.79/1.00 tff(1,plain,
% 0.79/1.00 ((g = h) <=> (h = g)),
% 0.79/1.00 inference(commutativity,[status(thm)],[])).
% 0.79/1.00 tff(2,plain,
% 0.79/1.00 (^[X: $i] : refl(defined(X, domain(X)) <=> defined(X, domain(X)))),
% 0.79/1.00 inference(bind,[status(th)],[])).
% 0.79/1.00 tff(3,plain,
% 0.79/1.00 (![X: $i] : defined(X, domain(X)) <=> ![X: $i] : defined(X, domain(X))),
% 0.79/1.00 inference(quant_intro,[status(thm)],[2])).
% 0.79/1.00 tff(4,plain,
% 0.79/1.00 (![X: $i] : defined(X, domain(X)) <=> ![X: $i] : defined(X, domain(X))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(5,axiom,(![X: $i] : defined(X, domain(X))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','mapping_from_x_to_its_domain')).
% 0.79/1.00 tff(6,plain,
% 0.79/1.00 (![X: $i] : defined(X, domain(X))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.79/1.00 tff(7,plain,(
% 0.79/1.00 ![X: $i] : defined(X, domain(X))),
% 0.79/1.00 inference(skolemize,[status(sab)],[6])).
% 0.79/1.00 tff(8,plain,
% 0.79/1.00 (![X: $i] : defined(X, domain(X))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.79/1.00 tff(9,plain,
% 0.79/1.00 ((~![X: $i] : defined(X, domain(X))) | defined(h, domain(h))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(10,plain,
% 0.79/1.00 (defined(h, domain(h))),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[9, 8])).
% 0.79/1.00 tff(11,plain,
% 0.79/1.00 (^[Y: $i, X: $i] : refl(((~defined(X, Y)) | product(X, Y, compose(X, Y))) <=> ((~defined(X, Y)) | product(X, Y, compose(X, Y))))),
% 0.79/1.00 inference(bind,[status(th)],[])).
% 0.79/1.00 tff(12,plain,
% 0.79/1.00 (![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)))),
% 0.79/1.00 inference(quant_intro,[status(thm)],[11])).
% 0.79/1.00 tff(13,plain,
% 0.79/1.00 (![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)))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(14,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')).
% 0.79/1.00 tff(15,plain,
% 0.79/1.00 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.79/1.00 tff(16,plain,(
% 0.79/1.00 ![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))),
% 0.79/1.00 inference(skolemize,[status(sab)],[15])).
% 0.79/1.00 tff(17,plain,
% 0.79/1.00 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.79/1.00 tff(18,plain,
% 0.79/1.00 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(h, domain(h))) | product(h, domain(h), compose(h, domain(h))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(h, domain(h))) | product(h, domain(h), compose(h, domain(h))))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(19,plain,
% 0.79/1.00 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(h, domain(h))) | product(h, domain(h), compose(h, domain(h))))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(20,plain,
% 0.79/1.00 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(h, domain(h))) | product(h, domain(h), compose(h, domain(h)))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[19, 18])).
% 0.79/1.00 tff(21,plain,
% 0.79/1.00 (product(h, domain(h), compose(h, domain(h)))),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[20, 17, 10])).
% 0.79/1.00 tff(22,plain,
% 0.79/1.00 (^[X: $i] : refl(product(X, domain(X), X) <=> product(X, domain(X), X))),
% 0.79/1.00 inference(bind,[status(th)],[])).
% 0.79/1.00 tff(23,plain,
% 0.79/1.00 (![X: $i] : product(X, domain(X), X) <=> ![X: $i] : product(X, domain(X), X)),
% 0.79/1.00 inference(quant_intro,[status(thm)],[22])).
% 0.79/1.00 tff(24,plain,
% 0.79/1.00 (![X: $i] : product(X, domain(X), X) <=> ![X: $i] : product(X, domain(X), X)),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(25,axiom,(![X: $i] : product(X, domain(X), X)), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','product_on_domain')).
% 0.79/1.00 tff(26,plain,
% 0.79/1.00 (![X: $i] : product(X, domain(X), X)),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[25, 24])).
% 0.79/1.00 tff(27,plain,(
% 0.79/1.00 ![X: $i] : product(X, domain(X), X)),
% 0.79/1.00 inference(skolemize,[status(sab)],[26])).
% 0.79/1.00 tff(28,plain,
% 0.79/1.00 (![X: $i] : product(X, domain(X), X)),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[27, 23])).
% 0.79/1.00 tff(29,plain,
% 0.79/1.00 ((~![X: $i] : product(X, domain(X), X)) | product(h, domain(h), h)),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(30,plain,
% 0.79/1.00 (product(h, domain(h), h)),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[29, 28])).
% 0.79/1.00 tff(31,plain,
% 0.79/1.00 (^[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)))),
% 0.79/1.00 inference(bind,[status(th)],[])).
% 0.79/1.00 tff(32,plain,
% 0.79/1.00 (![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))),
% 0.79/1.00 inference(quant_intro,[status(thm)],[31])).
% 0.79/1.00 tff(33,plain,
% 0.79/1.00 (![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))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(34,plain,
% 0.79/1.00 (^[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)))),
% 0.79/1.00 inference(bind,[status(th)],[])).
% 0.79/1.00 tff(35,plain,
% 0.79/1.00 (![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))),
% 0.79/1.00 inference(quant_intro,[status(thm)],[34])).
% 0.79/1.00 tff(36,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')).
% 0.79/1.00 tff(37,plain,
% 0.79/1.00 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.79/1.00 tff(38,plain,
% 0.79/1.00 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[37, 33])).
% 0.79/1.00 tff(39,plain,(
% 0.79/1.00 ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.79/1.00 inference(skolemize,[status(sab)],[38])).
% 0.79/1.00 tff(40,plain,
% 0.79/1.00 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[39, 32])).
% 0.79/1.00 tff(41,plain,
% 0.79/1.00 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(h, domain(h), h)) | (~product(h, domain(h), compose(h, domain(h)))) | (h = compose(h, domain(h))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(h, domain(h), h)) | (~product(h, domain(h), compose(h, domain(h)))) | (h = compose(h, domain(h))))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(42,plain,
% 0.79/1.00 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(h, domain(h), h)) | (~product(h, domain(h), compose(h, domain(h)))) | (h = compose(h, domain(h))))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(43,plain,
% 0.79/1.00 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(h, domain(h), h)) | (~product(h, domain(h), compose(h, domain(h)))) | (h = compose(h, domain(h)))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.79/1.00 tff(44,plain,
% 0.79/1.00 (h = compose(h, domain(h))),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[43, 40, 30, 21])).
% 0.79/1.00 tff(45,plain,
% 0.79/1.00 (compose(h, domain(h)) = h),
% 0.79/1.00 inference(symmetry,[status(thm)],[44])).
% 0.79/1.00 tff(46,plain,
% 0.79/1.00 ((~![X: $i] : product(X, domain(X), X)) | product(g, domain(g), g)),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(47,plain,
% 0.79/1.00 (product(g, domain(g), g)),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[46, 28])).
% 0.79/1.00 tff(48,plain,
% 0.79/1.00 (^[Z: $i, Y: $i, X: $i] : refl(((~product(X, Y, Z)) | defined(X, Y)) <=> ((~product(X, Y, Z)) | defined(X, Y)))),
% 0.79/1.00 inference(bind,[status(th)],[])).
% 0.79/1.00 tff(49,plain,
% 0.79/1.00 (![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))),
% 0.79/1.00 inference(quant_intro,[status(thm)],[48])).
% 0.79/1.00 tff(50,plain,
% 0.79/1.00 (![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))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(51,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')).
% 0.79/1.00 tff(52,plain,
% 0.79/1.00 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.79/1.00 tff(53,plain,(
% 0.79/1.00 ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))),
% 0.79/1.00 inference(skolemize,[status(sab)],[52])).
% 0.79/1.00 tff(54,plain,
% 0.79/1.00 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[53, 49])).
% 0.79/1.00 tff(55,plain,
% 0.79/1.00 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(g, domain(g), g)) | defined(g, domain(g)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(g, domain(g), g)) | defined(g, domain(g)))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(56,plain,
% 0.79/1.00 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(g, domain(g), g)) | defined(g, domain(g)))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(57,plain,
% 0.79/1.00 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(g, domain(g), g)) | defined(g, domain(g))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.79/1.00 tff(58,plain,
% 0.79/1.00 (defined(g, domain(g))),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[57, 54, 47])).
% 0.79/1.00 tff(59,plain,
% 0.79/1.00 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(g, domain(g))) | product(g, domain(g), compose(g, domain(g))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(g, domain(g))) | product(g, domain(g), compose(g, domain(g))))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(60,plain,
% 0.79/1.00 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(g, domain(g))) | product(g, domain(g), compose(g, domain(g))))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(61,plain,
% 0.79/1.00 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(g, domain(g))) | product(g, domain(g), compose(g, domain(g)))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[60, 59])).
% 0.79/1.00 tff(62,plain,
% 0.79/1.00 (product(g, domain(g), compose(g, domain(g)))),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[61, 17, 58])).
