%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : CAT003-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:00:18 AM UTC 2025
% Result : Unsatisfiable 0.39s 0.62s
% Output : Proof 0.48s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : CAT003-1 : TPTP v9.0.0. Released v1.0.0.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n015.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Jun 19 11:51:11 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.39/0.62 % SZS status Unsatisfiable
% 0.39/0.62 % SZS output start Proof
% 0.39/0.62 tff(product_type, type, (
% 0.39/0.62 product: ( $i * $i * $i ) > $o)).
% 0.39/0.62 tff(compose_type, type, (
% 0.39/0.62 compose: ( $i * $i ) > $i)).
% 0.39/0.62 tff(b_type, type, (
% 0.39/0.62 b: $i)).
% 0.39/0.62 tff(a_type, type, (
% 0.39/0.62 a: $i)).
% 0.39/0.62 tff(h_type, type, (
% 0.39/0.62 h: $i)).
% 0.39/0.62 tff(c_type, type, (
% 0.39/0.62 c: $i)).
% 0.39/0.62 tff(g_type, type, (
% 0.39/0.62 g: $i)).
% 0.39/0.62 tff(defined_type, type, (
% 0.39/0.62 defined: ( $i * $i ) > $o)).
% 0.39/0.62 tff(codomain_type, type, (
% 0.39/0.62 codomain: $i > $i)).
% 0.39/0.62 tff(d_type, type, (
% 0.39/0.62 d: $i)).
% 0.39/0.62 tff(identity_map_type, type, (
% 0.39/0.62 identity_map: $i > $o)).
% 0.39/0.62 tff(1,plain,
% 0.39/0.62 (product(a, b, c) <=> product(a, b, c)),
% 0.39/0.62 inference(rewrite,[status(thm)],[])).
% 0.39/0.62 tff(2,axiom,(product(a, b, c)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','ab_equals_c')).
% 0.39/0.62 tff(3,plain,
% 0.39/0.62 (product(a, b, c)),
% 0.39/0.62 inference(modus_ponens,[status(thm)],[2, 1])).
% 0.39/0.62 tff(4,plain,
% 0.39/0.62 (^[Z: $i, Y: $i, X: $i] : refl(((~product(X, Y, Z)) | defined(X, Y)) <=> ((~product(X, Y, Z)) | defined(X, Y)))),
% 0.39/0.62 inference(bind,[status(th)],[])).
% 0.39/0.62 tff(5,plain,
% 0.39/0.62 (![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))),
% 0.39/0.62 inference(quant_intro,[status(thm)],[4])).
% 0.39/0.62 tff(6,plain,
% 0.39/0.62 (![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))),
% 0.39/0.62 inference(rewrite,[status(thm)],[])).
% 0.39/0.62 tff(7,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')).
% 0.39/0.62 tff(8,plain,
% 0.39/0.62 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))),
% 0.39/0.62 inference(modus_ponens,[status(thm)],[7, 6])).
% 0.39/0.62 tff(9,plain,(
% 0.39/0.62 ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))),
% 0.39/0.62 inference(skolemize,[status(sab)],[8])).
% 0.39/0.62 tff(10,plain,
% 0.39/0.62 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))),
% 0.39/0.62 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.39/0.62 tff(11,plain,
% 0.39/0.62 (((~![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))),
% 0.39/0.62 inference(rewrite,[status(thm)],[])).
% 0.39/0.62 tff(12,plain,
% 0.39/0.62 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(a, b, c)) | defined(a, b))),
% 0.39/0.62 inference(quant_inst,[status(thm)],[])).
% 0.39/0.62 tff(13,plain,
% 0.39/0.62 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(a, b, c)) | defined(a, b)),
% 0.39/0.62 inference(modus_ponens,[status(thm)],[12, 11])).
% 0.39/0.62 tff(14,plain,
% 0.39/0.62 (defined(a, b)),
% 0.39/0.62 inference(unit_resolution,[status(thm)],[13, 10, 3])).
% 0.39/0.62 tff(15,plain,
% 0.39/0.62 (^[Y: $i, X: $i] : refl(((~defined(X, Y)) | product(X, Y, compose(X, Y))) <=> ((~defined(X, Y)) | product(X, Y, compose(X, Y))))),
% 0.39/0.62 inference(bind,[status(th)],[])).
% 0.39/0.62 tff(16,plain,
% 0.39/0.62 (![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)))),
% 0.39/0.62 inference(quant_intro,[status(thm)],[15])).
% 0.39/0.62 tff(17,plain,
% 0.39/0.62 (![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)))),
% 0.39/0.62 inference(rewrite,[status(thm)],[])).
% 0.39/0.62 tff(18,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')).
% 0.39/0.62 tff(19,plain,
% 0.39/0.62 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))),
% 0.39/0.62 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.39/0.62 tff(20,plain,(
% 0.39/0.62 ![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))),
% 0.39/0.62 inference(skolemize,[status(sab)],[19])).
% 0.39/0.62 tff(21,plain,
% 0.39/0.62 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))),
% 0.39/0.62 inference(modus_ponens,[status(thm)],[20, 16])).
% 0.39/0.62 tff(22,plain,
% 0.39/0.62 (((~![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)))),
% 0.45/0.62 inference(rewrite,[status(thm)],[])).
% 0.45/0.62 tff(23,plain,
% 0.45/0.62 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(a, b)) | product(a, b, compose(a, b)))),
% 0.45/0.62 inference(quant_inst,[status(thm)],[])).
% 0.45/0.62 tff(24,plain,
% 0.45/0.62 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(a, b)) | product(a, b, compose(a, b))),
% 0.45/0.62 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.45/0.62 tff(25,plain,
% 0.45/0.62 (product(a, b, compose(a, b))),
% 0.45/0.62 inference(unit_resolution,[status(thm)],[24, 21, 14])).
% 0.45/0.62 tff(26,plain,
% 0.45/0.62 (^[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)))),
% 0.45/0.62 inference(bind,[status(th)],[])).
% 0.45/0.62 tff(27,plain,
% 0.45/0.62 (![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))),
% 0.45/0.62 inference(quant_intro,[status(thm)],[26])).
% 0.45/0.62 tff(28,plain,
% 0.45/0.62 (![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))),
% 0.45/0.62 inference(rewrite,[status(thm)],[])).
% 0.45/0.62 tff(29,plain,
% 0.45/0.62 (^[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)))),
% 0.45/0.62 inference(bind,[status(th)],[])).
% 0.45/0.62 tff(30,plain,
% 0.45/0.62 (![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))),
% 0.45/0.62 inference(quant_intro,[status(thm)],[29])).
% 0.45/0.62 tff(31,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')).
% 0.45/0.62 tff(32,plain,
% 0.45/0.62 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.45/0.62 inference(modus_ponens,[status(thm)],[31, 30])).
