%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : CAT001-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n008.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 05:00:17 AM UTC 2025
% Result : Unsatisfiable 0.43s 0.59s
% Output : Proof 0.43s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : CAT001-1 : TPTP v9.0.0. Released v1.0.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n008.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 11:49:33 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.43/0.59 % SZS status Unsatisfiable
% 0.43/0.59 % SZS output start Proof
% 0.43/0.59 tff(product_type, type, (
% 0.43/0.59 product: ( $i * $i * $i ) > $o)).
% 0.43/0.59 tff(compose_type, type, (
% 0.43/0.59 compose: ( $i * $i ) > $i)).
% 0.43/0.59 tff(h_type, type, (
% 0.43/0.59 h: $i)).
% 0.43/0.59 tff(b_type, type, (
% 0.43/0.59 b: $i)).
% 0.43/0.59 tff(a_type, type, (
% 0.43/0.59 a: $i)).
% 0.43/0.59 tff(g_type, type, (
% 0.43/0.59 g: $i)).
% 0.43/0.59 tff(c_type, type, (
% 0.43/0.59 c: $i)).
% 0.43/0.59 tff(defined_type, type, (
% 0.43/0.59 defined: ( $i * $i ) > $o)).
% 0.43/0.59 tff(codomain_type, type, (
% 0.43/0.59 codomain: $i > $i)).
% 0.43/0.59 tff(d_type, type, (
% 0.43/0.59 d: $i)).
% 0.43/0.59 tff(identity_map_type, type, (
% 0.43/0.59 identity_map: $i > $o)).
% 0.43/0.59 tff(domain_type, type, (
% 0.43/0.59 domain: $i > $i)).
% 0.43/0.59 tff(1,plain,
% 0.43/0.59 (product(a, b, c) <=> product(a, b, c)),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(2,axiom,(product(a, b, c)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','ab_equals_c')).
% 0.43/0.59 tff(3,plain,
% 0.43/0.59 (product(a, b, c)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[2, 1])).
% 0.43/0.59 tff(4,plain,
% 0.43/0.59 (^[Z: $i, Y: $i, X: $i] : refl(((~product(X, Y, Z)) | defined(X, Y)) <=> ((~product(X, Y, Z)) | defined(X, Y)))),
% 0.43/0.59 inference(bind,[status(th)],[])).
% 0.43/0.59 tff(5,plain,
% 0.43/0.59 (![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))),
% 0.43/0.59 inference(quant_intro,[status(thm)],[4])).
% 0.43/0.59 tff(6,plain,
% 0.43/0.59 (![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))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(7,axiom,(![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax','associative_property1')).
% 0.43/0.59 tff(8,plain,
% 0.43/0.59 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[7, 6])).
% 0.43/0.59 tff(9,plain,(
% 0.43/0.59 ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))),
% 0.43/0.59 inference(skolemize,[status(sab)],[8])).
% 0.43/0.59 tff(10,plain,
% 0.43/0.59 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.43/0.59 tff(11,plain,
% 0.43/0.59 (((~![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))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(12,plain,
% 0.43/0.59 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(a, b, c)) | defined(a, b))),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(13,plain,
% 0.43/0.59 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(a, b, c)) | defined(a, b)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[12, 11])).
% 0.43/0.59 tff(14,plain,
% 0.43/0.59 (defined(a, b)),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[13, 10, 3])).
% 0.43/0.59 tff(15,plain,
% 0.43/0.59 (^[Y: $i, X: $i] : refl(((~defined(X, Y)) | product(X, Y, compose(X, Y))) <=> ((~defined(X, Y)) | product(X, Y, compose(X, Y))))),
% 0.43/0.59 inference(bind,[status(th)],[])).
% 0.43/0.59 tff(16,plain,
% 0.43/0.59 (![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)))),
% 0.43/0.59 inference(quant_intro,[status(thm)],[15])).
% 0.43/0.59 tff(17,plain,
% 0.43/0.59 (![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)))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(18,axiom,(![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))), file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax','closure_of_composition')).
% 0.43/0.59 tff(19,plain,
% 0.43/0.59 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.43/0.59 tff(20,plain,(
% 0.43/0.59 ![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))),
% 0.43/0.59 inference(skolemize,[status(sab)],[19])).
% 0.43/0.59 tff(21,plain,
% 0.43/0.59 (![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[20, 16])).
% 0.43/0.59 tff(22,plain,
% 0.43/0.59 (((~![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)))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(23,plain,
% 0.43/0.59 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(a, b)) | product(a, b, compose(a, b)))),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(24,plain,
% 0.43/0.59 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(a, b)) | product(a, b, compose(a, b))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.43/0.59 tff(25,plain,
% 0.43/0.59 (product(a, b, compose(a, b))),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[24, 21, 14])).
% 0.43/0.59 tff(26,plain,
% 0.43/0.59 (^[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)))),
% 0.43/0.59 inference(bind,[status(th)],[])).
% 0.43/0.59 tff(27,plain,
% 0.43/0.59 (![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))),
% 0.43/0.59 inference(quant_intro,[status(thm)],[26])).
% 0.43/0.59 tff(28,plain,
% 0.43/0.59 (![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))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(29,plain,
% 0.43/0.59 (^[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)))),
% 0.43/0.59 inference(bind,[status(th)],[])).
% 0.43/0.59 tff(30,plain,
% 0.43/0.59 (![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))),
% 0.43/0.59 inference(quant_intro,[status(thm)],[29])).
% 0.43/0.59 tff(31,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, Z)) | (~product(X, Y, W))) | (Z = W))), file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax','composition_is_well_defined')).
% 0.43/0.59 tff(32,plain,
% 0.43/0.59 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[31, 30])).
% 0.43/0.59 tff(33,plain,
% 0.43/0.59 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[32, 28])).
% 0.43/0.59 tff(34,plain,(
% 0.43/0.59 ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.43/0.59 inference(skolemize,[status(sab)],[33])).
% 0.43/0.59 tff(35,plain,
% 0.43/0.59 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[34, 27])).
% 0.43/0.59 tff(36,plain,
% 0.43/0.59 (((~![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)))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(37,plain,
% 0.43/0.59 ((~![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)))),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(38,plain,
% 0.43/0.59 ((~![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))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.43/0.59 tff(39,plain,
% 0.43/0.59 (c = compose(a, b)),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[38, 35, 3, 25])).
% 0.43/0.59 tff(40,plain,
% 0.43/0.59 (product(c, g, compose(a, compose(b, h))) <=> product(compose(a, b), g, compose(a, compose(b, h)))),
% 0.43/0.59 inference(monotonicity,[status(thm)],[39])).
% 0.43/0.59 tff(41,plain,
% 0.43/0.59 (product(compose(a, b), g, compose(a, compose(b, h))) <=> product(c, g, compose(a, compose(b, h)))),
% 0.43/0.59 inference(symmetry,[status(thm)],[40])).