% 0.79/1.00 tff(63,plain,
% 0.79/1.00 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(g, domain(g), g)) | (~product(g, domain(g), compose(g, domain(g)))) | (g = compose(g, domain(g))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(g, domain(g), g)) | (~product(g, domain(g), compose(g, domain(g)))) | (g = compose(g, domain(g))))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(64,plain,
% 0.79/1.00 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(g, domain(g), g)) | (~product(g, domain(g), compose(g, domain(g)))) | (g = compose(g, domain(g))))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(65,plain,
% 0.79/1.00 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(g, domain(g), g)) | (~product(g, domain(g), compose(g, domain(g)))) | (g = compose(g, domain(g)))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.79/1.00 tff(66,plain,
% 0.79/1.00 (g = compose(g, domain(g))),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[65, 40, 47, 62])).
% 0.79/1.00 tff(67,plain,
% 0.79/1.00 (compose(g, domain(g)) = g),
% 0.79/1.00 inference(symmetry,[status(thm)],[66])).
% 0.79/1.00 tff(68,plain,
% 0.79/1.00 ((compose(g, domain(g)) = compose(h, domain(h))) <=> (g = h)),
% 0.79/1.00 inference(monotonicity,[status(thm)],[67, 45])).
% 0.79/1.00 tff(69,plain,
% 0.79/1.00 ((compose(g, domain(g)) = compose(h, domain(h))) <=> (h = g)),
% 0.79/1.00 inference(transitivity,[status(thm)],[68, 1])).
% 0.79/1.00 tff(70,plain,
% 0.79/1.00 ((h = g) <=> (compose(g, domain(g)) = compose(h, domain(h)))),
% 0.79/1.00 inference(symmetry,[status(thm)],[69])).
% 0.79/1.00 tff(71,plain,
% 0.79/1.00 ((~(h = g)) <=> (~(compose(g, domain(g)) = compose(h, domain(h))))),
% 0.79/1.00 inference(monotonicity,[status(thm)],[70])).
% 0.79/1.00 tff(72,plain,
% 0.79/1.00 ((~(h = g)) <=> (~(h = g))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(73,axiom,(~(h = g)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_h_equals_g')).
% 0.79/1.00 tff(74,plain,
% 0.79/1.00 (~(h = g)),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.79/1.00 tff(75,plain,
% 0.79/1.00 (~(compose(g, domain(g)) = compose(h, domain(h)))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[74, 71])).
% 0.79/1.00 tff(76,plain,
% 0.79/1.00 (^[X: $i] : refl(defined(codomain(X), X) <=> defined(codomain(X), X))),
% 0.79/1.00 inference(bind,[status(th)],[])).
% 0.79/1.00 tff(77,plain,
% 0.79/1.00 (![X: $i] : defined(codomain(X), X) <=> ![X: $i] : defined(codomain(X), X)),
% 0.79/1.00 inference(quant_intro,[status(thm)],[76])).
% 0.79/1.00 tff(78,plain,
% 0.79/1.00 (![X: $i] : defined(codomain(X), X) <=> ![X: $i] : defined(codomain(X), X)),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(79,axiom,(![X: $i] : defined(codomain(X), X)), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','mapping_from_codomain_of_x_to_x')).
% 0.79/1.00 tff(80,plain,
% 0.79/1.00 (![X: $i] : defined(codomain(X), X)),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[79, 78])).
% 0.79/1.00 tff(81,plain,(
% 0.79/1.00 ![X: $i] : defined(codomain(X), X)),
% 0.79/1.00 inference(skolemize,[status(sab)],[80])).
% 0.79/1.00 tff(82,plain,
% 0.79/1.00 (![X: $i] : defined(codomain(X), X)),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[81, 77])).
% 0.79/1.00 tff(83,plain,
% 0.79/1.00 ((~![X: $i] : defined(codomain(X), X)) | defined(codomain(a), a)),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(84,plain,
% 0.79/1.00 (defined(codomain(a), a)),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[83, 82])).
% 0.79/1.00 tff(85,plain,
% 0.79/1.00 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(codomain(a), a)) | product(codomain(a), a, compose(codomain(a), a)))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(codomain(a), a)) | product(codomain(a), a, compose(codomain(a), a)))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(86,plain,
% 0.79/1.00 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(codomain(a), a)) | product(codomain(a), a, compose(codomain(a), a)))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(87,plain,
% 0.79/1.00 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(codomain(a), a)) | product(codomain(a), a, compose(codomain(a), a))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[86, 85])).
% 0.79/1.00 tff(88,plain,
% 0.79/1.00 (product(codomain(a), a, compose(codomain(a), a))),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[87, 17, 84])).
% 0.79/1.00 tff(89,plain,
% 0.79/1.00 (^[X: $i] : refl(product(codomain(X), X, X) <=> product(codomain(X), X, X))),
% 0.79/1.00 inference(bind,[status(th)],[])).
% 0.79/1.00 tff(90,plain,
% 0.79/1.00 (![X: $i] : product(codomain(X), X, X) <=> ![X: $i] : product(codomain(X), X, X)),
% 0.79/1.00 inference(quant_intro,[status(thm)],[89])).
% 0.79/1.00 tff(91,plain,
% 0.79/1.00 (![X: $i] : product(codomain(X), X, X) <=> ![X: $i] : product(codomain(X), X, X)),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(92,axiom,(![X: $i] : product(codomain(X), X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','product_on_codomain')).
% 0.79/1.00 tff(93,plain,
% 0.79/1.00 (![X: $i] : product(codomain(X), X, X)),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[92, 91])).
% 0.79/1.00 tff(94,plain,(
% 0.79/1.00 ![X: $i] : product(codomain(X), X, X)),
% 0.79/1.00 inference(skolemize,[status(sab)],[93])).
% 0.79/1.00 tff(95,plain,
% 0.79/1.00 (![X: $i] : product(codomain(X), X, X)),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[94, 90])).
% 0.79/1.00 tff(96,plain,
% 0.79/1.00 ((~![X: $i] : product(codomain(X), X, X)) | product(codomain(a), a, a)),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(97,plain,
% 0.79/1.00 (product(codomain(a), a, a)),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[96, 95])).
% 0.79/1.00 tff(98,plain,
% 0.79/1.00 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(codomain(a), a, a)) | (~product(codomain(a), a, compose(codomain(a), a))) | (a = compose(codomain(a), a)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(codomain(a), a, a)) | (~product(codomain(a), a, compose(codomain(a), a))) | (a = compose(codomain(a), a)))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(99,plain,
% 0.79/1.00 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(codomain(a), a, a)) | (~product(codomain(a), a, compose(codomain(a), a))) | (a = compose(codomain(a), a)))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(100,plain,
% 0.79/1.00 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(codomain(a), a, a)) | (~product(codomain(a), a, compose(codomain(a), a))) | (a = compose(codomain(a), a))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.79/1.00 tff(101,plain,
% 0.79/1.00 (a = compose(codomain(a), a)),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[100, 40, 97, 88])).
% 0.79/1.00 tff(102,plain,
% 0.79/1.00 (product(a, compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h)) <=> product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h))),
% 0.79/1.00 inference(monotonicity,[status(thm)],[101])).
% 0.79/1.00 tff(103,plain,
% 0.79/1.00 (product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h)) <=> product(a, compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h))),
% 0.79/1.00 inference(symmetry,[status(thm)],[102])).
% 0.79/1.00 tff(104,assumption,(~defined(compose(b, domain(b)), compose(g, domain(g)))), introduced(assumption)).
% 0.79/1.00 tff(105,plain,
% 0.79/1.00 ((~![X: $i] : product(codomain(X), X, X)) | product(codomain(compose(g, domain(g))), compose(g, domain(g)), compose(g, domain(g)))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(106,plain,
% 0.79/1.00 (product(codomain(compose(g, domain(g))), compose(g, domain(g)), compose(g, domain(g)))),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[105, 95])).
% 0.79/1.00 tff(107,plain,
% 0.79/1.00 (product(c, h, d) <=> product(c, h, d)),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(108,axiom,(product(c, h, d)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','ch_equals_d')).
% 0.79/1.00 tff(109,plain,
% 0.79/1.00 (product(c, h, d)),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[108, 107])).
% 0.79/1.00 tff(110,plain,
% 0.79/1.00 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(c, h, d)) | defined(c, h))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(c, h, d)) | defined(c, h))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(111,plain,
% 0.79/1.00 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(c, h, d)) | defined(c, h))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(112,plain,
% 0.79/1.00 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(c, h, d)) | defined(c, h)),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[111, 110])).
% 0.79/1.00 tff(113,plain,
% 0.79/1.00 (defined(c, h)),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[112, 54, 109])).
% 0.79/1.00 tff(114,plain,
% 0.79/1.00 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(c, h)) | product(c, h, compose(c, h)))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(c, h)) | product(c, h, compose(c, h)))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(115,plain,
% 0.79/1.00 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(c, h)) | product(c, h, compose(c, h)))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(116,plain,
% 0.79/1.00 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(c, h)) | product(c, h, compose(c, h))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.79/1.00 tff(117,plain,
% 0.79/1.00 (product(c, h, compose(c, h))),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[116, 17, 113])).
