%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP048-2 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n009.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 05:06:57 AM UTC 2025
% Result : Unsatisfiable 8.31s 8.47s
% Output : Proof 8.31s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GRP048-2 : TPTP v9.0.0. Released v1.0.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.32 % Computer : n009.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 300
% 0.12/0.32 % DateTime : Thu Jun 19 15:32:48 EDT 2025
% 0.12/0.33 % CPUTime :
% 8.31/8.47 % SZS status Unsatisfiable
% 8.31/8.47 % SZS output start Proof
% 8.31/8.47 tff(product_type, type, (
% 8.31/8.47 product: ( $i * $i * $i ) > $o)).
% 8.31/8.47 tff(inverse_type, type, (
% 8.31/8.47 inverse: $i > $i)).
% 8.31/8.47 tff(b_type, type, (
% 8.31/8.47 b: $i)).
% 8.31/8.47 tff(a_type, type, (
% 8.31/8.47 a: $i)).
% 8.31/8.47 tff(identity_type, type, (
% 8.31/8.47 identity: $i)).
% 8.31/8.47 tff(equalish_type, type, (
% 8.31/8.47 equalish: ( $i * $i ) > $o)).
% 8.31/8.47 tff(1,assumption,(~product(identity, inverse(a), inverse(b))), introduced(assumption)).
% 8.31/8.47 tff(2,assumption,(~product(inverse(b), identity, inverse(b))), introduced(assumption)).
% 8.31/8.47 tff(3,plain,
% 8.31/8.47 (^[X: $i] : refl(product(inverse(X), X, identity) <=> product(inverse(X), X, identity))),
% 8.31/8.47 inference(bind,[status(th)],[])).
% 8.31/8.47 tff(4,plain,
% 8.31/8.47 (![X: $i] : product(inverse(X), X, identity) <=> ![X: $i] : product(inverse(X), X, identity)),
% 8.31/8.47 inference(quant_intro,[status(thm)],[3])).
% 8.31/8.47 tff(5,plain,
% 8.31/8.47 (![X: $i] : product(inverse(X), X, identity) <=> ![X: $i] : product(inverse(X), X, identity)),
% 8.31/8.47 inference(rewrite,[status(thm)],[])).
% 8.31/8.47 tff(6,axiom,(![X: $i] : product(inverse(X), X, identity)), file('/export/starexec/sandbox/benchmark/Axioms/GRP005-0.ax','left_inverse')).
% 8.31/8.47 tff(7,plain,
% 8.31/8.47 (![X: $i] : product(inverse(X), X, identity)),
% 8.31/8.47 inference(modus_ponens,[status(thm)],[6, 5])).
% 8.31/8.47 tff(8,plain,(
% 8.31/8.47 ![X: $i] : product(inverse(X), X, identity)),
% 8.31/8.47 inference(skolemize,[status(sab)],[7])).
% 8.31/8.47 tff(9,plain,
% 8.31/8.47 (![X: $i] : product(inverse(X), X, identity)),
% 8.31/8.47 inference(modus_ponens,[status(thm)],[8, 4])).
% 8.31/8.47 tff(10,plain,
% 8.31/8.47 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(inverse(inverse(b))), inverse(inverse(b)), identity)),
% 8.31/8.47 inference(quant_inst,[status(thm)],[])).
% 8.31/8.47 tff(11,plain,
% 8.31/8.47 (product(inverse(inverse(inverse(b))), inverse(inverse(b)), identity)),
% 8.31/8.47 inference(unit_resolution,[status(thm)],[10, 9])).
% 8.31/8.47 tff(12,plain,
% 8.31/8.47 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(inverse(b)), inverse(b), identity)),
% 8.31/8.47 inference(quant_inst,[status(thm)],[])).
% 8.31/8.47 tff(13,plain,
% 8.31/8.47 (product(inverse(inverse(b)), inverse(b), identity)),
% 8.31/8.47 inference(unit_resolution,[status(thm)],[12, 9])).