% 0.43/0.59 tff(42,plain,
% 0.43/0.59 (^[X: $i] : refl(defined(codomain(X), X) <=> defined(codomain(X), X))),
% 0.43/0.59 inference(bind,[status(th)],[])).
% 0.43/0.59 tff(43,plain,
% 0.43/0.59 (![X: $i] : defined(codomain(X), X) <=> ![X: $i] : defined(codomain(X), X)),
% 0.43/0.59 inference(quant_intro,[status(thm)],[42])).
% 0.43/0.59 tff(44,plain,
% 0.43/0.59 (![X: $i] : defined(codomain(X), X) <=> ![X: $i] : defined(codomain(X), X)),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(45,axiom,(![X: $i] : defined(codomain(X), X)), file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax','mapping_from_codomain_of_x_to_x')).
% 0.43/0.59 tff(46,plain,
% 0.43/0.59 (![X: $i] : defined(codomain(X), X)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.43/0.59 tff(47,plain,(
% 0.43/0.59 ![X: $i] : defined(codomain(X), X)),
% 0.43/0.59 inference(skolemize,[status(sab)],[46])).
% 0.43/0.59 tff(48,plain,
% 0.43/0.59 (![X: $i] : defined(codomain(X), X)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[47, 43])).
% 0.43/0.59 tff(49,plain,
% 0.43/0.59 ((~![X: $i] : defined(codomain(X), X)) | defined(codomain(b), b)),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(50,plain,
% 0.43/0.59 (defined(codomain(b), b)),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[49, 48])).
% 0.43/0.59 tff(51,plain,
% 0.43/0.59 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(codomain(b), b)) | product(codomain(b), b, compose(codomain(b), b)))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(codomain(b), b)) | product(codomain(b), b, compose(codomain(b), b)))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(52,plain,
% 0.43/0.59 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(codomain(b), b)) | product(codomain(b), b, compose(codomain(b), b)))),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(53,plain,
% 0.43/0.59 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(codomain(b), b)) | product(codomain(b), b, compose(codomain(b), b))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.43/0.59 tff(54,plain,
% 0.43/0.59 (product(codomain(b), b, compose(codomain(b), b))),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[53, 21, 50])).
% 0.43/0.59 tff(55,plain,
% 0.43/0.59 (^[X: $i] : refl(product(codomain(X), X, X) <=> product(codomain(X), X, X))),
% 0.43/0.59 inference(bind,[status(th)],[])).
% 0.43/0.59 tff(56,plain,
% 0.43/0.59 (![X: $i] : product(codomain(X), X, X) <=> ![X: $i] : product(codomain(X), X, X)),
% 0.43/0.59 inference(quant_intro,[status(thm)],[55])).
% 0.43/0.59 tff(57,plain,
% 0.43/0.59 (![X: $i] : product(codomain(X), X, X) <=> ![X: $i] : product(codomain(X), X, X)),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(58,axiom,(![X: $i] : product(codomain(X), X, X)), file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax','product_on_codomain')).
% 0.43/0.59 tff(59,plain,
% 0.43/0.59 (![X: $i] : product(codomain(X), X, X)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.43/0.59 tff(60,plain,(
% 0.43/0.59 ![X: $i] : product(codomain(X), X, X)),
% 0.43/0.59 inference(skolemize,[status(sab)],[59])).
% 0.43/0.59 tff(61,plain,
% 0.43/0.59 (![X: $i] : product(codomain(X), X, X)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[60, 56])).
% 0.43/0.59 tff(62,plain,
% 0.43/0.59 ((~![X: $i] : product(codomain(X), X, X)) | product(codomain(b), b, b)),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(63,plain,
% 0.43/0.59 (product(codomain(b), b, b)),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[62, 61])).
% 0.43/0.59 tff(64,plain,
% 0.43/0.59 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(codomain(b), b, b)) | (~product(codomain(b), b, compose(codomain(b), b))) | (b = compose(codomain(b), b)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(codomain(b), b, b)) | (~product(codomain(b), b, compose(codomain(b), b))) | (b = compose(codomain(b), b)))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(65,plain,
% 0.43/0.59 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(codomain(b), b, b)) | (~product(codomain(b), b, compose(codomain(b), b))) | (b = compose(codomain(b), b)))),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(66,plain,
% 0.43/0.59 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(codomain(b), b, b)) | (~product(codomain(b), b, compose(codomain(b), b))) | (b = compose(codomain(b), b))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.43/0.59 tff(67,plain,
% 0.43/0.59 (b = compose(codomain(b), b)),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[66, 35, 63, 54])).
% 0.43/0.59 tff(68,plain,
% 0.43/0.59 (compose(codomain(b), b) = b),
% 0.43/0.59 inference(symmetry,[status(thm)],[67])).
% 0.43/0.59 tff(69,plain,
% 0.43/0.59 (product(a, compose(codomain(b), b), compose(a, b)) <=> product(a, b, compose(a, b))),
% 0.43/0.59 inference(monotonicity,[status(thm)],[68])).
% 0.43/0.59 tff(70,plain,
% 0.43/0.59 (product(a, b, compose(a, b)) <=> product(a, compose(codomain(b), b), compose(a, b))),
% 0.43/0.59 inference(symmetry,[status(thm)],[69])).
% 0.43/0.59 tff(71,plain,
% 0.43/0.59 (product(a, compose(codomain(b), b), compose(a, b))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[25, 70])).
% 0.43/0.59 tff(72,plain,
% 0.43/0.59 (product(b, h, d) <=> product(b, h, d)),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(73,axiom,(product(b, h, d)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','bh_equals_d')).
% 0.43/0.59 tff(74,plain,
% 0.43/0.59 (product(b, h, d)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.43/0.59 tff(75,plain,
% 0.43/0.59 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(b, h, d)) | defined(b, h))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(b, h, d)) | defined(b, h))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(76,plain,
% 0.43/0.59 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | ((~product(b, h, d)) | defined(b, h))),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(77,plain,
% 0.43/0.59 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | defined(X, Y))) | (~product(b, h, d)) | defined(b, h)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.43/0.59 tff(78,plain,
% 0.43/0.59 (defined(b, h)),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[77, 10, 74])).
% 0.43/0.59 tff(79,plain,
% 0.43/0.59 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(b, h)) | product(b, h, compose(b, h)))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(b, h)) | product(b, h, compose(b, h)))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(80,plain,
% 0.43/0.59 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(b, h)) | product(b, h, compose(b, h)))),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(81,plain,
% 0.43/0.59 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(b, h)) | product(b, h, compose(b, h))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[80, 79])).
% 0.43/0.59 tff(82,plain,
% 0.43/0.59 (product(b, h, compose(b, h))),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[81, 21, 78])).