% 0.45/0.62 tff(33,plain,
% 0.45/0.62 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.45/0.62 inference(modus_ponens,[status(thm)],[32, 28])).
% 0.45/0.62 tff(34,plain,(
% 0.45/0.62 ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.45/0.62 inference(skolemize,[status(sab)],[33])).
% 0.45/0.62 tff(35,plain,
% 0.45/0.62 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.45/0.62 inference(modus_ponens,[status(thm)],[34, 27])).
% 0.45/0.62 tff(36,plain,
% 0.45/0.62 (((~![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)))),
% 0.45/0.62 inference(rewrite,[status(thm)],[])).
% 0.45/0.62 tff(37,plain,
% 0.45/0.62 ((~![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)))),
% 0.45/0.63 inference(quant_inst,[status(thm)],[])).
% 0.45/0.63 tff(38,plain,
% 0.45/0.63 ((~![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))),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.45/0.63 tff(39,plain,
% 0.45/0.63 (c = compose(a, b)),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[38, 35, 3, 25])).
% 0.45/0.63 tff(40,plain,
% 0.45/0.63 (product(g, c, compose(compose(h, a), b)) <=> product(g, compose(a, b), compose(compose(h, a), b))),
% 0.45/0.63 inference(monotonicity,[status(thm)],[39])).
% 0.45/0.63 tff(41,plain,
% 0.45/0.63 (product(g, compose(a, b), compose(compose(h, a), b)) <=> product(g, c, compose(compose(h, a), b))),
% 0.45/0.63 inference(symmetry,[status(thm)],[40])).
% 0.45/0.63 tff(42,plain,
% 0.45/0.63 (product(h, a, d) <=> product(h, a, d)),
% 0.45/0.63 inference(rewrite,[status(thm)],[])).
% 0.45/0.63 tff(43,axiom,(product(h, a, d)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','ha_equals_d')).
% 0.45/0.63 tff(44,plain,
% 0.45/0.63 (product(h, a, d)),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.45/0.63 tff(45,plain,
% 0.45/0.63 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(h, a, d)) | defined(h, a))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(h, a, d)) | defined(h, a))),
% 0.45/0.63 inference(rewrite,[status(thm)],[])).
% 0.45/0.63 tff(46,plain,
% 0.45/0.63 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(h, a, d)) | defined(h, a))),
% 0.45/0.63 inference(quant_inst,[status(thm)],[])).
% 0.45/0.63 tff(47,plain,
% 0.45/0.63 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(h, a, d)) | defined(h, a)),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.45/0.63 tff(48,plain,
% 0.45/0.63 (defined(h, a)),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[47, 10, 44])).
% 0.45/0.63 tff(49,plain,
% 0.45/0.63 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(h, a)) | product(h, a, compose(h, a)))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(h, a)) | product(h, a, compose(h, a)))),
% 0.45/0.63 inference(rewrite,[status(thm)],[])).
% 0.45/0.63 tff(50,plain,
% 0.45/0.63 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(h, a)) | product(h, a, compose(h, a)))),
% 0.45/0.63 inference(quant_inst,[status(thm)],[])).
% 0.45/0.63 tff(51,plain,
% 0.45/0.63 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(h, a)) | product(h, a, compose(h, a))),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.45/0.63 tff(52,plain,
% 0.45/0.63 (product(h, a, compose(h, a))),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[51, 21, 48])).
% 0.45/0.63 tff(53,plain,
% 0.45/0.63 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(h, a, d)) | (~product(h, a, compose(h, a))) | (d = compose(h, a)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(h, a, d)) | (~product(h, a, compose(h, a))) | (d = compose(h, a)))),
% 0.45/0.63 inference(rewrite,[status(thm)],[])).
% 0.45/0.63 tff(54,plain,
% 0.45/0.63 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(h, a, d)) | (~product(h, a, compose(h, a))) | (d = compose(h, a)))),
% 0.45/0.63 inference(quant_inst,[status(thm)],[])).
% 0.45/0.63 tff(55,plain,
% 0.45/0.63 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(h, a, d)) | (~product(h, a, compose(h, a))) | (d = compose(h, a))),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.45/0.63 tff(56,plain,
% 0.45/0.63 (d = compose(h, a)),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[55, 35, 44, 52])).
% 0.45/0.63 tff(57,plain,
% 0.45/0.63 (compose(h, a) = d),
% 0.45/0.63 inference(symmetry,[status(thm)],[56])).
% 0.45/0.63 tff(58,plain,
% 0.45/0.63 (^[X: $i] : refl(defined(codomain(X), X) <=> defined(codomain(X), X))),
% 0.45/0.63 inference(bind,[status(th)],[])).
% 0.45/0.63 tff(59,plain,
% 0.45/0.63 (![X: $i] : defined(codomain(X), X) <=> ![X: $i] : defined(codomain(X), X)),
% 0.45/0.63 inference(quant_intro,[status(thm)],[58])).
% 0.45/0.63 tff(60,plain,
% 0.45/0.63 (![X: $i] : defined(codomain(X), X) <=> ![X: $i] : defined(codomain(X), X)),
% 0.45/0.63 inference(rewrite,[status(thm)],[])).
% 0.45/0.63 tff(61,axiom,(![X: $i] : defined(codomain(X), X)), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','mapping_from_codomain_of_x_to_x')).
% 0.45/0.63 tff(62,plain,
% 0.45/0.63 (![X: $i] : defined(codomain(X), X)),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[61, 60])).
% 0.45/0.63 tff(63,plain,(
% 0.45/0.63 ![X: $i] : defined(codomain(X), X)),
% 0.45/0.63 inference(skolemize,[status(sab)],[62])).
% 0.45/0.63 tff(64,plain,
% 0.45/0.63 (![X: $i] : defined(codomain(X), X)),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[63, 59])).
% 0.45/0.63 tff(65,plain,
% 0.45/0.63 ((~![X: $i] : defined(codomain(X), X)) | defined(codomain(a), a)),
% 0.45/0.63 inference(quant_inst,[status(thm)],[])).
% 0.45/0.63 tff(66,plain,
% 0.45/0.63 (defined(codomain(a), a)),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[65, 64])).
% 0.45/0.63 tff(67,plain,
% 0.45/0.63 (((~![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)))),
% 0.45/0.63 inference(rewrite,[status(thm)],[])).
% 0.45/0.63 tff(68,plain,
% 0.45/0.63 ((~![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)))),
% 0.45/0.63 inference(quant_inst,[status(thm)],[])).
% 0.45/0.63 tff(69,plain,
% 0.45/0.63 ((~![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))),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[68, 67])).
% 0.45/0.63 tff(70,plain,
% 0.45/0.63 (product(codomain(a), a, compose(codomain(a), a))),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[69, 21, 66])).