% 0.79/1.00 tff(118,plain,
% 0.79/1.00 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(c, h, d)) | (~product(c, h, compose(c, h))) | (d = compose(c, h)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(c, h, d)) | (~product(c, h, compose(c, h))) | (d = compose(c, h)))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(119,plain,
% 0.79/1.00 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(c, h, d)) | (~product(c, h, compose(c, h))) | (d = compose(c, h)))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(120,plain,
% 0.79/1.00 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(c, h, d)) | (~product(c, h, compose(c, h))) | (d = compose(c, h))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[119, 118])).
% 0.79/1.00 tff(121,plain,
% 0.79/1.00 (d = compose(c, h)),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[120, 40, 109, 117])).
% 0.79/1.00 tff(122,plain,
% 0.79/1.00 (compose(c, h) = d),
% 0.79/1.00 inference(symmetry,[status(thm)],[121])).
% 0.79/1.00 tff(123,plain,
% 0.79/1.00 (product(a, b, c) <=> product(a, b, c)),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(124,axiom,(product(a, b, c)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','ab_equals_c')).
% 0.79/1.00 tff(125,plain,
% 0.79/1.00 (product(a, b, c)),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[124, 123])).
% 0.79/1.00 tff(126,plain,
% 0.79/1.00 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(a, b, c)) | defined(a, b))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(a, b, c)) | defined(a, b))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(127,plain,
% 0.79/1.00 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(a, b, c)) | defined(a, b))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(128,plain,
% 0.79/1.00 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(a, b, c)) | defined(a, b)),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[127, 126])).
% 0.79/1.00 tff(129,plain,
% 0.79/1.00 (defined(a, b)),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[128, 54, 125])).
% 0.79/1.00 tff(130,plain,
% 0.79/1.00 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(a, b)) | product(a, b, compose(a, b)))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(a, b)) | product(a, b, compose(a, b)))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(131,plain,
% 0.79/1.00 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(a, b)) | product(a, b, compose(a, b)))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(132,plain,
% 0.79/1.00 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(a, b)) | product(a, b, compose(a, b))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[131, 130])).
% 0.79/1.00 tff(133,plain,
% 0.79/1.00 (product(a, b, compose(a, b))),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[132, 17, 129])).
% 0.79/1.00 tff(134,plain,
% 0.79/1.00 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(a, b, c)) | (~product(a, b, compose(a, b))) | (c = compose(a, b)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(a, b, c)) | (~product(a, b, compose(a, b))) | (c = compose(a, b)))),
% 0.79/1.00 inference(rewrite,[status(thm)],[])).
% 0.79/1.00 tff(135,plain,
% 0.79/1.00 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(a, b, c)) | (~product(a, b, compose(a, b))) | (c = compose(a, b)))),
% 0.79/1.00 inference(quant_inst,[status(thm)],[])).
% 0.79/1.00 tff(136,plain,
% 0.79/1.00 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(a, b, c)) | (~product(a, b, compose(a, b))) | (c = compose(a, b))),
% 0.79/1.00 inference(modus_ponens,[status(thm)],[135, 134])).
% 0.79/1.00 tff(137,plain,
% 0.79/1.00 (c = compose(a, b)),
% 0.79/1.00 inference(unit_resolution,[status(thm)],[136, 40, 125, 133])).
% 0.79/1.00 tff(138,plain,
% 0.79/1.00 (compose(a, b) = c),
% 0.79/1.00 inference(symmetry,[status(thm)],[137])).
% 0.85/1.00 tff(139,plain,
% 0.85/1.00 (product(compose(a, b), g, compose(c, h)) <=> product(c, g, d)),
% 0.85/1.00 inference(monotonicity,[status(thm)],[138, 122])).
% 0.85/1.00 tff(140,plain,
% 0.85/1.00 (product(c, g, d) <=> product(compose(a, b), g, compose(c, h))),
% 0.85/1.00 inference(symmetry,[status(thm)],[139])).
% 0.85/1.00 tff(141,plain,
% 0.85/1.00 (product(c, g, d) <=> product(c, g, d)),
% 0.85/1.00 inference(rewrite,[status(thm)],[])).
% 0.85/1.00 tff(142,axiom,(product(c, g, d)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','cg_equals_d')).
% 0.85/1.00 tff(143,plain,
% 0.85/1.00 (product(c, g, d)),
% 0.85/1.00 inference(modus_ponens,[status(thm)],[142, 141])).
% 0.85/1.00 tff(144,plain,
% 0.85/1.00 (product(compose(a, b), g, compose(c, h))),
% 0.85/1.00 inference(modus_ponens,[status(thm)],[143, 140])).
% 0.85/1.00 tff(145,plain,
% 0.85/1.00 (defined(compose(a, b), g) <=> defined(c, g)),
% 0.85/1.00 inference(monotonicity,[status(thm)],[138])).
% 0.85/1.00 tff(146,plain,
% 0.85/1.00 (defined(c, g) <=> defined(compose(a, b), g)),
% 0.85/1.00 inference(symmetry,[status(thm)],[145])).
% 0.85/1.00 tff(147,plain,
% 0.85/1.00 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(c, g, d)) | defined(c, g))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(c, g, d)) | defined(c, g))),
% 0.85/1.00 inference(rewrite,[status(thm)],[])).
% 0.85/1.00 tff(148,plain,
% 0.85/1.00 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(c, g, d)) | defined(c, g))),
% 0.85/1.00 inference(quant_inst,[status(thm)],[])).
% 0.85/1.00 tff(149,plain,
% 0.85/1.00 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(c, g, d)) | defined(c, g)),
% 0.85/1.00 inference(modus_ponens,[status(thm)],[148, 147])).
% 0.85/1.00 tff(150,plain,
% 0.85/1.00 (defined(c, g)),
% 0.85/1.00 inference(unit_resolution,[status(thm)],[149, 54, 143])).
% 0.85/1.00 tff(151,plain,
% 0.85/1.00 (defined(compose(a, b), g)),
% 0.85/1.00 inference(modus_ponens,[status(thm)],[150, 146])).
% 0.85/1.00 tff(152,plain,
% 0.85/1.00 (^[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))))),
% 0.85/1.00 inference(bind,[status(th)],[])).
% 0.85/1.00 tff(153,plain,
% 0.85/1.00 (![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)))),
% 0.85/1.00 inference(quant_intro,[status(thm)],[152])).
% 0.85/1.00 tff(154,plain,
% 0.85/1.00 (![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)))),
% 0.85/1.00 inference(rewrite,[status(thm)],[])).
% 0.85/1.00 tff(155,plain,
% 0.85/1.00 (^[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)))))),
% 0.85/1.00 inference(bind,[status(th)],[])).
% 0.85/1.00 tff(156,plain,
% 0.85/1.00 (![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)))),
% 0.85/1.01 inference(quant_intro,[status(thm)],[155])).
% 0.85/1.01 tff(157,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')).
% 0.85/1.01 tff(158,plain,
% 0.85/1.01 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.85/1.01 inference(modus_ponens,[status(thm)],[157, 156])).
% 0.85/1.01 tff(159,plain,
% 0.85/1.01 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.85/1.01 inference(modus_ponens,[status(thm)],[158, 154])).
% 0.85/1.01 tff(160,plain,(
% 0.85/1.01 ![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.85/1.01 inference(skolemize,[status(sab)],[159])).
% 0.85/1.01 tff(161,plain,
% 0.85/1.01 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.85/1.01 inference(modus_ponens,[status(thm)],[160, 153])).
% 0.85/1.01 tff(162,plain,
% 0.85/1.01 (((~![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(a, b), g)) | (~product(compose(a, b), g, compose(c, h))) | defined(compose(c, h), domain(g)) | (~product(g, domain(g), g)))) <=> ((~![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(a, b), g)) | (~product(compose(a, b), g, compose(c, h))) | defined(compose(c, h), domain(g)) | (~product(g, domain(g), g)))),
% 0.85/1.01 inference(rewrite,[status(thm)],[])).
% 0.85/1.01 tff(163,plain,
% 0.85/1.01 (((~defined(compose(a, b), g)) | defined(compose(c, h), domain(g)) | (~product(g, domain(g), g)) | (~product(compose(a, b), g, compose(c, h)))) <=> ((~defined(compose(a, b), g)) | (~product(compose(a, b), g, compose(c, h))) | defined(compose(c, h), domain(g)) | (~product(g, domain(g), g)))),
% 0.85/1.01 inference(rewrite,[status(thm)],[])).
% 0.85/1.01 tff(164,plain,
% 0.85/1.01 (((~![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(a, b), g)) | defined(compose(c, h), domain(g)) | (~product(g, domain(g), g)) | (~product(compose(a, b), g, compose(c, h))))) <=> ((~![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(a, b), g)) | (~product(compose(a, b), g, compose(c, h))) | defined(compose(c, h), domain(g)) | (~product(g, domain(g), g))))),
% 0.85/1.01 inference(monotonicity,[status(thm)],[163])).