% 0.43/0.59 tff(83,plain,
% 0.43/0.59 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(b, h, d)) | (~product(b, h, compose(b, h))) | (d = compose(b, h)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(b, h, d)) | (~product(b, h, compose(b, h))) | (d = compose(b, h)))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(84,plain,
% 0.43/0.59 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | ((~product(b, h, d)) | (~product(b, h, compose(b, h))) | (d = compose(b, h)))),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(85,plain,
% 0.43/0.59 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | (~product(X, Y, W)) | (Z = W))) | (~product(b, h, d)) | (~product(b, h, compose(b, h))) | (d = compose(b, h))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.43/0.59 tff(86,plain,
% 0.43/0.59 (d = compose(b, h)),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[85, 35, 74, 82])).
% 0.43/0.59 tff(87,plain,
% 0.43/0.59 (compose(b, h) = d),
% 0.43/0.59 inference(symmetry,[status(thm)],[86])).
% 0.43/0.59 tff(88,plain,
% 0.43/0.59 (product(compose(codomain(b), b), g, compose(b, h)) <=> product(b, g, d)),
% 0.43/0.59 inference(monotonicity,[status(thm)],[68, 87])).
% 0.43/0.59 tff(89,plain,
% 0.43/0.59 (product(b, g, d) <=> product(compose(codomain(b), b), g, compose(b, h))),
% 0.43/0.59 inference(symmetry,[status(thm)],[88])).
% 0.43/0.59 tff(90,plain,
% 0.43/0.59 (product(b, g, d) <=> product(b, g, d)),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(91,axiom,(product(b, g, d)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','bg_equals_d')).
% 0.43/0.59 tff(92,plain,
% 0.43/0.59 (product(b, g, d)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.43/0.59 tff(93,plain,
% 0.43/0.59 (product(compose(codomain(b), b), g, compose(b, h))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[92, 89])).
% 0.43/0.59 tff(94,plain,
% 0.43/0.59 (^[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))))),
% 0.43/0.59 inference(bind,[status(th)],[])).
% 0.43/0.59 tff(95,plain,
% 0.43/0.59 (![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)))),
% 0.43/0.59 inference(quant_intro,[status(thm)],[94])).
% 0.43/0.59 tff(96,plain,
% 0.43/0.59 (![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)))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(97,plain,
% 0.43/0.59 (^[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)))))),
% 0.43/0.59 inference(bind,[status(th)],[])).
% 0.43/0.59 tff(98,plain,
% 0.43/0.59 (![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)))),
% 0.43/0.59 inference(quant_intro,[status(thm)],[97])).
% 0.43/0.59 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/sandbox/benchmark/Axioms/CAT001-0.ax','category_theory_axiom1')).
% 0.43/0.59 tff(100,plain,
% 0.43/0.59 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.43/0.59 tff(101,plain,
% 0.43/0.59 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[100, 96])).
% 0.43/0.59 tff(102,plain,(
% 0.43/0.59 ![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.43/0.59 inference(skolemize,[status(sab)],[101])).
% 0.43/0.59 tff(103,plain,
% 0.43/0.59 (![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[102, 95])).
% 0.43/0.59 tff(104,plain,
% 0.43/0.59 (((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | ((~defined(b, h)) | (~product(b, h, compose(b, h))) | defined(codomain(b), compose(b, h)) | (~product(codomain(b), b, 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(b, h)) | (~product(b, h, compose(b, h))) | defined(codomain(b), compose(b, h)) | (~product(codomain(b), b, b)))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(105,plain,
% 0.43/0.59 ((defined(codomain(b), compose(b, h)) | (~defined(b, h)) | (~product(b, h, compose(b, h))) | (~product(codomain(b), b, b))) <=> ((~defined(b, h)) | (~product(b, h, compose(b, h))) | defined(codomain(b), compose(b, h)) | (~product(codomain(b), b, b)))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(106,plain,
% 0.43/0.59 (((~![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(b), compose(b, h)) | (~defined(b, h)) | (~product(b, h, compose(b, h))) | (~product(codomain(b), b, 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(b, h)) | (~product(b, h, compose(b, h))) | defined(codomain(b), compose(b, h)) | (~product(codomain(b), b, b))))),
% 0.43/0.60 inference(monotonicity,[status(thm)],[105])).
% 0.43/0.60 tff(107,plain,
% 0.43/0.60 (((~![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(b), compose(b, h)) | (~defined(b, h)) | (~product(b, h, compose(b, h))) | (~product(codomain(b), b, 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(b, h)) | (~product(b, h, compose(b, h))) | defined(codomain(b), compose(b, h)) | (~product(codomain(b), b, b)))),
% 0.43/0.60 inference(transitivity,[status(thm)],[106, 104])).
% 0.43/0.60 tff(108,plain,
% 0.43/0.60 ((~![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(b), compose(b, h)) | (~defined(b, h)) | (~product(b, h, compose(b, h))) | (~product(codomain(b), b, b)))),
% 0.43/0.60 inference(quant_inst,[status(thm)],[])).
% 0.43/0.60 tff(109,plain,
% 0.43/0.60 ((~![Xy: $i, Z: $i, Y: $i, X: $i, Yz: $i] : (defined(X, Yz) | (~defined(Xy, Z)) | (~product(Y, Z, Yz)) | (~product(X, Y, Xy)))) | (~defined(b, h)) | (~product(b, h, compose(b, h))) | defined(codomain(b), compose(b, h)) | (~product(codomain(b), b, b))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[108, 107])).
% 0.43/0.60 tff(110,plain,
% 0.43/0.60 (defined(codomain(b), compose(b, h))),
% 0.43/0.60 inference(unit_resolution,[status(thm)],[109, 103, 78, 82, 63])).
% 0.43/0.60 tff(111,plain,
% 0.43/0.60 (^[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))))),
% 0.43/0.60 inference(bind,[status(th)],[])).
% 0.43/0.60 tff(112,plain,
% 0.43/0.60 (![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)))),
% 0.43/0.60 inference(quant_intro,[status(thm)],[111])).
% 0.43/0.60 tff(113,plain,
% 0.43/0.60 (![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)))),
% 0.43/0.60 inference(rewrite,[status(thm)],[])).
% 0.43/0.60 tff(114,plain,
% 0.43/0.60 (^[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)))))),
% 0.43/0.60 inference(bind,[status(th)],[])).
% 0.43/0.60 tff(115,plain,
% 0.43/0.60 (![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)))),
% 0.43/0.60 inference(quant_intro,[status(thm)],[114])).
% 0.43/0.60 tff(116,axiom,(![Z: $i, Y: $i, X: $i, Yz: $i] : (((~product(Y, Z, Yz)) | (~defined(X, Yz))) | defined(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax','category_theory_axiom3')).
% 0.43/0.60 tff(117,plain,
% 0.43/0.60 (![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[116, 115])).
% 0.43/0.60 tff(118,plain,
% 0.43/0.60 (![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[117, 113])).
% 0.43/0.60 tff(119,plain,(
% 0.43/0.60 ![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))),
% 0.43/0.60 inference(skolemize,[status(sab)],[118])).
% 0.43/0.60 tff(120,plain,
% 0.43/0.60 (![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[119, 112])).