% 0.45/0.63 tff(71,plain,
% 0.45/0.63 (^[X: $i] : refl(product(codomain(X), X, X) <=> product(codomain(X), X, X))),
% 0.45/0.63 inference(bind,[status(th)],[])).
% 0.45/0.63 tff(72,plain,
% 0.45/0.63 (![X: $i] : product(codomain(X), X, X) <=> ![X: $i] : product(codomain(X), X, X)),
% 0.45/0.63 inference(quant_intro,[status(thm)],[71])).
% 0.45/0.63 tff(73,plain,
% 0.45/0.63 (![X: $i] : product(codomain(X), X, X) <=> ![X: $i] : product(codomain(X), X, X)),
% 0.45/0.63 inference(rewrite,[status(thm)],[])).
% 0.45/0.63 tff(74,axiom,(![X: $i] : product(codomain(X), X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','product_on_codomain')).
% 0.45/0.63 tff(75,plain,
% 0.45/0.63 (![X: $i] : product(codomain(X), X, X)),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[74, 73])).
% 0.45/0.63 tff(76,plain,(
% 0.45/0.63 ![X: $i] : product(codomain(X), X, X)),
% 0.45/0.63 inference(skolemize,[status(sab)],[75])).
% 0.45/0.63 tff(77,plain,
% 0.45/0.63 (![X: $i] : product(codomain(X), X, X)),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[76, 72])).
% 0.45/0.63 tff(78,plain,
% 0.45/0.63 ((~![X: $i] : product(codomain(X), X, X)) | product(codomain(a), a, a)),
% 0.45/0.63 inference(quant_inst,[status(thm)],[])).
% 0.45/0.63 tff(79,plain,
% 0.45/0.63 (product(codomain(a), a, a)),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[78, 77])).
% 0.45/0.63 tff(80,plain,
% 0.45/0.63 (((~![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)))),
% 0.45/0.63 inference(rewrite,[status(thm)],[])).
% 0.45/0.63 tff(81,plain,
% 0.45/0.63 ((~![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)))),
% 0.45/0.63 inference(quant_inst,[status(thm)],[])).
% 0.45/0.63 tff(82,plain,
% 0.45/0.63 ((~![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))),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[81, 80])).
% 0.45/0.63 tff(83,plain,
% 0.45/0.63 (a = compose(codomain(a), a)),
% 0.45/0.63 inference(unit_resolution,[status(thm)],[82, 35, 79, 70])).
% 0.45/0.63 tff(84,plain,
% 0.45/0.63 (compose(codomain(a), a) = a),
% 0.45/0.63 inference(symmetry,[status(thm)],[83])).
% 0.45/0.63 tff(85,plain,
% 0.45/0.63 (product(g, compose(codomain(a), a), compose(h, a)) <=> product(g, a, d)),
% 0.45/0.63 inference(monotonicity,[status(thm)],[84, 57])).
% 0.45/0.63 tff(86,plain,
% 0.45/0.63 (product(g, a, d) <=> product(g, compose(codomain(a), a), compose(h, a))),
% 0.45/0.63 inference(symmetry,[status(thm)],[85])).
% 0.45/0.63 tff(87,plain,
% 0.45/0.63 (product(g, a, d) <=> product(g, a, d)),
% 0.45/0.63 inference(rewrite,[status(thm)],[])).
% 0.45/0.63 tff(88,axiom,(product(g, a, d)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','ga_equals_d')).
% 0.45/0.63 tff(89,plain,
% 0.45/0.63 (product(g, a, d)),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.45/0.63 tff(90,plain,
% 0.45/0.63 (product(g, compose(codomain(a), a), compose(h, a))),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[89, 86])).
% 0.45/0.63 tff(91,plain,
% 0.45/0.63 (product(compose(codomain(a), a), b, compose(a, b)) <=> product(a, b, compose(a, b))),
% 0.45/0.63 inference(monotonicity,[status(thm)],[84])).
% 0.45/0.63 tff(92,plain,
% 0.45/0.63 (product(a, b, compose(a, b)) <=> product(compose(codomain(a), a), b, compose(a, b))),
% 0.45/0.63 inference(symmetry,[status(thm)],[91])).
% 0.45/0.63 tff(93,plain,
% 0.45/0.63 (product(compose(codomain(a), a), b, compose(a, b))),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[25, 92])).
% 0.45/0.63 tff(94,plain,
% 0.45/0.63 (^[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))))),
% 0.45/0.63 inference(bind,[status(th)],[])).
% 0.45/0.63 tff(95,plain,
% 0.45/0.63 (![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)))),
% 0.45/0.63 inference(quant_intro,[status(thm)],[94])).
% 0.45/0.63 tff(96,plain,
% 0.45/0.63 (![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)))),
% 0.45/0.63 inference(rewrite,[status(thm)],[])).
% 0.45/0.63 tff(97,plain,
% 0.45/0.63 (^[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)))))),
% 0.45/0.63 inference(bind,[status(th)],[])).
% 0.45/0.63 tff(98,plain,
% 0.45/0.63 (![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)))),
% 0.45/0.63 inference(quant_intro,[status(thm)],[97])).
% 0.45/0.63 tff(99,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')).
% 0.45/0.63 tff(100,plain,
% 0.45/0.63 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.45/0.63 tff(101,plain,
% 0.45/0.63 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[100, 96])).
% 0.45/0.63 tff(102,plain,(
% 0.45/0.63 ![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.45/0.63 inference(skolemize,[status(sab)],[101])).
% 0.45/0.63 tff(103,plain,
% 0.45/0.63 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.45/0.63 inference(modus_ponens,[status(thm)],[102, 95])).
% 0.45/0.63 tff(104,plain,
% 0.45/0.63 (((~![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(a, b)) | (~product(a, b, compose(a, b))) | (~product(codomain(a), a, a)) | defined(codomain(a), compose(a, b)))) <=> ((~![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(a, b)) | (~product(a, b, compose(a, b))) | (~product(codomain(a), a, a)) | defined(codomain(a), compose(a, b)))),
% 0.45/0.63 inference(rewrite,[status(thm)],[])).
% 0.45/0.63 tff(105,plain,
% 0.45/0.63 ((defined(codomain(a), compose(a, b)) | (~defined(a, b)) | (~product(a, b, compose(a, b))) | (~product(codomain(a), a, a))) <=> ((~defined(a, b)) | (~product(a, b, compose(a, b))) | (~product(codomain(a), a, a)) | defined(codomain(a), compose(a, b)))),
% 0.45/0.63 inference(rewrite,[status(thm)],[])).
% 0.45/0.63 tff(106,plain,
% 0.45/0.63 (((~![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(codomain(a), compose(a, b)) | (~defined(a, b)) | (~product(a, b, compose(a, b))) | (~product(codomain(a), a, a)))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~defined(a, b)) | (~product(a, b, compose(a, b))) | (~product(codomain(a), a, a)) | defined(codomain(a), compose(a, b))))),
% 0.45/0.64 inference(monotonicity,[status(thm)],[105])).