% 0.85/1.01 tff(165,plain,
% 0.85/1.01 (((~![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(a, b), g)) | defined(compose(c, h), domain(g)) | (~product(g, domain(g), g)) | (~product(compose(a, b), g, compose(c, h))))) <=> ((~![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(a, b), g)) | (~product(compose(a, b), g, compose(c, h))) | defined(compose(c, h), domain(g)) | (~product(g, domain(g), g)))),
% 0.85/1.01 inference(transitivity,[status(thm)],[164, 162])).
% 0.85/1.01 tff(166,plain,
% 0.85/1.01 ((~![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(a, b), g)) | defined(compose(c, h), domain(g)) | (~product(g, domain(g), g)) | (~product(compose(a, b), g, compose(c, h))))),
% 0.85/1.01 inference(quant_inst,[status(thm)],[])).
% 0.85/1.01 tff(167,plain,
% 0.85/1.01 ((~![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(a, b), g)) | (~product(compose(a, b), g, compose(c, h))) | defined(compose(c, h), domain(g)) | (~product(g, domain(g), g))),
% 0.85/1.01 inference(modus_ponens,[status(thm)],[166, 165])).
% 0.85/1.01 tff(168,plain,
% 0.85/1.01 (defined(compose(c, h), domain(g))),
% 0.85/1.01 inference(unit_resolution,[status(thm)],[167, 161, 151, 144, 47])).
% 0.85/1.01 tff(169,plain,
% 0.85/1.01 (^[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))))),
% 0.85/1.01 inference(bind,[status(th)],[])).
% 0.85/1.01 tff(170,plain,
% 0.85/1.01 (![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)))),
% 0.85/1.01 inference(quant_intro,[status(thm)],[169])).
% 0.85/1.01 tff(171,plain,
% 0.85/1.01 (![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)))),
% 0.85/1.01 inference(rewrite,[status(thm)],[])).
% 0.85/1.01 tff(172,plain,
% 0.85/1.01 (^[Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : trans(monotonicity(rewrite((((~product(X, Y, Xy)) | (~product(Y, Z, Yz))) | (~defined(Xy, Z))) <=> ((~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))), (((((~product(X, Y, Xy)) | (~product(Y, Z, Yz))) | (~defined(Xy, Z))) | defined(X, Yz)) <=> (((~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy))) | defined(X, Yz)))), rewrite((((~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy))) | defined(X, Yz)) <=> (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))), (((((~product(X, Y, Xy)) | (~product(Y, Z, Yz))) | (~defined(Xy, Z))) | defined(X, Yz)) <=> (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))))),
% 0.85/1.01 inference(bind,[status(th)],[])).
% 0.85/1.01 tff(173,plain,
% 0.85/1.01 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((((~product(X, Y, Xy)) | (~product(Y, Z, Yz))) | (~defined(Xy, Z))) | defined(X, Yz)) <=> ![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.85/1.01 inference(quant_intro,[status(thm)],[172])).
% 0.85/1.01 tff(174,axiom,(![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((((~product(X, Y, Xy)) | (~product(Y, Z, Yz))) | (~defined(Xy, Z))) | defined(X, Yz))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','category_theory_axiom1')).
% 0.85/1.01 tff(175,plain,
% 0.85/1.01 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.85/1.01 inference(modus_ponens,[status(thm)],[174, 173])).
% 0.85/1.01 tff(176,plain,
% 0.85/1.01 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.85/1.01 inference(modus_ponens,[status(thm)],[175, 171])).
% 0.85/1.01 tff(177,plain,(
% 0.85/1.01 ![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.85/1.01 inference(skolemize,[status(sab)],[176])).
% 0.85/1.01 tff(178,plain,
% 0.85/1.01 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.85/1.01 inference(modus_ponens,[status(thm)],[177, 170])).
% 0.85/1.01 tff(179,plain,
% 0.85/1.01 (((~![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(a, b), g, compose(c, h))) | (~product(g, domain(g), compose(g, domain(g)))) | defined(compose(a, b), compose(g, domain(g))) | (~defined(compose(c, h), domain(g))))) <=> ((~![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(a, b), g, compose(c, h))) | (~product(g, domain(g), compose(g, domain(g)))) | defined(compose(a, b), compose(g, domain(g))) | (~defined(compose(c, h), domain(g))))),
% 0.85/1.01 inference(rewrite,[status(thm)],[])).
% 0.85/1.01 tff(180,plain,
% 0.85/1.01 ((defined(compose(a, b), compose(g, domain(g))) | (~defined(compose(c, h), domain(g))) | (~product(g, domain(g), compose(g, domain(g)))) | (~product(compose(a, b), g, compose(c, h)))) <=> ((~product(compose(a, b), g, compose(c, h))) | (~product(g, domain(g), compose(g, domain(g)))) | defined(compose(a, b), compose(g, domain(g))) | (~defined(compose(c, h), domain(g))))),
% 0.85/1.01 inference(rewrite,[status(thm)],[])).
% 0.85/1.01 tff(181,plain,
% 0.85/1.01 (((~![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(a, b), compose(g, domain(g))) | (~defined(compose(c, h), domain(g))) | (~product(g, domain(g), compose(g, domain(g)))) | (~product(compose(a, b), g, compose(c, h))))) <=> ((~![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(a, b), g, compose(c, h))) | (~product(g, domain(g), compose(g, domain(g)))) | defined(compose(a, b), compose(g, domain(g))) | (~defined(compose(c, h), domain(g)))))),
% 0.85/1.01 inference(monotonicity,[status(thm)],[180])).
% 0.85/1.01 tff(182,plain,
% 0.85/1.01 (((~![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(a, b), compose(g, domain(g))) | (~defined(compose(c, h), domain(g))) | (~product(g, domain(g), compose(g, domain(g)))) | (~product(compose(a, b), g, compose(c, h))))) <=> ((~![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(a, b), g, compose(c, h))) | (~product(g, domain(g), compose(g, domain(g)))) | defined(compose(a, b), compose(g, domain(g))) | (~defined(compose(c, h), domain(g))))),
% 0.85/1.01 inference(transitivity,[status(thm)],[181, 179])).
% 0.85/1.01 tff(183,plain,
% 0.85/1.01 ((~![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(a, b), compose(g, domain(g))) | (~defined(compose(c, h), domain(g))) | (~product(g, domain(g), compose(g, domain(g)))) | (~product(compose(a, b), g, compose(c, h))))),
% 0.85/1.01 inference(quant_inst,[status(thm)],[])).
% 0.85/1.01 tff(184,plain,
% 0.85/1.01 ((~![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(a, b), g, compose(c, h))) | (~product(g, domain(g), compose(g, domain(g)))) | defined(compose(a, b), compose(g, domain(g))) | (~defined(compose(c, h), domain(g)))),
% 0.85/1.01 inference(modus_ponens,[status(thm)],[183, 182])).
% 0.85/1.01 tff(185,plain,
% 0.85/1.01 (defined(compose(a, b), compose(g, domain(g)))),
% 0.85/1.01 inference(unit_resolution,[status(thm)],[184, 178, 144, 168, 62])).
% 0.85/1.01 tff(186,plain,
% 0.85/1.01 (^[Z: $i, Y: $i, X: $i, Yz: $i] : refl(((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz))) <=> ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz))))),
% 0.85/1.01 inference(bind,[status(th)],[])).
% 0.85/1.01 tff(187,plain,
% 0.85/1.01 (![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz))) <=> ![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))),
% 0.85/1.01 inference(quant_intro,[status(thm)],[186])).
% 0.85/1.01 tff(188,plain,
% 0.85/1.01 (![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz))) <=> ![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))),
% 0.85/1.01 inference(rewrite,[status(thm)],[])).
% 0.85/1.01 tff(189,plain,
% 0.85/1.01 (^[Z: $i, Y: $i, X: $i, Yz: $i] : trans(monotonicity(rewrite(((~product(Y, Z, Yz)) | (~defined(X, Yz))) <=> ((~defined(X, Yz)) | (~product(Y, Z, Yz)))), ((((~product(Y, Z, Yz)) | (~defined(X, Yz))) | defined(X, Y)) <=> (((~defined(X, Yz)) | (~product(Y, Z, Yz))) | defined(X, Y)))), rewrite((((~defined(X, Yz)) | (~product(Y, Z, Yz))) | defined(X, Y)) <=> ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))), ((((~product(Y, Z, Yz)) | (~defined(X, Yz))) | defined(X, Y)) <=> ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))))),
% 0.85/1.01 inference(bind,[status(th)],[])).
% 0.85/1.01 tff(190,plain,
% 0.85/1.01 (![Z: $i, Y: $i, X: $i, Yz: $i] : (((~product(Y, Z, Yz)) | (~defined(X, Yz))) | defined(X, Y)) <=> ![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))),
% 0.85/1.01 inference(quant_intro,[status(thm)],[189])).
% 0.85/1.01 tff(191,axiom,(![Z: $i, Y: $i, X: $i, Yz: $i] : (((~product(Y, Z, Yz)) | (~defined(X, Yz))) | defined(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','category_theory_axiom3')).