% 0.43/0.60 tff(121,plain,
% 0.43/0.60 (((~![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))) | ((~defined(a, b)) | defined(a, codomain(b)) | (~product(codomain(b), b, b)))) <=> ((~![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))) | (~defined(a, b)) | defined(a, codomain(b)) | (~product(codomain(b), b, b)))),
% 0.43/0.60 inference(rewrite,[status(thm)],[])).
% 0.43/0.60 tff(122,plain,
% 0.43/0.60 ((~![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))) | ((~defined(a, b)) | defined(a, codomain(b)) | (~product(codomain(b), b, b)))),
% 0.43/0.60 inference(quant_inst,[status(thm)],[])).
% 0.43/0.60 tff(123,plain,
% 0.43/0.60 ((~![Z: $i, Y: $i, X: $i, Yz: $i] : ((~defined(X, Yz)) | defined(X, Y) | (~product(Y, Z, Yz)))) | (~defined(a, b)) | defined(a, codomain(b)) | (~product(codomain(b), b, b))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[122, 121])).
% 0.43/0.60 tff(124,plain,
% 0.43/0.60 (defined(a, codomain(b))),
% 0.43/0.60 inference(unit_resolution,[status(thm)],[123, 120, 14, 63])).
% 0.43/0.60 tff(125,plain,
% 0.43/0.60 (^[X: $i] : refl(identity_map(codomain(X)) <=> identity_map(codomain(X)))),
% 0.43/0.60 inference(bind,[status(th)],[])).
% 0.43/0.60 tff(126,plain,
% 0.43/0.60 (![X: $i] : identity_map(codomain(X)) <=> ![X: $i] : identity_map(codomain(X))),
% 0.43/0.60 inference(quant_intro,[status(thm)],[125])).
% 0.43/0.60 tff(127,plain,
% 0.43/0.60 (![X: $i] : identity_map(codomain(X)) <=> ![X: $i] : identity_map(codomain(X))),
% 0.43/0.60 inference(rewrite,[status(thm)],[])).
% 0.43/0.60 tff(128,axiom,(![X: $i] : identity_map(codomain(X))), file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax','codomain_is_an_identity_map')).
% 0.43/0.60 tff(129,plain,
% 0.43/0.60 (![X: $i] : identity_map(codomain(X))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[128, 127])).
% 0.43/0.60 tff(130,plain,(
% 0.43/0.60 ![X: $i] : identity_map(codomain(X))),
% 0.43/0.60 inference(skolemize,[status(sab)],[129])).
% 0.43/0.60 tff(131,plain,
% 0.43/0.60 (![X: $i] : identity_map(codomain(X))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[130, 126])).
% 0.43/0.60 tff(132,plain,
% 0.43/0.60 ((~![X: $i] : identity_map(codomain(X))) | identity_map(codomain(b))),
% 0.43/0.60 inference(quant_inst,[status(thm)],[])).
% 0.43/0.60 tff(133,plain,
% 0.43/0.60 (identity_map(codomain(b))),
% 0.43/0.60 inference(unit_resolution,[status(thm)],[132, 131])).
% 0.43/0.60 tff(134,assumption,(~defined(a, compose(b, h))), introduced(assumption)).
% 0.43/0.60 tff(135,plain,
% 0.43/0.60 (^[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))))),
% 0.43/0.60 inference(bind,[status(th)],[])).
% 0.43/0.60 tff(136,plain,
% 0.43/0.60 (![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)))),
% 0.43/0.60 inference(quant_intro,[status(thm)],[135])).
% 0.43/0.60 tff(137,plain,
% 0.43/0.60 (![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)))),
% 0.43/0.60 inference(rewrite,[status(thm)],[])).
% 0.43/0.60 tff(138,plain,
% 0.43/0.60 (^[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)))))),
% 0.43/0.60 inference(bind,[status(th)],[])).
% 0.43/0.60 tff(139,plain,
% 0.43/0.60 (![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)))),
% 0.43/0.60 inference(quant_intro,[status(thm)],[138])).
% 0.43/0.60 tff(140,axiom,(![Z: $i, Y: $i, X: $i] : ((((~defined(X, Y)) | (~defined(Y, Z))) | (~identity_map(Y))) | defined(X, Z))), file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax','category_theory_axiom6')).
% 0.43/0.60 tff(141,plain,
% 0.43/0.60 (![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[140, 139])).
% 0.43/0.60 tff(142,plain,
% 0.43/0.60 (![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[141, 137])).
% 0.43/0.60 tff(143,plain,(
% 0.43/0.60 ![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))),
% 0.43/0.60 inference(skolemize,[status(sab)],[142])).
% 0.43/0.60 tff(144,plain,
% 0.43/0.60 (![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[143, 136])).
% 0.43/0.60 tff(145,plain,
% 0.43/0.60 (((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | (defined(a, compose(b, h)) | (~defined(codomain(b), compose(b, h))) | (~identity_map(codomain(b))) | (~defined(a, codomain(b))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | defined(a, compose(b, h)) | (~defined(codomain(b), compose(b, h))) | (~identity_map(codomain(b))) | (~defined(a, codomain(b))))),
% 0.43/0.60 inference(rewrite,[status(thm)],[])).
% 0.43/0.60 tff(146,plain,
% 0.43/0.60 (((~defined(a, codomain(b))) | defined(a, compose(b, h)) | (~identity_map(codomain(b))) | (~defined(codomain(b), compose(b, h)))) <=> (defined(a, compose(b, h)) | (~defined(codomain(b), compose(b, h))) | (~identity_map(codomain(b))) | (~defined(a, codomain(b))))),
% 0.43/0.60 inference(rewrite,[status(thm)],[])).
% 0.43/0.60 tff(147,plain,
% 0.43/0.60 (((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | ((~defined(a, codomain(b))) | defined(a, compose(b, h)) | (~identity_map(codomain(b))) | (~defined(codomain(b), compose(b, h))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | (defined(a, compose(b, h)) | (~defined(codomain(b), compose(b, h))) | (~identity_map(codomain(b))) | (~defined(a, codomain(b)))))),
% 0.43/0.60 inference(monotonicity,[status(thm)],[146])).
% 0.43/0.60 tff(148,plain,
% 0.43/0.60 (((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | ((~defined(a, codomain(b))) | defined(a, compose(b, h)) | (~identity_map(codomain(b))) | (~defined(codomain(b), compose(b, h))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | defined(a, compose(b, h)) | (~defined(codomain(b), compose(b, h))) | (~identity_map(codomain(b))) | (~defined(a, codomain(b))))),
% 0.43/0.60 inference(transitivity,[status(thm)],[147, 145])).
% 0.43/0.60 tff(149,plain,
% 0.43/0.60 ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | ((~defined(a, codomain(b))) | defined(a, compose(b, h)) | (~identity_map(codomain(b))) | (~defined(codomain(b), compose(b, h))))),
% 0.43/0.60 inference(quant_inst,[status(thm)],[])).