% 0.45/0.64 tff(107,plain,
% 0.45/0.64 (((~![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(codomain(a), compose(a, b)) | (~defined(a, b)) | (~product(a, b, compose(a, b))) | (~product(codomain(a), a, a)))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | (~defined(a, b)) | (~product(a, b, compose(a, b))) | (~product(codomain(a), a, a)) | defined(codomain(a), compose(a, b)))),
% 0.45/0.64 inference(transitivity,[status(thm)],[106, 104])).
% 0.45/0.64 tff(108,plain,
% 0.45/0.64 ((~![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(codomain(a), compose(a, b)) | (~defined(a, b)) | (~product(a, b, compose(a, b))) | (~product(codomain(a), a, a)))),
% 0.45/0.64 inference(quant_inst,[status(thm)],[])).
% 0.45/0.64 tff(109,plain,
% 0.45/0.64 ((~![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(a, b)) | (~product(a, b, compose(a, b))) | (~product(codomain(a), a, a)) | defined(codomain(a), compose(a, b))),
% 0.45/0.64 inference(modus_ponens,[status(thm)],[108, 107])).
% 0.45/0.64 tff(110,plain,
% 0.45/0.64 (defined(codomain(a), compose(a, b))),
% 0.45/0.64 inference(unit_resolution,[status(thm)],[109, 103, 14, 25, 79])).
% 0.45/0.64 tff(111,plain,
% 0.45/0.64 (^[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))))),
% 0.45/0.64 inference(bind,[status(th)],[])).
% 0.45/0.64 tff(112,plain,
% 0.45/0.64 (![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)))),
% 0.45/0.64 inference(quant_intro,[status(thm)],[111])).
% 0.45/0.64 tff(113,plain,
% 0.45/0.64 (![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)))),
% 0.45/0.64 inference(rewrite,[status(thm)],[])).
% 0.45/0.64 tff(114,plain,
% 0.45/0.64 (^[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)))))),
% 0.45/0.64 inference(bind,[status(th)],[])).
% 0.45/0.64 tff(115,plain,
% 0.45/0.64 (![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)))),
% 0.45/0.64 inference(quant_intro,[status(thm)],[114])).
% 0.45/0.64 tff(116,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')).
% 0.45/0.64 tff(117,plain,
% 0.45/0.64 (![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))),
% 0.45/0.64 inference(modus_ponens,[status(thm)],[116, 115])).
% 0.45/0.64 tff(118,plain,
% 0.45/0.64 (![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))),
% 0.45/0.64 inference(modus_ponens,[status(thm)],[117, 113])).
% 0.45/0.64 tff(119,plain,(
% 0.45/0.64 ![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))),
% 0.45/0.64 inference(skolemize,[status(sab)],[118])).
% 0.45/0.64 tff(120,plain,
% 0.45/0.64 (![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))),
% 0.45/0.64 inference(modus_ponens,[status(thm)],[119, 112])).
% 0.45/0.64 tff(121,plain,
% 0.45/0.64 (((~![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))) | ((~defined(h, a)) | (~product(codomain(a), a, a)) | defined(h, codomain(a)))) <=> ((~![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))) | (~defined(h, a)) | (~product(codomain(a), a, a)) | defined(h, codomain(a)))),
% 0.45/0.64 inference(rewrite,[status(thm)],[])).
% 0.45/0.64 tff(122,plain,
% 0.45/0.64 (((~defined(h, a)) | defined(h, codomain(a)) | (~product(codomain(a), a, a))) <=> ((~defined(h, a)) | (~product(codomain(a), a, a)) | defined(h, codomain(a)))),
% 0.45/0.64 inference(rewrite,[status(thm)],[])).
% 0.45/0.64 tff(123,plain,
% 0.45/0.64 (((~![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))) | ((~defined(h, a)) | defined(h, codomain(a)) | (~product(codomain(a), a, a)))) <=> ((~![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))) | ((~defined(h, a)) | (~product(codomain(a), a, a)) | defined(h, codomain(a))))),
% 0.45/0.64 inference(monotonicity,[status(thm)],[122])).
% 0.45/0.64 tff(124,plain,
% 0.45/0.64 (((~![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))) | ((~defined(h, a)) | defined(h, codomain(a)) | (~product(codomain(a), a, a)))) <=> ((~![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))) | (~defined(h, a)) | (~product(codomain(a), a, a)) | defined(h, codomain(a)))),
% 0.45/0.64 inference(transitivity,[status(thm)],[123, 121])).
% 0.45/0.64 tff(125,plain,
% 0.45/0.64 ((~![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))) | ((~defined(h, a)) | defined(h, codomain(a)) | (~product(codomain(a), a, a)))),
% 0.45/0.64 inference(quant_inst,[status(thm)],[])).
% 0.45/0.64 tff(126,plain,
% 0.45/0.64 ((~![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))) | (~defined(h, a)) | (~product(codomain(a), a, a)) | defined(h, codomain(a))),
% 0.45/0.64 inference(modus_ponens,[status(thm)],[125, 124])).
% 0.45/0.64 tff(127,plain,
% 0.45/0.64 (defined(h, codomain(a))),
% 0.45/0.64 inference(unit_resolution,[status(thm)],[126, 120, 48, 79])).
% 0.45/0.64 tff(128,plain,
% 0.45/0.64 (^[X: $i] : refl(identity_map(codomain(X)) <=> identity_map(codomain(X)))),
% 0.45/0.64 inference(bind,[status(th)],[])).
% 0.45/0.64 tff(129,plain,
% 0.45/0.64 (![X: $i] : identity_map(codomain(X)) <=> ![X: $i] : identity_map(codomain(X))),
% 0.45/0.64 inference(quant_intro,[status(thm)],[128])).
% 0.45/0.64 tff(130,plain,
% 0.45/0.64 (![X: $i] : identity_map(codomain(X)) <=> ![X: $i] : identity_map(codomain(X))),
% 0.45/0.64 inference(rewrite,[status(thm)],[])).
% 0.45/0.64 tff(131,axiom,(![X: $i] : identity_map(codomain(X))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','codomain_is_an_identity_map')).
% 0.45/0.64 tff(132,plain,
% 0.45/0.64 (![X: $i] : identity_map(codomain(X))),
% 0.45/0.64 inference(modus_ponens,[status(thm)],[131, 130])).
% 0.45/0.64 tff(133,plain,(
% 0.45/0.64 ![X: $i] : identity_map(codomain(X))),
% 0.45/0.64 inference(skolemize,[status(sab)],[132])).
% 0.45/0.64 tff(134,plain,
% 0.45/0.64 (![X: $i] : identity_map(codomain(X))),
% 0.45/0.64 inference(modus_ponens,[status(thm)],[133, 129])).