% 0.85/1.01 tff(192,plain,
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% 0.85/1.01 inference(modus_ponens,[status(thm)],[191, 190])).
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% 0.85/1.02 inference(bind,[status(th)],[])).
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% 0.85/1.02 inference(rewrite,[status(thm)],[])).
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% 0.85/1.02 inference(rewrite,[status(thm)],[])).
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% 0.85/1.02 inference(transitivity,[status(thm)],[292, 290])).
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% 0.85/1.02 inference(unit_resolution,[status(thm)],[295, 178, 289, 106, 287])).
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% 0.85/1.02 inference(quant_inst,[status(thm)],[])).
% 0.85/1.02 tff(298,plain,
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% 0.85/1.02 inference(unit_resolution,[status(thm)],[297, 28])).
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% 0.85/1.02 inference(rewrite,[status(thm)],[])).
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% 0.85/1.02 inference(quant_inst,[status(thm)],[])).
% 0.85/1.02 tff(301,plain,
% 0.85/1.02 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(b, domain(b), b)) | defined(b, domain(b))),
% 0.85/1.02 inference(modus_ponens,[status(thm)],[300, 299])).
% 0.85/1.02 tff(302,plain,
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% 0.85/1.02 inference(unit_resolution,[status(thm)],[301, 54, 298])).
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% 0.85/1.02 inference(rewrite,[status(thm)],[])).
% 0.85/1.02 tff(304,plain,
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% 0.85/1.02 inference(quant_inst,[status(thm)],[])).
% 0.85/1.02 tff(305,plain,
% 0.85/1.02 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(b, domain(b))) | product(b, domain(b), compose(b, domain(b)))),
% 0.85/1.02 inference(modus_ponens,[status(thm)],[304, 303])).
% 0.85/1.02 tff(306,plain,
% 0.85/1.02 (product(b, domain(b), compose(b, domain(b)))),
% 0.85/1.02 inference(unit_resolution,[status(thm)],[305, 17, 302])).
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% 0.85/1.02 inference(rewrite,[status(thm)],[])).
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% 0.85/1.02 inference(quant_inst,[status(thm)],[])).
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% 0.85/1.02 inference(modus_ponens,[status(thm)],[308, 307])).
% 0.85/1.02 tff(310,plain,
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% 0.85/1.02 tff(311,plain,
% 0.85/1.02 (compose(b, domain(b)) = b),
% 0.85/1.02 inference(symmetry,[status(thm)],[310])).
% 0.85/1.02 tff(312,plain,
% 0.85/1.02 (compose(codomain(a), a) = a),
% 0.85/1.02 inference(symmetry,[status(thm)],[101])).
% 0.85/1.02 tff(313,plain,
% 0.85/1.02 (product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b)) <=> product(a, b, compose(a, b))),
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% 0.85/1.02 tff(314,plain,
% 0.85/1.02 (product(a, b, compose(a, b)) <=> product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))),
% 0.85/1.02 inference(symmetry,[status(thm)],[313])).
% 0.85/1.02 tff(315,plain,
% 0.85/1.02 (product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))),
% 0.85/1.02 inference(modus_ponens,[status(thm)],[133, 314])).
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% 0.85/1.02 inference(rewrite,[status(thm)],[])).
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% 0.85/1.03 (defined(compose(b, domain(b)), codomain(compose(h, domain(h))))),
% 0.85/1.03 inference(unit_resolution,[status(thm)],[322, 315])).
% 0.85/1.03 tff(324,plain,
% 0.85/1.03 (^[Z: $i, Y: $i, X: $i] : refl(((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z))) <=> ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z))))),
% 0.85/1.03 inference(bind,[status(th)],[])).
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% 0.85/1.03 inference(quant_intro,[status(thm)],[324])).
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% 0.85/1.03 (![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z))) <=> ![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))),
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% 0.85/1.03 inference(bind,[status(th)],[])).
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% 0.85/1.03 inference(rewrite,[status(thm)],[])).
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% 0.85/1.03 inference(rewrite,[status(thm)],[])).
% 0.85/1.03 tff(336,plain,
% 0.85/1.03 (((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | ((~defined(compose(b, domain(b)), codomain(compose(h, domain(h))))) | defined(compose(b, domain(b)), compose(g, domain(g))) | (~identity_map(codomain(compose(h, domain(h))))) | (~defined(codomain(compose(h, domain(h))), compose(g, domain(g)))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | ((~identity_map(codomain(compose(h, domain(h))))) | defined(compose(b, domain(b)), compose(g, domain(g))) | (~defined(codomain(compose(h, domain(h))), compose(g, domain(g)))) | (~defined(compose(b, domain(b)), codomain(compose(h, domain(h)))))))),
% 0.85/1.03 inference(monotonicity,[status(thm)],[335])).
% 0.85/1.03 tff(337,plain,
% 0.85/1.03 (((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | ((~defined(compose(b, domain(b)), codomain(compose(h, domain(h))))) | defined(compose(b, domain(b)), compose(g, domain(g))) | (~identity_map(codomain(compose(h, domain(h))))) | (~defined(codomain(compose(h, domain(h))), compose(g, domain(g)))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | (~identity_map(codomain(compose(h, domain(h))))) | defined(compose(b, domain(b)), compose(g, domain(g))) | (~defined(codomain(compose(h, domain(h))), compose(g, domain(g)))) | (~defined(compose(b, domain(b)), codomain(compose(h, domain(h))))))),
% 0.85/1.03 inference(transitivity,[status(thm)],[336, 334])).
% 0.85/1.03 tff(338,plain,
% 0.85/1.03 ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | ((~defined(compose(b, domain(b)), codomain(compose(h, domain(h))))) | defined(compose(b, domain(b)), compose(g, domain(g))) | (~identity_map(codomain(compose(h, domain(h))))) | (~defined(codomain(compose(h, domain(h))), compose(g, domain(g)))))),
% 0.85/1.03 inference(quant_inst,[status(thm)],[])).
% 0.85/1.03 tff(339,plain,
% 0.85/1.03 ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | (~identity_map(codomain(compose(h, domain(h))))) | defined(compose(b, domain(b)), compose(g, domain(g))) | (~defined(codomain(compose(h, domain(h))), compose(g, domain(g)))) | (~defined(compose(b, domain(b)), codomain(compose(h, domain(h)))))),
% 0.85/1.03 inference(modus_ponens,[status(thm)],[338, 337])).
% 0.85/1.03 tff(340,plain,
% 0.85/1.03 ($false),
% 0.85/1.03 inference(unit_resolution,[status(thm)],[339, 333, 208, 323, 296, 104])).
% 0.85/1.03 tff(341,plain,(defined(compose(b, domain(b)), compose(g, domain(g)))), inference(lemma,lemma(discharge,[]))).
% 0.85/1.03 tff(342,plain,
% 0.85/1.03 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(compose(b, domain(b)), compose(g, domain(g)))) | product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g)))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(compose(b, domain(b)), compose(g, domain(g)))) | product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g)))))),
% 0.85/1.03 inference(rewrite,[status(thm)],[])).
% 0.85/1.03 tff(343,plain,
% 0.85/1.03 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(compose(b, domain(b)), compose(g, domain(g)))) | product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g)))))),
% 0.85/1.03 inference(quant_inst,[status(thm)],[])).
% 0.85/1.03 tff(344,plain,
% 0.85/1.03 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(compose(b, domain(b)), compose(g, domain(g)))) | product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))),
% 0.85/1.03 inference(modus_ponens,[status(thm)],[343, 342])).
% 0.85/1.03 tff(345,plain,
% 0.85/1.03 (product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))),
% 0.85/1.03 inference(unit_resolution,[status(thm)],[344, 17, 341])).
% 0.85/1.03 tff(346,plain,
% 0.85/1.03 (product(compose(a, b), compose(g, domain(g)), compose(c, h)) <=> product(c, g, d)),
% 0.85/1.03 inference(monotonicity,[status(thm)],[138, 67, 122])).
% 0.85/1.03 tff(347,plain,
% 0.85/1.03 (product(c, g, d) <=> product(compose(a, b), compose(g, domain(g)), compose(c, h))),
% 0.85/1.03 inference(symmetry,[status(thm)],[346])).
% 0.85/1.03 tff(348,plain,
% 0.85/1.03 (product(compose(a, b), compose(g, domain(g)), compose(c, h))),
% 0.85/1.03 inference(modus_ponens,[status(thm)],[143, 347])).
% 0.85/1.03 tff(349,plain,
% 0.85/1.03 (^[Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : refl((product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy))) <=> (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy))))),
% 0.85/1.03 inference(bind,[status(th)],[])).
% 0.85/1.03 tff(350,plain,
% 0.85/1.03 (![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy))) <=> ![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))),
% 0.85/1.03 inference(quant_intro,[status(thm)],[349])).
% 0.85/1.03 tff(351,plain,
% 0.85/1.03 (![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy))) <=> ![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))),
% 0.85/1.03 inference(rewrite,[status(thm)],[])).