% 0.43/0.60 tff(150,plain,
% 0.43/0.60 ((~![Z: $i, Y: $i, X: $i] : ((~defined(X, Y)) | defined(X, Z) | (~identity_map(Y)) | (~defined(Y, Z)))) | defined(a, compose(b, h)) | (~defined(codomain(b), compose(b, h))) | (~identity_map(codomain(b))) | (~defined(a, codomain(b)))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[149, 148])).
% 0.43/0.60 tff(151,plain,
% 0.43/0.60 ($false),
% 0.43/0.60 inference(unit_resolution,[status(thm)],[150, 144, 134, 133, 124, 110])).
% 0.43/0.60 tff(152,plain,(defined(a, compose(b, h))), inference(lemma,lemma(discharge,[]))).
% 0.43/0.60 tff(153,plain,
% 0.43/0.60 (((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(a, compose(b, h))) | product(a, compose(b, h), compose(a, compose(b, h))))) <=> ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(a, compose(b, h))) | product(a, compose(b, h), compose(a, compose(b, h))))),
% 0.43/0.60 inference(rewrite,[status(thm)],[])).
% 0.43/0.60 tff(154,plain,
% 0.43/0.60 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | ((~defined(a, compose(b, h))) | product(a, compose(b, h), compose(a, compose(b, h))))),
% 0.43/0.60 inference(quant_inst,[status(thm)],[])).
% 0.43/0.60 tff(155,plain,
% 0.43/0.60 ((~![Y: $i, X: $i] : ((~defined(X, Y)) | product(X, Y, compose(X, Y)))) | (~defined(a, compose(b, h))) | product(a, compose(b, h), compose(a, compose(b, h)))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[154, 153])).
% 0.43/0.60 tff(156,plain,
% 0.43/0.60 (product(a, compose(b, h), compose(a, compose(b, h)))),
% 0.43/0.60 inference(unit_resolution,[status(thm)],[155, 21, 152])).
% 0.43/0.60 tff(157,plain,
% 0.43/0.60 (^[Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : refl(((~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)) | (~product(X, Yz, Xyz))))),
% 0.43/0.60 inference(bind,[status(th)],[])).
% 0.43/0.60 tff(158,plain,
% 0.43/0.60 (![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz))) <=> ![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))),
% 0.43/0.60 inference(quant_intro,[status(thm)],[157])).
% 0.43/0.60 tff(159,plain,
% 0.43/0.60 (![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz))) <=> ![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))),
% 0.43/0.60 inference(rewrite,[status(thm)],[])).
% 0.43/0.60 tff(160,plain,
% 0.43/0.60 (^[Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~product(Y, Z, Yz)) | (~product(X, Yz, Xyz))) <=> ((~product(Y, Z, Yz)) | (~product(X, Yz, Xyz)))), ((((~product(Y, Z, Yz)) | (~product(X, Yz, Xyz))) | (~product(X, Y, Xy))) <=> (((~product(Y, Z, Yz)) | (~product(X, Yz, Xyz))) | (~product(X, Y, Xy))))), rewrite((((~product(Y, Z, Yz)) | (~product(X, Yz, Xyz))) | (~product(X, Y, Xy))) <=> ((~product(Y, Z, Yz)) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))), ((((~product(Y, Z, Yz)) | (~product(X, Yz, Xyz))) | (~product(X, Y, Xy))) <=> ((~product(Y, Z, Yz)) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz))))), (((((~product(Y, Z, Yz)) | (~product(X, Yz, Xyz))) | (~product(X, Y, Xy))) | product(Xy, Z, Xyz)) <=> (((~product(Y, Z, Yz)) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz))) | product(Xy, Z, Xyz)))), rewrite((((~product(Y, Z, Yz)) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz))) | product(Xy, Z, Xyz)) <=> ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))), (((((~product(Y, Z, Yz)) | (~product(X, Yz, Xyz))) | (~product(X, Y, Xy))) | product(Xy, Z, Xyz)) <=> ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))))),
% 0.43/0.60 inference(bind,[status(th)],[])).
% 0.43/0.60 tff(161,plain,
% 0.43/0.60 (![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((((~product(Y, Z, Yz)) | (~product(X, Yz, Xyz))) | (~product(X, Y, Xy))) | product(Xy, Z, Xyz)) <=> ![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))),
% 0.43/0.60 inference(quant_intro,[status(thm)],[160])).
% 0.43/0.60 tff(162,axiom,(![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((((~product(Y, Z, Yz)) | (~product(X, Yz, Xyz))) | (~product(X, Y, Xy))) | product(Xy, Z, Xyz))), file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax','category_theory_axiom5')).
% 0.43/0.60 tff(163,plain,
% 0.43/0.60 (![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[162, 161])).
% 0.43/0.60 tff(164,plain,
% 0.43/0.60 (![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[163, 159])).
% 0.43/0.60 tff(165,plain,(
% 0.43/0.60 ![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))),
% 0.43/0.60 inference(skolemize,[status(sab)],[164])).
% 0.43/0.60 tff(166,plain,
% 0.43/0.60 (![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[165, 158])).
% 0.43/0.60 tff(167,plain,
% 0.43/0.60 (((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | ((~product(compose(codomain(b), b), g, compose(b, h))) | (~product(a, compose(b, h), compose(a, compose(b, h)))) | (~product(a, compose(codomain(b), b), compose(a, b))) | product(compose(a, b), g, compose(a, compose(b, h))))) <=> ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | (~product(compose(codomain(b), b), g, compose(b, h))) | (~product(a, compose(b, h), compose(a, compose(b, h)))) | (~product(a, compose(codomain(b), b), compose(a, b))) | product(compose(a, b), g, compose(a, compose(b, h))))),
% 0.43/0.60 inference(rewrite,[status(thm)],[])).
% 0.43/0.60 tff(168,plain,
% 0.43/0.60 (((~product(compose(codomain(b), b), g, compose(b, h))) | product(compose(a, b), g, compose(a, compose(b, h))) | (~product(a, compose(codomain(b), b), compose(a, b))) | (~product(a, compose(b, h), compose(a, compose(b, h))))) <=> ((~product(compose(codomain(b), b), g, compose(b, h))) | (~product(a, compose(b, h), compose(a, compose(b, h)))) | (~product(a, compose(codomain(b), b), compose(a, b))) | product(compose(a, b), g, compose(a, compose(b, h))))),
% 0.43/0.60 inference(rewrite,[status(thm)],[])).
% 0.43/0.60 tff(169,plain,
% 0.43/0.60 (((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | ((~product(compose(codomain(b), b), g, compose(b, h))) | product(compose(a, b), g, compose(a, compose(b, h))) | (~product(a, compose(codomain(b), b), compose(a, b))) | (~product(a, compose(b, h), compose(a, compose(b, h)))))) <=> ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | ((~product(compose(codomain(b), b), g, compose(b, h))) | (~product(a, compose(b, h), compose(a, compose(b, h)))) | (~product(a, compose(codomain(b), b), compose(a, b))) | product(compose(a, b), g, compose(a, compose(b, h)))))),
% 0.43/0.60 inference(monotonicity,[status(thm)],[168])).