% 0.45/0.64 tff(135,plain,
% 0.45/0.64 ((~![X: $i] : identity_map(codomain(X))) | identity_map(codomain(a))),
% 0.45/0.64 inference(quant_inst,[status(thm)],[])).
% 0.45/0.64 tff(136,plain,
% 0.45/0.64 (identity_map(codomain(a))),
% 0.45/0.64 inference(unit_resolution,[status(thm)],[135, 134])).
% 0.45/0.64 tff(137,assumption,(~defined(h, compose(a, b))), introduced(assumption)).
% 0.45/0.64 tff(138,plain,
% 0.45/0.64 (^[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))))),
% 0.45/0.64 inference(bind,[status(th)],[])).
% 0.45/0.64 tff(139,plain,
% 0.45/0.64 (![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)))),
% 0.45/0.64 inference(quant_intro,[status(thm)],[138])).
% 0.45/0.64 tff(140,plain,
% 0.45/0.64 (![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)))),
% 0.45/0.64 inference(rewrite,[status(thm)],[])).
% 0.45/0.64 tff(141,plain,
% 0.45/0.64 (^[Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite((((~defined(X, Y)) | (~defined(Y, Z))) | (~identity_map(Y))) <=> ((~defined(X, Y)) | (~identity_map(Y)) | (~defined(Y, Z)))), (((((~defined(X, Y)) | (~defined(Y, Z))) | (~identity_map(Y))) | defined(X, Z)) <=> (((~defined(X, Y)) | (~identity_map(Y)) | (~defined(Y, Z))) | defined(X, Z)))), rewrite((((~defined(X, Y)) | (~identity_map(Y)) | (~defined(Y, Z))) | defined(X, Z)) <=> ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))), (((((~defined(X, Y)) | (~defined(Y, Z))) | (~identity_map(Y))) | defined(X, Z)) <=> ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))))),
% 0.45/0.64 inference(bind,[status(th)],[])).
% 0.45/0.64 tff(142,plain,
% 0.45/0.64 (![Z: $i, Y: $i, X: $i] : ((((~defined(X, Y)) | (~defined(Y, Z))) | (~identity_map(Y))) | defined(X, Z)) <=> ![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))),
% 0.45/0.64 inference(quant_intro,[status(thm)],[141])).
% 0.45/0.64 tff(143,axiom,(![Z: $i, Y: $i, X: $i] : ((((~defined(X, Y)) | (~defined(Y, Z))) | (~identity_map(Y))) | defined(X, Z))), file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax','category_theory_axiom6')).
% 0.45/0.64 tff(144,plain,
% 0.45/0.64 (![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))),
% 0.45/0.64 inference(modus_ponens,[status(thm)],[143, 142])).
% 0.45/0.64 tff(145,plain,
% 0.45/0.64 (![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))),
% 0.45/0.64 inference(modus_ponens,[status(thm)],[144, 140])).
% 0.45/0.64 tff(146,plain,(
% 0.45/0.64 ![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))),
% 0.45/0.64 inference(skolemize,[status(sab)],[145])).
% 0.45/0.64 tff(147,plain,
% 0.45/0.64 (![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))),
% 0.45/0.64 inference(modus_ponens,[status(thm)],[146, 139])).
% 0.45/0.64 tff(148,plain,
% 0.45/0.64 (((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | (defined(h, compose(a, b)) | (~defined(h, codomain(a))) | (~identity_map(codomain(a))) | (~defined(codomain(a), compose(a, b))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | defined(h, compose(a, b)) | (~defined(h, codomain(a))) | (~identity_map(codomain(a))) | (~defined(codomain(a), compose(a, b))))),
% 0.45/0.64 inference(rewrite,[status(thm)],[])).
% 0.45/0.64 tff(149,plain,
% 0.45/0.64 (((~defined(h, codomain(a))) | defined(h, compose(a, b)) | (~identity_map(codomain(a))) | (~defined(codomain(a), compose(a, b)))) <=> (defined(h, compose(a, b)) | (~defined(h, codomain(a))) | (~identity_map(codomain(a))) | (~defined(codomain(a), compose(a, b))))),
% 0.45/0.64 inference(rewrite,[status(thm)],[])).
% 0.45/0.64 tff(150,plain,
% 0.45/0.64 (((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | ((~defined(h, codomain(a))) | defined(h, compose(a, b)) | (~identity_map(codomain(a))) | (~defined(codomain(a), compose(a, b))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | (defined(h, compose(a, b)) | (~defined(h, codomain(a))) | (~identity_map(codomain(a))) | (~defined(codomain(a), compose(a, b)))))),
% 0.45/0.64 inference(monotonicity,[status(thm)],[149])).
% 0.45/0.64 tff(151,plain,
% 0.45/0.64 (((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | ((~defined(h, codomain(a))) | defined(h, compose(a, b)) | (~identity_map(codomain(a))) | (~defined(codomain(a), compose(a, b))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | defined(h, compose(a, b)) | (~defined(h, codomain(a))) | (~identity_map(codomain(a))) | (~defined(codomain(a), compose(a, b))))),
% 0.45/0.65 inference(transitivity,[status(thm)],[150, 148])).
% 0.45/0.65 tff(152,plain,
% 0.45/0.65 ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | ((~defined(h, codomain(a))) | defined(h, compose(a, b)) | (~identity_map(codomain(a))) | (~defined(codomain(a), compose(a, b))))),
% 0.45/0.65 inference(quant_inst,[status(thm)],[])).
% 0.45/0.65 tff(153,plain,
% 0.45/0.65 ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | defined(h, compose(a, b)) | (~defined(h, codomain(a))) | (~identity_map(codomain(a))) | (~defined(codomain(a), compose(a, b)))),
% 0.45/0.65 inference(modus_ponens,[status(thm)],[152, 151])).
% 0.45/0.65 tff(154,plain,
% 0.45/0.65 ($false),
% 0.45/0.65 inference(unit_resolution,[status(thm)],[153, 147, 137, 136, 127, 110])).
% 0.45/0.65 tff(155,plain,(defined(h, compose(a, b))), inference(lemma,lemma(discharge,[]))).
% 0.45/0.65 tff(156,plain,
% 0.45/0.65 (^[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))))),
% 0.45/0.65 inference(bind,[status(th)],[])).
% 0.45/0.65 tff(157,plain,
% 0.45/0.65 (![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)))),
% 0.45/0.65 inference(quant_intro,[status(thm)],[156])).
% 0.45/0.65 tff(158,plain,
% 0.45/0.65 (![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)))),
% 0.45/0.65 inference(rewrite,[status(thm)],[])).
% 0.45/0.65 tff(159,plain,
% 0.45/0.65 (^[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)))))),
% 0.45/0.65 inference(bind,[status(th)],[])).