% 0.85/1.03 tff(352,plain,
% 0.85/1.03 (^[Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : trans(monotonicity(rewrite((((~product(X, Y, Xy)) | (~product(Xy, Z, Xyz))) | (~product(Y, Z, Yz))) <=> ((~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))), (((((~product(X, Y, Xy)) | (~product(Xy, Z, Xyz))) | (~product(Y, Z, Yz))) | product(X, Yz, Xyz)) <=> (((~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy))) | product(X, Yz, Xyz)))), rewrite((((~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy))) | product(X, Yz, Xyz)) <=> (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))), (((((~product(X, Y, Xy)) | (~product(Xy, Z, Xyz))) | (~product(Y, Z, Yz))) | product(X, Yz, Xyz)) <=> (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))))),
% 0.85/1.03 inference(bind,[status(th)],[])).
% 0.85/1.03 tff(353,plain,
% 0.85/1.03 (![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((((~product(X, Y, Xy)) | (~product(Xy, Z, Xyz))) | (~product(Y, Z, Yz))) | product(X, Yz, Xyz)) <=> ![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))),
% 0.85/1.03 inference(quant_intro,[status(thm)],[352])).
% 0.85/1.03 tff(354,axiom,(![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((((~product(X, Y, Xy)) | (~product(Xy, Z, Xyz))) | (~product(Y, Z, Yz))) | product(X, Yz, Xyz))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','category_theory_axiom2')).
% 0.85/1.03 tff(355,plain,
% 0.85/1.03 (![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))),
% 0.85/1.03 inference(modus_ponens,[status(thm)],[354, 353])).
% 0.85/1.03 tff(356,plain,
% 0.85/1.03 (![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))),
% 0.85/1.03 inference(modus_ponens,[status(thm)],[355, 351])).
% 0.85/1.03 tff(357,plain,(
% 0.85/1.03 ![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))),
% 0.85/1.03 inference(skolemize,[status(sab)],[356])).
% 0.85/1.03 tff(358,plain,
% 0.85/1.03 (![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))),
% 0.85/1.03 inference(modus_ponens,[status(thm)],[357, 350])).
% 0.85/1.03 tff(359,plain,
% 0.85/1.03 (((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | ((~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))) | (~product(compose(a, b), compose(g, domain(g)), compose(c, h))) | (~product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h)))) <=> ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))) | (~product(compose(a, b), compose(g, domain(g)), compose(c, h))) | (~product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h)))),
% 0.85/1.03 inference(rewrite,[status(thm)],[])).
% 0.85/1.03 tff(360,plain,
% 0.85/1.03 ((product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h)) | (~product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(compose(a, b), compose(g, domain(g)), compose(c, h))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b)))) <=> ((~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))) | (~product(compose(a, b), compose(g, domain(g)), compose(c, h))) | (~product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h)))),
% 0.85/1.03 inference(rewrite,[status(thm)],[])).
% 0.85/1.03 tff(361,plain,
% 0.85/1.03 (((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | (product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h)) | (~product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(compose(a, b), compose(g, domain(g)), compose(c, h))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))))) <=> ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | ((~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))) | (~product(compose(a, b), compose(g, domain(g)), compose(c, h))) | (~product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h))))),
% 0.85/1.03 inference(monotonicity,[status(thm)],[360])).
% 0.85/1.03 tff(362,plain,
% 0.85/1.03 (((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | (product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h)) | (~product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(compose(a, b), compose(g, domain(g)), compose(c, h))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))))) <=> ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))) | (~product(compose(a, b), compose(g, domain(g)), compose(c, h))) | (~product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h)))),
% 0.85/1.03 inference(transitivity,[status(thm)],[361, 359])).
% 0.85/1.03 tff(363,plain,
% 0.85/1.03 ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | (product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h)) | (~product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(compose(a, b), compose(g, domain(g)), compose(c, h))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))))),
% 0.85/1.03 inference(quant_inst,[status(thm)],[])).
% 0.85/1.03 tff(364,plain,
% 0.85/1.03 ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))) | (~product(compose(a, b), compose(g, domain(g)), compose(c, h))) | (~product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h))),
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% 0.85/1.03 tff(365,plain,
% 0.85/1.03 (product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h))),
% 0.85/1.03 inference(unit_resolution,[status(thm)],[364, 358, 315, 348, 345])).
% 0.85/1.03 tff(366,plain,
% 0.85/1.03 (product(a, compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h))),
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% 0.85/1.03 tff(367,plain,
% 0.85/1.03 (product(a, compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h)) <=> product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h))),
% 0.85/1.03 inference(monotonicity,[status(thm)],[101])).
% 0.85/1.03 tff(368,plain,
% 0.85/1.03 (product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h)) <=> product(a, compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h))),
% 0.85/1.03 inference(symmetry,[status(thm)],[367])).
% 0.85/1.03 tff(369,assumption,(~defined(compose(b, domain(b)), compose(h, domain(h)))), introduced(assumption)).
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% 0.85/1.03 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(codomain(compose(h, domain(h))), compose(h, domain(h)), compose(h, domain(h)))) | defined(codomain(compose(h, domain(h))), compose(h, domain(h))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(codomain(compose(h, domain(h))), compose(h, domain(h)), compose(h, domain(h)))) | defined(codomain(compose(h, domain(h))), compose(h, domain(h))))),
% 0.85/1.03 inference(rewrite,[status(thm)],[])).
% 0.85/1.03 tff(371,plain,
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% 0.85/1.03 inference(quant_inst,[status(thm)],[])).
% 0.85/1.03 tff(372,plain,
% 0.85/1.03 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(codomain(compose(h, domain(h))), compose(h, domain(h)), compose(h, domain(h)))) | defined(codomain(compose(h, domain(h))), compose(h, domain(h)))),
% 0.85/1.03 inference(modus_ponens,[status(thm)],[371, 370])).
% 0.85/1.03 tff(373,plain,
% 0.85/1.03 (defined(codomain(compose(h, domain(h))), compose(h, domain(h)))),
% 0.85/1.03 inference(unit_resolution,[status(thm)],[372, 54, 210])).
% 0.85/1.03 tff(374,plain,
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% 0.85/1.03 inference(rewrite,[status(thm)],[])).
% 0.85/1.03 tff(375,plain,
% 0.85/1.03 (((~defined(compose(b, domain(b)), codomain(compose(h, domain(h))))) | defined(compose(b, domain(b)), compose(h, domain(h))) | (~identity_map(codomain(compose(h, domain(h))))) | (~defined(codomain(compose(h, domain(h))), compose(h, domain(h))))) <=> ((~defined(codomain(compose(h, domain(h))), compose(h, domain(h)))) | (~identity_map(codomain(compose(h, domain(h))))) | defined(compose(b, domain(b)), compose(h, domain(h))) | (~defined(compose(b, domain(b)), codomain(compose(h, domain(h))))))),
% 0.85/1.03 inference(rewrite,[status(thm)],[])).
% 0.85/1.03 tff(376,plain,
% 0.85/1.03 (((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | ((~defined(compose(b, domain(b)), codomain(compose(h, domain(h))))) | defined(compose(b, domain(b)), compose(h, domain(h))) | (~identity_map(codomain(compose(h, domain(h))))) | (~defined(codomain(compose(h, domain(h))), compose(h, domain(h)))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | ((~defined(codomain(compose(h, domain(h))), compose(h, domain(h)))) | (~identity_map(codomain(compose(h, domain(h))))) | defined(compose(b, domain(b)), compose(h, domain(h))) | (~defined(compose(b, domain(b)), codomain(compose(h, domain(h)))))))),
% 0.85/1.03 inference(monotonicity,[status(thm)],[375])).
% 0.85/1.03 tff(377,plain,
% 0.85/1.03 (((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | ((~defined(compose(b, domain(b)), codomain(compose(h, domain(h))))) | defined(compose(b, domain(b)), compose(h, domain(h))) | (~identity_map(codomain(compose(h, domain(h))))) | (~defined(codomain(compose(h, domain(h))), compose(h, domain(h)))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | (~defined(codomain(compose(h, domain(h))), compose(h, domain(h)))) | (~identity_map(codomain(compose(h, domain(h))))) | defined(compose(b, domain(b)), compose(h, domain(h))) | (~defined(compose(b, domain(b)), codomain(compose(h, domain(h))))))),
% 0.85/1.04 inference(transitivity,[status(thm)],[376, 374])).
% 0.85/1.04 tff(378,plain,
% 0.85/1.04 ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | ((~defined(compose(b, domain(b)), codomain(compose(h, domain(h))))) | defined(compose(b, domain(b)), compose(h, domain(h))) | (~identity_map(codomain(compose(h, domain(h))))) | (~defined(codomain(compose(h, domain(h))), compose(h, domain(h)))))),
% 0.85/1.04 inference(quant_inst,[status(thm)],[])).
% 0.85/1.04 tff(379,plain,
% 0.85/1.04 ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | (~defined(codomain(compose(h, domain(h))), compose(h, domain(h)))) | (~identity_map(codomain(compose(h, domain(h))))) | defined(compose(b, domain(b)), compose(h, domain(h))) | (~defined(compose(b, domain(b)), codomain(compose(h, domain(h)))))),
% 0.85/1.04 inference(modus_ponens,[status(thm)],[378, 377])).