% 0.43/0.60 tff(170,plain,
% 0.43/0.60 (((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | ((~product(compose(codomain(b), b), g, compose(b, h))) | product(compose(a, b), g, compose(a, compose(b, h))) | (~product(a, compose(codomain(b), b), compose(a, b))) | (~product(a, compose(b, h), compose(a, compose(b, h)))))) <=> ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | (~product(compose(codomain(b), b), g, compose(b, h))) | (~product(a, compose(b, h), compose(a, compose(b, h)))) | (~product(a, compose(codomain(b), b), compose(a, b))) | product(compose(a, b), g, compose(a, compose(b, h))))),
% 0.43/0.60 inference(transitivity,[status(thm)],[169, 167])).
% 0.43/0.60 tff(171,plain,
% 0.43/0.60 ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | ((~product(compose(codomain(b), b), g, compose(b, h))) | product(compose(a, b), g, compose(a, compose(b, h))) | (~product(a, compose(codomain(b), b), compose(a, b))) | (~product(a, compose(b, h), compose(a, compose(b, h)))))),
% 0.43/0.60 inference(quant_inst,[status(thm)],[])).
% 0.43/0.60 tff(172,plain,
% 0.43/0.60 ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | (~product(compose(codomain(b), b), g, compose(b, h))) | (~product(a, compose(b, h), compose(a, compose(b, h)))) | (~product(a, compose(codomain(b), b), compose(a, b))) | product(compose(a, b), g, compose(a, compose(b, h)))),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[171, 170])).
% 0.43/0.61 tff(173,plain,
% 0.43/0.61 ((~product(compose(codomain(b), b), g, compose(b, h))) | (~product(a, compose(codomain(b), b), compose(a, b))) | product(compose(a, b), g, compose(a, compose(b, h)))),
% 0.43/0.61 inference(unit_resolution,[status(thm)],[172, 166, 156])).
% 0.43/0.61 tff(174,plain,
% 0.43/0.61 (product(compose(a, b), g, compose(a, compose(b, h)))),
% 0.43/0.61 inference(unit_resolution,[status(thm)],[173, 93, 71])).
% 0.43/0.61 tff(175,plain,
% 0.43/0.61 (product(c, g, compose(a, compose(b, h)))),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[174, 41])).
% 0.43/0.61 tff(176,plain,
% 0.43/0.61 (product(c, compose(h, domain(h)), compose(a, compose(b, h))) <=> product(compose(a, b), compose(h, domain(h)), compose(a, compose(b, h)))),
% 0.43/0.61 inference(monotonicity,[status(thm)],[39])).
% 0.43/0.61 tff(177,plain,
% 0.43/0.61 (product(compose(a, b), compose(h, domain(h)), compose(a, compose(b, h))) <=> product(c, compose(h, domain(h)), compose(a, compose(b, h)))),
% 0.43/0.61 inference(symmetry,[status(thm)],[176])).
% 0.43/0.61 tff(178,plain,
% 0.43/0.61 (^[X: $i] : refl(defined(X, domain(X)) <=> defined(X, domain(X)))),
% 0.43/0.61 inference(bind,[status(th)],[])).
% 0.43/0.61 tff(179,plain,
% 0.43/0.61 (![X: $i] : defined(X, domain(X)) <=> ![X: $i] : defined(X, domain(X))),
% 0.43/0.61 inference(quant_intro,[status(thm)],[178])).
% 0.43/0.61 tff(180,plain,
% 0.43/0.61 (![X: $i] : defined(X, domain(X)) <=> ![X: $i] : defined(X, domain(X))),
% 0.43/0.61 inference(rewrite,[status(thm)],[])).
% 0.43/0.61 tff(181,axiom,(![X: $i] : defined(X, domain(X))), file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax','mapping_from_x_to_its_domain')).
% 0.43/0.61 tff(182,plain,
% 0.43/0.61 (![X: $i] : defined(X, domain(X))),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[181, 180])).
% 0.43/0.61 tff(183,plain,(
% 0.43/0.61 ![X: $i] : defined(X, domain(X))),
% 0.43/0.61 inference(skolemize,[status(sab)],[182])).
% 0.43/0.61 tff(184,plain,
% 0.43/0.61 (![X: $i] : defined(X, domain(X))),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[183, 179])).
% 0.43/0.61 tff(185,plain,
% 0.43/0.61 ((~![X: $i] : defined(X, domain(X))) | defined(h, domain(h))),
% 0.43/0.61 inference(quant_inst,[status(thm)],[])).
% 0.43/0.61 tff(186,plain,
% 0.43/0.61 (defined(h, domain(h))),
% 0.43/0.61 inference(unit_resolution,[status(thm)],[185, 184])).
% 0.43/0.61 tff(187,plain,
% 0.43/0.61 (((~![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))))),
% 0.43/0.61 inference(rewrite,[status(thm)],[])).
% 0.43/0.61 tff(188,plain,
% 0.43/0.61 ((~![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))))),
% 0.43/0.61 inference(quant_inst,[status(thm)],[])).
% 0.43/0.61 tff(189,plain,
% 0.43/0.61 ((~![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)))),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[188, 187])).
% 0.43/0.61 tff(190,plain,
% 0.43/0.61 (product(h, domain(h), compose(h, domain(h)))),
% 0.43/0.61 inference(unit_resolution,[status(thm)],[189, 21, 186])).
% 0.43/0.61 tff(191,plain,
% 0.43/0.61 (^[X: $i] : refl(product(X, domain(X), X) <=> product(X, domain(X), X))),
% 0.43/0.61 inference(bind,[status(th)],[])).
% 0.43/0.61 tff(192,plain,
% 0.43/0.61 (![X: $i] : product(X, domain(X), X) <=> ![X: $i] : product(X, domain(X), X)),
% 0.43/0.61 inference(quant_intro,[status(thm)],[191])).
% 0.43/0.61 tff(193,plain,
% 0.43/0.61 (![X: $i] : product(X, domain(X), X) <=> ![X: $i] : product(X, domain(X), X)),
% 0.43/0.61 inference(rewrite,[status(thm)],[])).
% 0.43/0.61 tff(194,axiom,(![X: $i] : product(X, domain(X), X)), file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax','product_on_domain')).
% 0.43/0.61 tff(195,plain,
% 0.43/0.61 (![X: $i] : product(X, domain(X), X)),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[194, 193])).
% 0.43/0.61 tff(196,plain,(
% 0.43/0.61 ![X: $i] : product(X, domain(X), X)),
% 0.43/0.61 inference(skolemize,[status(sab)],[195])).
% 0.43/0.61 tff(197,plain,
% 0.43/0.61 (![X: $i] : product(X, domain(X), X)),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[196, 192])).
% 0.43/0.61 tff(198,plain,
% 0.43/0.61 ((~![X: $i] : product(X, domain(X), X)) | product(h, domain(h), h)),
% 0.43/0.61 inference(quant_inst,[status(thm)],[])).