% 0.45/0.65 tff(160,plain,
% 0.45/0.65 (![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)))),
% 0.45/0.65 inference(quant_intro,[status(thm)],[159])).
% 0.45/0.65 tff(161,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')).
% 0.45/0.65 tff(162,plain,
% 0.45/0.65 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.45/0.65 inference(modus_ponens,[status(thm)],[161, 160])).
% 0.45/0.65 tff(163,plain,
% 0.45/0.65 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.45/0.65 inference(modus_ponens,[status(thm)],[162, 158])).
% 0.45/0.65 tff(164,plain,(
% 0.45/0.65 ![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.45/0.65 inference(skolemize,[status(sab)],[163])).
% 0.45/0.65 tff(165,plain,
% 0.45/0.65 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.45/0.65 inference(modus_ponens,[status(thm)],[164, 157])).
% 0.45/0.65 tff(166,plain,
% 0.45/0.65 (((~![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(a, b, compose(a, b))) | (~defined(h, compose(a, b))) | defined(compose(h, a), b) | (~product(h, a, compose(h, a))))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | (~product(a, b, compose(a, b))) | (~defined(h, compose(a, b))) | defined(compose(h, a), b) | (~product(h, a, compose(h, a))))),
% 0.45/0.65 inference(rewrite,[status(thm)],[])).
% 0.45/0.65 tff(167,plain,
% 0.45/0.65 (((~defined(h, compose(a, b))) | defined(compose(h, a), b) | (~product(a, b, compose(a, b))) | (~product(h, a, compose(h, a)))) <=> ((~product(a, b, compose(a, b))) | (~defined(h, compose(a, b))) | defined(compose(h, a), b) | (~product(h, a, compose(h, a))))),
% 0.45/0.65 inference(rewrite,[status(thm)],[])).
% 0.45/0.65 tff(168,plain,
% 0.45/0.65 (((~![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(h, compose(a, b))) | defined(compose(h, a), b) | (~product(a, b, compose(a, b))) | (~product(h, a, compose(h, a))))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~product(a, b, compose(a, b))) | (~defined(h, compose(a, b))) | defined(compose(h, a), b) | (~product(h, a, compose(h, a)))))),
% 0.45/0.65 inference(monotonicity,[status(thm)],[167])).
% 0.45/0.65 tff(169,plain,
% 0.45/0.65 (((~![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(h, compose(a, b))) | defined(compose(h, a), b) | (~product(a, b, compose(a, b))) | (~product(h, a, compose(h, a))))) <=> ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(Xy, Z) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | (~product(a, b, compose(a, b))) | (~defined(h, compose(a, b))) | defined(compose(h, a), b) | (~product(h, a, compose(h, a))))),
% 0.45/0.65 inference(transitivity,[status(thm)],[168, 166])).
% 0.45/0.65 tff(170,plain,
% 0.45/0.65 ((~![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(h, compose(a, b))) | defined(compose(h, a), b) | (~product(a, b, compose(a, b))) | (~product(h, a, compose(h, a))))),
% 0.45/0.65 inference(quant_inst,[status(thm)],[])).
% 0.45/0.65 tff(171,plain,
% 0.45/0.65 ((~![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(a, b, compose(a, b))) | (~defined(h, compose(a, b))) | defined(compose(h, a), b) | (~product(h, a, compose(h, a)))),
% 0.45/0.65 inference(modus_ponens,[status(thm)],[170, 169])).
% 0.45/0.65 tff(172,plain,
% 0.45/0.65 ((~defined(h, compose(a, b))) | defined(compose(h, a), b)),
% 0.45/0.65 inference(unit_resolution,[status(thm)],[171, 165, 25, 52])).
% 0.45/0.65 tff(173,plain,
% 0.45/0.65 (defined(compose(h, a), b)),
% 0.45/0.65 inference(unit_resolution,[status(thm)],[172, 155])).
% 0.45/0.65 tff(174,plain,
% 0.45/0.65 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(compose(h, a), b)) | product(compose(h, a), b, compose(compose(h, a), b)))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(compose(h, a), b)) | product(compose(h, a), b, compose(compose(h, a), b)))),
% 0.45/0.65 inference(rewrite,[status(thm)],[])).
% 0.45/0.65 tff(175,plain,
% 0.45/0.65 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(compose(h, a), b)) | product(compose(h, a), b, compose(compose(h, a), b)))),
% 0.45/0.65 inference(quant_inst,[status(thm)],[])).
% 0.45/0.65 tff(176,plain,
% 0.45/0.65 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(compose(h, a), b)) | product(compose(h, a), b, compose(compose(h, a), b))),
% 0.45/0.65 inference(modus_ponens,[status(thm)],[175, 174])).
% 0.48/0.65 tff(177,plain,
% 0.48/0.65 (product(compose(h, a), b, compose(compose(h, a), b))),
% 0.48/0.65 inference(unit_resolution,[status(thm)],[176, 21, 173])).
% 0.48/0.65 tff(178,plain,
% 0.48/0.65 (^[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))))),
% 0.48/0.65 inference(bind,[status(th)],[])).
% 0.48/0.65 tff(179,plain,
% 0.48/0.65 (![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)))),
% 0.48/0.65 inference(quant_intro,[status(thm)],[178])).
% 0.48/0.65 tff(180,plain,
% 0.48/0.65 (![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)))),
% 0.48/0.65 inference(rewrite,[status(thm)],[])).
% 0.48/0.65 tff(181,plain,
% 0.48/0.65 (^[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)))))),
% 0.48/0.65 inference(bind,[status(th)],[])).
% 0.48/0.65 tff(182,plain,
% 0.48/0.65 (![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)))),
% 0.48/0.65 inference(quant_intro,[status(thm)],[181])).
% 0.48/0.65 tff(183,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')).
% 0.48/0.65 tff(184,plain,
% 0.48/0.65 (![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)))),
% 0.48/0.65 inference(modus_ponens,[status(thm)],[183, 182])).
% 0.48/0.65 tff(185,plain,
% 0.48/0.65 (![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)))),
% 0.48/0.65 inference(modus_ponens,[status(thm)],[184, 180])).
% 0.48/0.65 tff(186,plain,(
% 0.48/0.65 ![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)))),
% 0.48/0.65 inference(skolemize,[status(sab)],[185])).
% 0.48/0.65 tff(187,plain,
% 0.48/0.65 (![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)))),
% 0.48/0.65 inference(modus_ponens,[status(thm)],[186, 179])).
% 0.48/0.65 tff(188,plain,
% 0.48/0.65 (((~![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), b, compose(a, b))) | (~product(g, compose(codomain(a), a), compose(h, a))) | product(g, compose(a, b), compose(compose(h, a), b)) | (~product(compose(h, a), b, compose(compose(h, 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), b, compose(a, b))) | (~product(g, compose(codomain(a), a), compose(h, a))) | product(g, compose(a, b), compose(compose(h, a), b)) | (~product(compose(h, a), b, compose(compose(h, a), b))))),
% 0.48/0.66 inference(rewrite,[status(thm)],[])).