% 0.85/1.04 tff(380,plain,
% 0.85/1.04 ($false),
% 0.85/1.04 inference(unit_resolution,[status(thm)],[379, 333, 373, 208, 323, 369])).
% 0.85/1.04 tff(381,plain,(defined(compose(b, domain(b)), compose(h, domain(h)))), inference(lemma,lemma(discharge,[]))).
% 0.85/1.04 tff(382,plain,
% 0.85/1.04 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(compose(b, domain(b)), compose(h, domain(h)))) | product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h)))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(compose(b, domain(b)), compose(h, domain(h)))) | product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h)))))),
% 0.85/1.04 inference(rewrite,[status(thm)],[])).
% 0.85/1.04 tff(383,plain,
% 0.85/1.04 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(compose(b, domain(b)), compose(h, domain(h)))) | product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h)))))),
% 0.85/1.04 inference(quant_inst,[status(thm)],[])).
% 0.85/1.04 tff(384,plain,
% 0.85/1.04 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(compose(b, domain(b)), compose(h, domain(h)))) | product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h))))),
% 0.85/1.04 inference(modus_ponens,[status(thm)],[383, 382])).
% 0.85/1.04 tff(385,plain,
% 0.85/1.04 (product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h))))),
% 0.85/1.04 inference(unit_resolution,[status(thm)],[384, 17, 381])).
% 0.85/1.04 tff(386,plain,
% 0.85/1.04 (product(compose(a, b), compose(h, domain(h)), compose(c, h)) <=> product(c, h, compose(c, h))),
% 0.85/1.04 inference(monotonicity,[status(thm)],[138, 45])).
% 0.85/1.04 tff(387,plain,
% 0.85/1.04 (product(c, h, compose(c, h)) <=> product(compose(a, b), compose(h, domain(h)), compose(c, h))),
% 0.85/1.04 inference(symmetry,[status(thm)],[386])).
% 0.85/1.04 tff(388,plain,
% 0.85/1.04 (product(compose(a, b), compose(h, domain(h)), compose(c, h))),
% 0.85/1.04 inference(modus_ponens,[status(thm)],[117, 387])).
% 0.85/1.04 tff(389,plain,
% 0.85/1.04 (((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | ((~product(compose(a, b), compose(h, domain(h)), compose(c, h))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))) | product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h)) | (~product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h))))))) <=> ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | (~product(compose(a, b), compose(h, domain(h)), compose(c, h))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))) | product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h)) | (~product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h))))))),
% 0.85/1.04 inference(rewrite,[status(thm)],[])).
% 0.85/1.04 tff(390,plain,
% 0.85/1.04 ((product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h)) | (~product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h))))) | (~product(compose(a, b), compose(h, domain(h)), compose(c, h))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b)))) <=> ((~product(compose(a, b), compose(h, domain(h)), compose(c, h))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))) | product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h)) | (~product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h))))))),
% 0.85/1.04 inference(rewrite,[status(thm)],[])).
% 0.85/1.04 tff(391,plain,
% 0.85/1.04 (((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | (product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h)) | (~product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h))))) | (~product(compose(a, b), compose(h, domain(h)), compose(c, h))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))))) <=> ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | ((~product(compose(a, b), compose(h, domain(h)), compose(c, h))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))) | product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h)) | (~product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h)))))))),
% 0.85/1.04 inference(monotonicity,[status(thm)],[390])).
% 0.85/1.04 tff(392,plain,
% 0.85/1.04 (((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | (product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h)) | (~product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h))))) | (~product(compose(a, b), compose(h, domain(h)), compose(c, h))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))))) <=> ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | (~product(compose(a, b), compose(h, domain(h)), compose(c, h))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))) | product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h)) | (~product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h))))))),
% 0.85/1.04 inference(transitivity,[status(thm)],[391, 389])).
% 0.85/1.04 tff(393,plain,
% 0.85/1.04 ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | (product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h)) | (~product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h))))) | (~product(compose(a, b), compose(h, domain(h)), compose(c, h))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))))),
% 0.85/1.04 inference(quant_inst,[status(thm)],[])).
% 0.85/1.04 tff(394,plain,
% 0.85/1.04 ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : (product(X, Yz, Xyz) | (~product(Y, Z, Yz)) | (~product(Xy, Z, Xyz)) | (~product(X, Y, Xy)))) | (~product(compose(a, b), compose(h, domain(h)), compose(c, h))) | (~product(compose(codomain(a), a), compose(b, domain(b)), compose(a, b))) | product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h)) | (~product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h)))))),
% 0.85/1.04 inference(modus_ponens,[status(thm)],[393, 392])).
% 0.85/1.04 tff(395,plain,
% 0.85/1.04 (product(compose(codomain(a), a), compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h))),
% 0.85/1.04 inference(unit_resolution,[status(thm)],[394, 358, 388, 315, 385])).
% 0.85/1.04 tff(396,plain,
% 0.85/1.04 (product(a, compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h))),
% 0.85/1.04 inference(modus_ponens,[status(thm)],[395, 368])).
% 0.85/1.04 tff(397,plain,
% 0.85/1.04 (^[W: $i, Y: $i, X: $i] : refl(((X = Y) | (~product(a, Y, W)) | (~product(a, X, W))) <=> ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W))))),
% 0.85/1.04 inference(bind,[status(th)],[])).
% 0.85/1.04 tff(398,plain,
% 0.85/1.04 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W))) <=> ![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))),
% 0.85/1.04 inference(quant_intro,[status(thm)],[397])).
% 0.85/1.04 tff(399,plain,
% 0.85/1.04 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W))) <=> ![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))),
% 0.85/1.04 inference(rewrite,[status(thm)],[])).
% 0.85/1.04 tff(400,plain,
% 0.85/1.04 (^[W: $i, Y: $i, X: $i] : rewrite((((~product(a, X, W)) | (~product(a, Y, W))) | (X = Y)) <=> ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W))))),
% 0.85/1.04 inference(bind,[status(th)],[])).
% 0.85/1.04 tff(401,plain,
% 0.85/1.04 (![W: $i, Y: $i, X: $i] : (((~product(a, X, W)) | (~product(a, Y, W))) | (X = Y)) <=> ![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))),
% 0.85/1.04 inference(quant_intro,[status(thm)],[400])).
% 0.85/1.04 tff(402,axiom,(![W: $i, Y: $i, X: $i] : (((~product(a, X, W)) | (~product(a, Y, W))) | (X = Y))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','cancellation_for_product1')).
% 0.85/1.04 tff(403,plain,
% 0.85/1.04 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))),
% 0.85/1.04 inference(modus_ponens,[status(thm)],[402, 401])).
% 0.85/1.04 tff(404,plain,
% 0.85/1.04 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))),
% 0.85/1.04 inference(modus_ponens,[status(thm)],[403, 399])).
% 0.85/1.04 tff(405,plain,(
% 0.85/1.04 ![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))),
% 0.85/1.04 inference(skolemize,[status(sab)],[404])).
% 0.85/1.04 tff(406,plain,
% 0.85/1.04 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))),
% 0.85/1.04 inference(modus_ponens,[status(thm)],[405, 398])).
% 0.85/1.04 tff(407,plain,
% 0.85/1.04 (((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))) | ((compose(compose(b, domain(b)), compose(h, domain(h))) = compose(compose(b, domain(b)), compose(g, domain(g)))) | (~product(a, compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h))) | (~product(a, compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h))))) <=> ((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))) | (compose(compose(b, domain(b)), compose(h, domain(h))) = compose(compose(b, domain(b)), compose(g, domain(g)))) | (~product(a, compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h))) | (~product(a, compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h))))),
% 0.85/1.04 inference(rewrite,[status(thm)],[])).
% 0.85/1.04 tff(408,plain,
% 0.85/1.04 (((compose(compose(b, domain(b)), compose(h, domain(h))) = compose(compose(b, domain(b)), compose(g, domain(g)))) | (~product(a, compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h))) | (~product(a, compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h)))) <=> ((compose(compose(b, domain(b)), compose(h, domain(h))) = compose(compose(b, domain(b)), compose(g, domain(g)))) | (~product(a, compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h))) | (~product(a, compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h))))),
% 0.85/1.04 inference(rewrite,[status(thm)],[])).
% 0.85/1.04 tff(409,plain,
% 0.85/1.04 (((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))) | ((compose(compose(b, domain(b)), compose(h, domain(h))) = compose(compose(b, domain(b)), compose(g, domain(g)))) | (~product(a, compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h))) | (~product(a, compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h))))) <=> ((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))) | ((compose(compose(b, domain(b)), compose(h, domain(h))) = compose(compose(b, domain(b)), compose(g, domain(g)))) | (~product(a, compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h))) | (~product(a, compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h)))))),
% 0.85/1.04 inference(monotonicity,[status(thm)],[408])).