% 0.43/0.61 tff(199,plain,
% 0.43/0.61 (product(h, domain(h), h)),
% 0.43/0.61 inference(unit_resolution,[status(thm)],[198, 197])).
% 0.43/0.61 tff(200,plain,
% 0.43/0.61 (((~![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))))),
% 0.43/0.61 inference(rewrite,[status(thm)],[])).
% 0.43/0.61 tff(201,plain,
% 0.43/0.61 ((~![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))))),
% 0.43/0.61 inference(quant_inst,[status(thm)],[])).
% 0.43/0.61 tff(202,plain,
% 0.43/0.61 ((~![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)))),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[201, 200])).
% 0.43/0.61 tff(203,plain,
% 0.43/0.61 (h = compose(h, domain(h))),
% 0.43/0.61 inference(unit_resolution,[status(thm)],[202, 35, 199, 190])).
% 0.43/0.61 tff(204,plain,
% 0.43/0.61 (compose(h, domain(h)) = h),
% 0.43/0.61 inference(symmetry,[status(thm)],[203])).
% 0.43/0.61 tff(205,plain,
% 0.43/0.61 (product(compose(codomain(b), b), compose(h, domain(h)), compose(b, h)) <=> product(b, h, compose(b, h))),
% 0.43/0.61 inference(monotonicity,[status(thm)],[68, 204])).
% 0.43/0.61 tff(206,plain,
% 0.43/0.61 (product(b, h, compose(b, h)) <=> product(compose(codomain(b), b), compose(h, domain(h)), compose(b, h))),
% 0.43/0.61 inference(symmetry,[status(thm)],[205])).
% 0.43/0.61 tff(207,plain,
% 0.43/0.61 (product(compose(codomain(b), b), compose(h, domain(h)), compose(b, h))),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[82, 206])).
% 0.43/0.61 tff(208,plain,
% 0.43/0.61 (((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | ((~product(compose(codomain(b), b), compose(h, domain(h)), compose(b, h))) | (~product(a, compose(b, h), compose(a, compose(b, h)))) | (~product(a, compose(codomain(b), b), compose(a, b))) | product(compose(a, b), compose(h, domain(h)), compose(a, compose(b, h))))) <=> ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | (~product(compose(codomain(b), b), compose(h, domain(h)), compose(b, h))) | (~product(a, compose(b, h), compose(a, compose(b, h)))) | (~product(a, compose(codomain(b), b), compose(a, b))) | product(compose(a, b), compose(h, domain(h)), compose(a, compose(b, h))))),
% 0.43/0.61 inference(rewrite,[status(thm)],[])).
% 0.43/0.61 tff(209,plain,
% 0.43/0.61 (((~product(compose(codomain(b), b), compose(h, domain(h)), compose(b, h))) | product(compose(a, b), compose(h, domain(h)), compose(a, compose(b, h))) | (~product(a, compose(codomain(b), b), compose(a, b))) | (~product(a, compose(b, h), compose(a, compose(b, h))))) <=> ((~product(compose(codomain(b), b), compose(h, domain(h)), compose(b, h))) | (~product(a, compose(b, h), compose(a, compose(b, h)))) | (~product(a, compose(codomain(b), b), compose(a, b))) | product(compose(a, b), compose(h, domain(h)), compose(a, compose(b, h))))),
% 0.43/0.61 inference(rewrite,[status(thm)],[])).
% 0.43/0.61 tff(210,plain,
% 0.43/0.61 (((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | ((~product(compose(codomain(b), b), compose(h, domain(h)), compose(b, h))) | product(compose(a, b), compose(h, domain(h)), compose(a, compose(b, h))) | (~product(a, compose(codomain(b), b), compose(a, b))) | (~product(a, compose(b, h), compose(a, compose(b, h)))))) <=> ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | ((~product(compose(codomain(b), b), compose(h, domain(h)), compose(b, h))) | (~product(a, compose(b, h), compose(a, compose(b, h)))) | (~product(a, compose(codomain(b), b), compose(a, b))) | product(compose(a, b), compose(h, domain(h)), compose(a, compose(b, h)))))),
% 0.43/0.61 inference(monotonicity,[status(thm)],[209])).
% 0.43/0.61 tff(211,plain,
% 0.43/0.61 (((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | ((~product(compose(codomain(b), b), compose(h, domain(h)), compose(b, h))) | product(compose(a, b), compose(h, domain(h)), compose(a, compose(b, h))) | (~product(a, compose(codomain(b), b), compose(a, b))) | (~product(a, compose(b, h), compose(a, compose(b, h)))))) <=> ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | (~product(compose(codomain(b), b), compose(h, domain(h)), compose(b, h))) | (~product(a, compose(b, h), compose(a, compose(b, h)))) | (~product(a, compose(codomain(b), b), compose(a, b))) | product(compose(a, b), compose(h, domain(h)), compose(a, compose(b, h))))),
% 0.43/0.61 inference(transitivity,[status(thm)],[210, 208])).
% 0.43/0.61 tff(212,plain,
% 0.43/0.61 ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | ((~product(compose(codomain(b), b), compose(h, domain(h)), compose(b, h))) | product(compose(a, b), compose(h, domain(h)), compose(a, compose(b, h))) | (~product(a, compose(codomain(b), b), compose(a, b))) | (~product(a, compose(b, h), compose(a, compose(b, h)))))),
% 0.43/0.61 inference(quant_inst,[status(thm)],[])).
% 0.43/0.61 tff(213,plain,
% 0.43/0.61 ((~![Xy: $i, Z: $i, Y: $i, Xyz: $i, X: $i, Yz: $i] : ((~product(Y, Z, Yz)) | product(Xy, Z, Xyz) | (~product(X, Y, Xy)) | (~product(X, Yz, Xyz)))) | (~product(compose(codomain(b), b), compose(h, domain(h)), compose(b, h))) | (~product(a, compose(b, h), compose(a, compose(b, h)))) | (~product(a, compose(codomain(b), b), compose(a, b))) | product(compose(a, b), compose(h, domain(h)), compose(a, compose(b, h)))),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[212, 211])).
% 0.43/0.61 tff(214,plain,
% 0.43/0.61 ((~product(compose(codomain(b), b), compose(h, domain(h)), compose(b, h))) | (~product(a, compose(codomain(b), b), compose(a, b))) | product(compose(a, b), compose(h, domain(h)), compose(a, compose(b, h)))),
% 0.43/0.61 inference(unit_resolution,[status(thm)],[213, 166, 156])).
% 0.43/0.61 tff(215,plain,
% 0.43/0.61 (product(compose(a, b), compose(h, domain(h)), compose(a, compose(b, h)))),
% 0.43/0.61 inference(unit_resolution,[status(thm)],[214, 71, 207])).
% 0.43/0.61 tff(216,plain,
% 0.43/0.61 (product(c, compose(h, domain(h)), compose(a, compose(b, h)))),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[215, 177])).
% 0.43/0.61 tff(217,plain,
% 0.43/0.61 ((g = h) <=> (h = g)),
% 0.43/0.61 inference(commutativity,[status(thm)],[])).