% 0.48/0.66 tff(189,plain,
% 0.48/0.66 ((product(g, compose(a, b), compose(compose(h, a), b)) | (~product(compose(codomain(a), a), b, compose(a, b))) | (~product(compose(h, a), b, compose(compose(h, a), b))) | (~product(g, compose(codomain(a), a), compose(h, a)))) <=> ((~product(compose(codomain(a), a), b, compose(a, b))) | (~product(g, compose(codomain(a), a), compose(h, a))) | product(g, compose(a, b), compose(compose(h, a), b)) | (~product(compose(h, a), b, compose(compose(h, a), b))))),
% 0.48/0.66 inference(rewrite,[status(thm)],[])).
% 0.48/0.66 tff(190,plain,
% 0.48/0.66 (((~![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(g, compose(a, b), compose(compose(h, a), b)) | (~product(compose(codomain(a), a), b, compose(a, b))) | (~product(compose(h, a), b, compose(compose(h, a), b))) | (~product(g, compose(codomain(a), a), compose(h, a))))) <=> ((~![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), b, compose(a, b))) | (~product(g, compose(codomain(a), a), compose(h, a))) | product(g, compose(a, b), compose(compose(h, a), b)) | (~product(compose(h, a), b, compose(compose(h, a), b)))))),
% 0.48/0.66 inference(monotonicity,[status(thm)],[189])).
% 0.48/0.66 tff(191,plain,
% 0.48/0.66 (((~![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(g, compose(a, b), compose(compose(h, a), b)) | (~product(compose(codomain(a), a), b, compose(a, b))) | (~product(compose(h, a), b, compose(compose(h, a), b))) | (~product(g, compose(codomain(a), a), compose(h, a))))) <=> ((~![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), b, compose(a, b))) | (~product(g, compose(codomain(a), a), compose(h, a))) | product(g, compose(a, b), compose(compose(h, a), b)) | (~product(compose(h, a), b, compose(compose(h, a), b))))),
% 0.48/0.66 inference(transitivity,[status(thm)],[190, 188])).
% 0.48/0.66 tff(192,plain,
% 0.48/0.66 ((~![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(g, compose(a, b), compose(compose(h, a), b)) | (~product(compose(codomain(a), a), b, compose(a, b))) | (~product(compose(h, a), b, compose(compose(h, a), b))) | (~product(g, compose(codomain(a), a), compose(h, a))))),
% 0.48/0.66 inference(quant_inst,[status(thm)],[])).
% 0.48/0.66 tff(193,plain,
% 0.48/0.66 ((~![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), b, compose(a, b))) | (~product(g, compose(codomain(a), a), compose(h, a))) | product(g, compose(a, b), compose(compose(h, a), b)) | (~product(compose(h, a), b, compose(compose(h, a), b)))),
% 0.48/0.66 inference(modus_ponens,[status(thm)],[192, 191])).
% 0.48/0.66 tff(194,plain,
% 0.48/0.66 ((~product(g, compose(codomain(a), a), compose(h, a))) | product(g, compose(a, b), compose(compose(h, a), b))),
% 0.48/0.66 inference(unit_resolution,[status(thm)],[193, 187, 177, 93])).
% 0.48/0.66 tff(195,plain,
% 0.48/0.66 (product(g, compose(a, b), compose(compose(h, a), b))),
% 0.48/0.66 inference(unit_resolution,[status(thm)],[194, 90])).
% 0.48/0.66 tff(196,plain,
% 0.48/0.66 (product(g, c, compose(compose(h, a), b))),
% 0.48/0.66 inference(modus_ponens,[status(thm)],[195, 41])).
% 0.48/0.66 tff(197,plain,
% 0.48/0.66 (product(h, c, compose(compose(h, a), b)) <=> product(h, compose(a, b), compose(compose(h, a), b))),
% 0.48/0.66 inference(monotonicity,[status(thm)],[39])).
% 0.48/0.66 tff(198,plain,
% 0.48/0.66 (product(h, compose(a, b), compose(compose(h, a), b)) <=> product(h, c, compose(compose(h, a), b))),
% 0.48/0.66 inference(symmetry,[status(thm)],[197])).
% 0.48/0.66 tff(199,plain,
% 0.48/0.66 (product(h, compose(codomain(a), a), compose(h, a)) <=> product(h, a, compose(h, a))),
% 0.48/0.66 inference(monotonicity,[status(thm)],[84])).
% 0.48/0.66 tff(200,plain,
% 0.48/0.66 (product(h, a, compose(h, a)) <=> product(h, compose(codomain(a), a), compose(h, a))),
% 0.48/0.66 inference(symmetry,[status(thm)],[199])).
% 0.48/0.66 tff(201,plain,
% 0.48/0.66 (product(h, compose(codomain(a), a), compose(h, a))),
% 0.48/0.66 inference(modus_ponens,[status(thm)],[52, 200])).
% 0.48/0.66 tff(202,plain,
% 0.48/0.66 (((~![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), b, compose(a, b))) | (~product(h, compose(codomain(a), a), compose(h, a))) | product(h, compose(a, b), compose(compose(h, a), b)) | (~product(compose(h, a), b, compose(compose(h, 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), b, compose(a, b))) | (~product(h, compose(codomain(a), a), compose(h, a))) | product(h, compose(a, b), compose(compose(h, a), b)) | (~product(compose(h, a), b, compose(compose(h, a), b))))),
% 0.48/0.66 inference(rewrite,[status(thm)],[])).
% 0.48/0.66 tff(203,plain,
% 0.48/0.66 ((product(h, compose(a, b), compose(compose(h, a), b)) | (~product(compose(codomain(a), a), b, compose(a, b))) | (~product(compose(h, a), b, compose(compose(h, a), b))) | (~product(h, compose(codomain(a), a), compose(h, a)))) <=> ((~product(compose(codomain(a), a), b, compose(a, b))) | (~product(h, compose(codomain(a), a), compose(h, a))) | product(h, compose(a, b), compose(compose(h, a), b)) | (~product(compose(h, a), b, compose(compose(h, a), b))))),
% 0.48/0.66 inference(rewrite,[status(thm)],[])).
% 0.48/0.66 tff(204,plain,
% 0.48/0.66 (((~![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(h, compose(a, b), compose(compose(h, a), b)) | (~product(compose(codomain(a), a), b, compose(a, b))) | (~product(compose(h, a), b, compose(compose(h, a), b))) | (~product(h, compose(codomain(a), a), compose(h, a))))) <=> ((~![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), b, compose(a, b))) | (~product(h, compose(codomain(a), a), compose(h, a))) | product(h, compose(a, b), compose(compose(h, a), b)) | (~product(compose(h, a), b, compose(compose(h, a), b)))))),
% 0.48/0.66 inference(monotonicity,[status(thm)],[203])).