% 0.85/1.04 tff(410,plain,
% 0.85/1.04 (((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))) | ((compose(compose(b, domain(b)), compose(h, domain(h))) = compose(compose(b, domain(b)), compose(g, domain(g)))) | (~product(a, compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h))) | (~product(a, compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h))))) <=> ((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))) | (compose(compose(b, domain(b)), compose(h, domain(h))) = compose(compose(b, domain(b)), compose(g, domain(g)))) | (~product(a, compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h))) | (~product(a, compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h))))),
% 0.85/1.04 inference(transitivity,[status(thm)],[409, 407])).
% 0.85/1.04 tff(411,plain,
% 0.85/1.04 ((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))) | ((compose(compose(b, domain(b)), compose(h, domain(h))) = compose(compose(b, domain(b)), compose(g, domain(g)))) | (~product(a, compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h))) | (~product(a, compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h))))),
% 0.85/1.04 inference(quant_inst,[status(thm)],[])).
% 0.85/1.04 tff(412,plain,
% 0.85/1.04 ((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(a, Y, W)) | (~product(a, X, W)))) | (compose(compose(b, domain(b)), compose(h, domain(h))) = compose(compose(b, domain(b)), compose(g, domain(g)))) | (~product(a, compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h))) | (~product(a, compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h)))),
% 0.85/1.04 inference(modus_ponens,[status(thm)],[411, 410])).
% 0.85/1.04 tff(413,plain,
% 0.85/1.04 ((compose(compose(b, domain(b)), compose(h, domain(h))) = compose(compose(b, domain(b)), compose(g, domain(g)))) | (~product(a, compose(compose(b, domain(b)), compose(h, domain(h))), compose(c, h))) | (~product(a, compose(compose(b, domain(b)), compose(g, domain(g))), compose(c, h)))),
% 0.85/1.04 inference(unit_resolution,[status(thm)],[412, 406])).
% 0.85/1.04 tff(414,plain,
% 0.85/1.04 (compose(compose(b, domain(b)), compose(h, domain(h))) = compose(compose(b, domain(b)), compose(g, domain(g)))),
% 0.85/1.04 inference(unit_resolution,[status(thm)],[413, 396, 366])).
% 0.85/1.04 tff(415,plain,
% 0.85/1.04 (compose(compose(b, domain(b)), compose(g, domain(g))) = compose(compose(b, domain(b)), compose(h, domain(h)))),
% 0.85/1.04 inference(symmetry,[status(thm)],[414])).
% 0.85/1.04 tff(416,plain,
% 0.85/1.04 (product(b, compose(h, domain(h)), compose(compose(b, domain(b)), compose(g, domain(g)))) <=> product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h))))),
% 0.85/1.04 inference(monotonicity,[status(thm)],[310, 415])).
% 0.85/1.04 tff(417,plain,
% 0.85/1.04 (product(compose(b, domain(b)), compose(h, domain(h)), compose(compose(b, domain(b)), compose(h, domain(h)))) <=> product(b, compose(h, domain(h)), compose(compose(b, domain(b)), compose(g, domain(g))))),
% 0.85/1.04 inference(symmetry,[status(thm)],[416])).
% 0.85/1.04 tff(418,plain,
% 0.85/1.04 (product(b, compose(h, domain(h)), compose(compose(b, domain(b)), compose(g, domain(g))))),
% 0.85/1.04 inference(modus_ponens,[status(thm)],[385, 417])).
% 0.85/1.04 tff(419,plain,
% 0.85/1.04 (product(b, compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g)))) <=> product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))),
% 0.85/1.04 inference(monotonicity,[status(thm)],[310])).
% 0.85/1.04 tff(420,plain,
% 0.85/1.04 (product(compose(b, domain(b)), compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g)))) <=> product(b, compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))),
% 0.85/1.04 inference(symmetry,[status(thm)],[419])).
% 0.85/1.04 tff(421,plain,
% 0.85/1.04 (product(b, compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))),
% 0.85/1.04 inference(modus_ponens,[status(thm)],[345, 420])).
% 0.85/1.04 tff(422,plain,
% 0.85/1.04 (^[W: $i, Y: $i, X: $i] : refl(((X = Y) | (~product(b, Y, W)) | (~product(b, X, W))) <=> ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W))))),
% 0.85/1.04 inference(bind,[status(th)],[])).
% 0.85/1.04 tff(423,plain,
% 0.85/1.04 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W))) <=> ![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))),
% 0.85/1.04 inference(quant_intro,[status(thm)],[422])).
% 0.85/1.04 tff(424,plain,
% 0.85/1.04 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W))) <=> ![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))),
% 0.85/1.04 inference(rewrite,[status(thm)],[])).
% 0.85/1.04 tff(425,plain,
% 0.85/1.04 (^[W: $i, Y: $i, X: $i] : rewrite((((~product(b, X, W)) | (~product(b, Y, W))) | (X = Y)) <=> ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W))))),
% 0.85/1.04 inference(bind,[status(th)],[])).
% 0.85/1.04 tff(426,plain,
% 0.85/1.04 (![W: $i, Y: $i, X: $i] : (((~product(b, X, W)) | (~product(b, Y, W))) | (X = Y)) <=> ![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))),
% 0.85/1.04 inference(quant_intro,[status(thm)],[425])).
% 0.85/1.04 tff(427,axiom,(![W: $i, Y: $i, X: $i] : (((~product(b, X, W)) | (~product(b, Y, W))) | (X = Y))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','cancellation_for_product2')).
% 0.85/1.04 tff(428,plain,
% 0.85/1.04 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))),
% 0.85/1.04 inference(modus_ponens,[status(thm)],[427, 426])).
% 0.85/1.04 tff(429,plain,
% 0.85/1.04 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))),
% 0.85/1.04 inference(modus_ponens,[status(thm)],[428, 424])).
% 0.85/1.04 tff(430,plain,(
% 0.85/1.04 ![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))),
% 0.85/1.04 inference(skolemize,[status(sab)],[429])).
% 0.85/1.04 tff(431,plain,
% 0.85/1.04 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))),
% 0.85/1.04 inference(modus_ponens,[status(thm)],[430, 423])).
% 0.85/1.04 tff(432,plain,
% 0.85/1.04 (((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))) | ((~product(b, compose(h, domain(h)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(b, compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (compose(g, domain(g)) = compose(h, domain(h))))) <=> ((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))) | (~product(b, compose(h, domain(h)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(b, compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (compose(g, domain(g)) = compose(h, domain(h))))),
% 0.85/1.04 inference(rewrite,[status(thm)],[])).
% 0.85/1.04 tff(433,plain,
% 0.85/1.04 (((compose(g, domain(g)) = compose(h, domain(h))) | (~product(b, compose(h, domain(h)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(b, compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g)))))) <=> ((~product(b, compose(h, domain(h)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(b, compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (compose(g, domain(g)) = compose(h, domain(h))))),
% 0.85/1.04 inference(rewrite,[status(thm)],[])).
% 0.85/1.04 tff(434,plain,
% 0.85/1.04 (((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))) | ((compose(g, domain(g)) = compose(h, domain(h))) | (~product(b, compose(h, domain(h)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(b, compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))))) <=> ((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))) | ((~product(b, compose(h, domain(h)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(b, compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (compose(g, domain(g)) = compose(h, domain(h)))))),
% 0.85/1.04 inference(monotonicity,[status(thm)],[433])).
% 0.85/1.05 tff(435,plain,
% 0.85/1.05 (((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))) | ((compose(g, domain(g)) = compose(h, domain(h))) | (~product(b, compose(h, domain(h)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(b, compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))))) <=> ((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))) | (~product(b, compose(h, domain(h)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(b, compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (compose(g, domain(g)) = compose(h, domain(h))))),
% 0.85/1.05 inference(transitivity,[status(thm)],[434, 432])).
% 0.85/1.05 tff(436,plain,
% 0.85/1.05 ((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))) | ((compose(g, domain(g)) = compose(h, domain(h))) | (~product(b, compose(h, domain(h)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(b, compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))))),
% 0.85/1.05 inference(quant_inst,[status(thm)],[])).
% 0.85/1.05 tff(437,plain,
% 0.85/1.05 ((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(b, Y, W)) | (~product(b, X, W)))) | (~product(b, compose(h, domain(h)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(b, compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (compose(g, domain(g)) = compose(h, domain(h)))),
% 0.85/1.05 inference(modus_ponens,[status(thm)],[436, 435])).
% 0.85/1.05 tff(438,plain,
% 0.85/1.05 ((~product(b, compose(h, domain(h)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (~product(b, compose(g, domain(g)), compose(compose(b, domain(b)), compose(g, domain(g))))) | (compose(g, domain(g)) = compose(h, domain(h)))),
% 0.85/1.05 inference(unit_resolution,[status(thm)],[437, 431])).
% 0.85/1.05 tff(439,plain,
% 0.85/1.05 ($false),
% 0.85/1.05 inference(unit_resolution,[status(thm)],[438, 421, 418, 75])).
% 0.85/1.05 % SZS output end Proof
% 0.85/1.05 % E exiting
%------------------------------------------------------------------------------