% 0.43/0.61 tff(218,plain,
% 0.43/0.61 ((g = compose(h, domain(h))) <=> (g = h)),
% 0.43/0.61 inference(monotonicity,[status(thm)],[204])).
% 0.43/0.61 tff(219,plain,
% 0.43/0.61 ((g = compose(h, domain(h))) <=> (h = g)),
% 0.43/0.61 inference(transitivity,[status(thm)],[218, 217])).
% 0.43/0.61 tff(220,plain,
% 0.43/0.61 ((h = g) <=> (g = compose(h, domain(h)))),
% 0.43/0.61 inference(symmetry,[status(thm)],[219])).
% 0.43/0.61 tff(221,plain,
% 0.43/0.61 ((~(h = g)) <=> (~(g = compose(h, domain(h))))),
% 0.43/0.61 inference(monotonicity,[status(thm)],[220])).
% 0.43/0.61 tff(222,plain,
% 0.43/0.61 ((~(h = g)) <=> (~(h = g))),
% 0.43/0.61 inference(rewrite,[status(thm)],[])).
% 0.43/0.61 tff(223,axiom,(~(h = g)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_h_equals_g')).
% 0.43/0.61 tff(224,plain,
% 0.43/0.61 (~(h = g)),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[223, 222])).
% 0.43/0.61 tff(225,plain,
% 0.43/0.61 (~(g = compose(h, domain(h)))),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[224, 221])).
% 0.43/0.61 tff(226,plain,
% 0.43/0.61 (^[X3: $i, X2: $i, X1: $i] : refl(((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2))) <=> ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2))))),
% 0.43/0.61 inference(bind,[status(th)],[])).
% 0.43/0.61 tff(227,plain,
% 0.43/0.61 (![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2))) <=> ![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))),
% 0.43/0.61 inference(quant_intro,[status(thm)],[226])).
% 0.43/0.61 tff(228,plain,
% 0.43/0.61 (![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2))) <=> ![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))),
% 0.43/0.61 inference(rewrite,[status(thm)],[])).
% 0.43/0.61 tff(229,plain,
% 0.43/0.61 (^[X3: $i, X2: $i, X1: $i] : rewrite((((~product(c, X1, X2)) | (~product(c, X3, X2))) | (X1 = X3)) <=> ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2))))),
% 0.43/0.61 inference(bind,[status(th)],[])).
% 0.43/0.61 tff(230,plain,
% 0.43/0.61 (![X3: $i, X2: $i, X1: $i] : (((~product(c, X1, X2)) | (~product(c, X3, X2))) | (X1 = X3)) <=> ![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))),
% 0.43/0.61 inference(quant_intro,[status(thm)],[229])).
% 0.43/0.61 tff(231,axiom,(![X3: $i, X2: $i, X1: $i] : (((~product(c, X1, X2)) | (~product(c, X3, X2))) | (X1 = X3))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','cancellation_for_product')).
% 0.43/0.61 tff(232,plain,
% 0.43/0.61 (![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[231, 230])).
% 0.43/0.61 tff(233,plain,
% 0.43/0.61 (![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[232, 228])).
% 0.43/0.61 tff(234,plain,(
% 0.43/0.61 ![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))),
% 0.43/0.61 inference(skolemize,[status(sab)],[233])).
% 0.43/0.61 tff(235,plain,
% 0.43/0.61 (![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[234, 227])).
% 0.43/0.61 tff(236,plain,
% 0.43/0.61 (((~![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))) | ((~product(c, compose(h, domain(h)), compose(a, compose(b, h)))) | (g = compose(h, domain(h))) | (~product(c, g, compose(a, compose(b, h)))))) <=> ((~![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))) | (~product(c, compose(h, domain(h)), compose(a, compose(b, h)))) | (g = compose(h, domain(h))) | (~product(c, g, compose(a, compose(b, h)))))),
% 0.43/0.61 inference(rewrite,[status(thm)],[])).
% 0.43/0.61 tff(237,plain,
% 0.43/0.61 (((g = compose(h, domain(h))) | (~product(c, compose(h, domain(h)), compose(a, compose(b, h)))) | (~product(c, g, compose(a, compose(b, h))))) <=> ((~product(c, compose(h, domain(h)), compose(a, compose(b, h)))) | (g = compose(h, domain(h))) | (~product(c, g, compose(a, compose(b, h)))))),
% 0.43/0.61 inference(rewrite,[status(thm)],[])).
% 0.43/0.61 tff(238,plain,
% 0.43/0.61 (((~![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))) | ((g = compose(h, domain(h))) | (~product(c, compose(h, domain(h)), compose(a, compose(b, h)))) | (~product(c, g, compose(a, compose(b, h)))))) <=> ((~![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))) | ((~product(c, compose(h, domain(h)), compose(a, compose(b, h)))) | (g = compose(h, domain(h))) | (~product(c, g, compose(a, compose(b, h))))))),
% 0.43/0.61 inference(monotonicity,[status(thm)],[237])).
% 0.43/0.61 tff(239,plain,
% 0.43/0.61 (((~![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))) | ((g = compose(h, domain(h))) | (~product(c, compose(h, domain(h)), compose(a, compose(b, h)))) | (~product(c, g, compose(a, compose(b, h)))))) <=> ((~![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))) | (~product(c, compose(h, domain(h)), compose(a, compose(b, h)))) | (g = compose(h, domain(h))) | (~product(c, g, compose(a, compose(b, h)))))),
% 0.43/0.61 inference(transitivity,[status(thm)],[238, 236])).
% 0.43/0.61 tff(240,plain,
% 0.43/0.61 ((~![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))) | ((g = compose(h, domain(h))) | (~product(c, compose(h, domain(h)), compose(a, compose(b, h)))) | (~product(c, g, compose(a, compose(b, h)))))),
% 0.43/0.61 inference(quant_inst,[status(thm)],[])).
% 0.43/0.61 tff(241,plain,
% 0.43/0.61 ((~![X3: $i, X2: $i, X1: $i] : ((X1 = X3) | (~product(c, X3, X2)) | (~product(c, X1, X2)))) | (~product(c, compose(h, domain(h)), compose(a, compose(b, h)))) | (g = compose(h, domain(h))) | (~product(c, g, compose(a, compose(b, h))))),
% 0.43/0.61 inference(modus_ponens,[status(thm)],[240, 239])).
% 0.43/0.61 tff(242,plain,
% 0.43/0.61 ((~product(c, compose(h, domain(h)), compose(a, compose(b, h)))) | (g = compose(h, domain(h))) | (~product(c, g, compose(a, compose(b, h))))),
% 0.43/0.61 inference(unit_resolution,[status(thm)],[241, 235])).
% 0.43/0.61 tff(243,plain,
% 0.43/0.61 ($false),
% 0.43/0.61 inference(unit_resolution,[status(thm)],[242, 225, 216, 175])).
% 0.43/0.61 % SZS output end Proof
% 0.43/0.62 % E exiting
%------------------------------------------------------------------------------