% 0.48/0.66 tff(205,plain,
% 0.48/0.66 (((~![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(h, compose(a, b), compose(compose(h, a), b)) | (~product(compose(codomain(a), a), b, compose(a, b))) | (~product(compose(h, a), b, compose(compose(h, a), b))) | (~product(h, compose(codomain(a), a), compose(h, a))))) <=> ((~![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), b, compose(a, b))) | (~product(h, compose(codomain(a), a), compose(h, a))) | product(h, compose(a, b), compose(compose(h, a), b)) | (~product(compose(h, a), b, compose(compose(h, a), b))))),
% 0.48/0.66 inference(transitivity,[status(thm)],[204, 202])).
% 0.48/0.66 tff(206,plain,
% 0.48/0.66 ((~![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(h, compose(a, b), compose(compose(h, a), b)) | (~product(compose(codomain(a), a), b, compose(a, b))) | (~product(compose(h, a), b, compose(compose(h, a), b))) | (~product(h, compose(codomain(a), a), compose(h, a))))),
% 0.48/0.66 inference(quant_inst,[status(thm)],[])).
% 0.48/0.66 tff(207,plain,
% 0.48/0.66 ((~![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), b, compose(a, b))) | (~product(h, compose(codomain(a), a), compose(h, a))) | product(h, compose(a, b), compose(compose(h, a), b)) | (~product(compose(h, a), b, compose(compose(h, a), b)))),
% 0.48/0.67 inference(modus_ponens,[status(thm)],[206, 205])).
% 0.48/0.67 tff(208,plain,
% 0.48/0.67 ((~product(h, compose(codomain(a), a), compose(h, a))) | product(h, compose(a, b), compose(compose(h, a), b))),
% 0.48/0.67 inference(unit_resolution,[status(thm)],[207, 187, 177, 93])).
% 0.48/0.67 tff(209,plain,
% 0.48/0.67 (product(h, compose(a, b), compose(compose(h, a), b))),
% 0.48/0.67 inference(unit_resolution,[status(thm)],[208, 201])).
% 0.48/0.67 tff(210,plain,
% 0.48/0.67 (product(h, c, compose(compose(h, a), b))),
% 0.48/0.67 inference(modus_ponens,[status(thm)],[209, 198])).
% 0.48/0.67 tff(211,plain,
% 0.48/0.67 ((~(h = g)) <=> (~(h = g))),
% 0.48/0.67 inference(rewrite,[status(thm)],[])).
% 0.48/0.67 tff(212,axiom,(~(h = g)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_h_equals_g')).
% 0.48/0.67 tff(213,plain,
% 0.48/0.67 (~(h = g)),
% 0.48/0.67 inference(modus_ponens,[status(thm)],[212, 211])).
% 0.48/0.67 tff(214,plain,
% 0.48/0.67 (^[W: $i, Y: $i, X: $i] : refl(((X = Y) | (~product(Y, c, W)) | (~product(X, c, W))) <=> ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W))))),
% 0.48/0.67 inference(bind,[status(th)],[])).
% 0.48/0.67 tff(215,plain,
% 0.48/0.67 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W))) <=> ![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W)))),
% 0.48/0.67 inference(quant_intro,[status(thm)],[214])).
% 0.48/0.67 tff(216,plain,
% 0.48/0.67 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W))) <=> ![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W)))),
% 0.48/0.67 inference(rewrite,[status(thm)],[])).
% 0.48/0.67 tff(217,plain,
% 0.48/0.67 (^[W: $i, Y: $i, X: $i] : rewrite((((~product(X, c, W)) | (~product(Y, c, W))) | (X = Y)) <=> ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W))))),
% 0.48/0.67 inference(bind,[status(th)],[])).
% 0.48/0.67 tff(218,plain,
% 0.48/0.67 (![W: $i, Y: $i, X: $i] : (((~product(X, c, W)) | (~product(Y, c, W))) | (X = Y)) <=> ![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W)))),
% 0.48/0.67 inference(quant_intro,[status(thm)],[217])).
% 0.48/0.67 tff(219,axiom,(![W: $i, Y: $i, X: $i] : (((~product(X, c, W)) | (~product(Y, c, W))) | (X = Y))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','cancellation_for_product')).
% 0.48/0.67 tff(220,plain,
% 0.48/0.67 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W)))),
% 0.48/0.67 inference(modus_ponens,[status(thm)],[219, 218])).
% 0.48/0.67 tff(221,plain,
% 0.48/0.67 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W)))),
% 0.48/0.67 inference(modus_ponens,[status(thm)],[220, 216])).
% 0.48/0.67 tff(222,plain,(
% 0.48/0.67 ![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W)))),
% 0.48/0.67 inference(skolemize,[status(sab)],[221])).
% 0.48/0.67 tff(223,plain,
% 0.48/0.67 (![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W)))),
% 0.48/0.67 inference(modus_ponens,[status(thm)],[222, 215])).
% 0.48/0.67 tff(224,plain,
% 0.48/0.67 (((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W)))) | ((h = g) | (~product(g, c, compose(compose(h, a), b))) | (~product(h, c, compose(compose(h, a), b))))) <=> ((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W)))) | (h = g) | (~product(g, c, compose(compose(h, a), b))) | (~product(h, c, compose(compose(h, a), b))))),
% 0.48/0.67 inference(rewrite,[status(thm)],[])).
% 0.48/0.67 tff(225,plain,
% 0.48/0.67 ((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W)))) | ((h = g) | (~product(g, c, compose(compose(h, a), b))) | (~product(h, c, compose(compose(h, a), b))))),
% 0.48/0.67 inference(quant_inst,[status(thm)],[])).
% 0.48/0.67 tff(226,plain,
% 0.48/0.67 ((~![W: $i, Y: $i, X: $i] : ((X = Y) | (~product(Y, c, W)) | (~product(X, c, W)))) | (h = g) | (~product(g, c, compose(compose(h, a), b))) | (~product(h, c, compose(compose(h, a), b)))),
% 0.48/0.67 inference(modus_ponens,[status(thm)],[225, 224])).
% 0.48/0.67 tff(227,plain,
% 0.48/0.67 ((~product(g, c, compose(compose(h, a), b))) | (~product(h, c, compose(compose(h, a), b)))),
% 0.48/0.67 inference(unit_resolution,[status(thm)],[226, 223, 213])).
% 0.48/0.67 tff(228,plain,
% 0.48/0.67 ($false),
% 0.48/0.67 inference(unit_resolution,[status(thm)],[227, 210, 196])).
% 0.48/0.67 % SZS output end Proof
% 0.48/0.67 % E exiting
%------------------------------------------------------------------------------