% 8.31/8.47 tff(14,plain,
% 8.31/8.47 (^[X: $i] : refl(product(identity, X, X) <=> product(identity, X, X))),
% 8.31/8.47 inference(bind,[status(th)],[])).
% 8.31/8.47 tff(15,plain,
% 8.31/8.47 (![X: $i] : product(identity, X, X) <=> ![X: $i] : product(identity, X, X)),
% 8.31/8.47 inference(quant_intro,[status(thm)],[14])).
% 8.31/8.47 tff(16,plain,
% 8.31/8.47 (![X: $i] : product(identity, X, X) <=> ![X: $i] : product(identity, X, X)),
% 8.31/8.47 inference(rewrite,[status(thm)],[])).
% 8.31/8.47 tff(17,axiom,(![X: $i] : product(identity, X, X)), file('/export/starexec/sandbox/benchmark/Axioms/GRP005-0.ax','left_identity')).
% 8.31/8.47 tff(18,plain,
% 8.31/8.47 (![X: $i] : product(identity, X, X)),
% 8.31/8.47 inference(modus_ponens,[status(thm)],[17, 16])).
% 8.31/8.47 tff(19,plain,(
% 8.31/8.47 ![X: $i] : product(identity, X, X)),
% 8.31/8.47 inference(skolemize,[status(sab)],[18])).
% 8.31/8.47 tff(20,plain,
% 8.31/8.47 (![X: $i] : product(identity, X, X)),
% 8.31/8.47 inference(modus_ponens,[status(thm)],[19, 15])).
% 8.31/8.47 tff(21,plain,
% 8.31/8.47 ((~![X: $i] : product(identity, X, X)) | product(identity, inverse(b), inverse(b))),
% 8.31/8.47 inference(quant_inst,[status(thm)],[])).
% 8.31/8.47 tff(22,plain,
% 8.31/8.47 (product(identity, inverse(b), inverse(b))),
% 8.31/8.47 inference(unit_resolution,[status(thm)],[21, 20])).
% 8.31/8.47 tff(23,plain,
% 8.31/8.47 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : refl((product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))) <=> (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))))),
% 8.31/8.47 inference(bind,[status(th)],[])).
% 8.31/8.47 tff(24,plain,
% 8.31/8.47 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 8.31/8.47 inference(quant_intro,[status(thm)],[23])).
% 8.31/8.47 tff(25,plain,
% 8.31/8.47 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(26,plain,
% 8.31/8.48 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : trans(monotonicity(rewrite((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) <=> ((~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))), (((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) | product(X, V, W)) <=> (((~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))) | product(X, V, W)))), rewrite((((~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))) | product(X, V, W)) <=> (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))), (((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) | product(X, V, W)) <=> (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))))),
% 8.31/8.48 inference(bind,[status(th)],[])).
% 8.31/8.48 tff(27,plain,
% 8.31/8.48 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) | product(X, V, W)) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 8.31/8.48 inference(quant_intro,[status(thm)],[26])).
% 8.31/8.48 tff(28,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) | product(X, V, W))), file('/export/starexec/sandbox/benchmark/Axioms/GRP005-0.ax','associativity1')).
% 8.31/8.48 tff(29,plain,
% 8.31/8.48 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[28, 27])).
% 8.31/8.48 tff(30,plain,
% 8.31/8.48 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[29, 25])).
% 8.31/8.48 tff(31,plain,(
% 8.31/8.48 ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 8.31/8.48 inference(skolemize,[status(sab)],[30])).
% 8.31/8.48 tff(32,plain,
% 8.31/8.48 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[31, 24])).
% 8.31/8.48 tff(33,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(identity, inverse(b), inverse(b))) | (~product(inverse(inverse(b)), inverse(b), identity)) | (~product(inverse(inverse(inverse(b))), inverse(inverse(b)), identity)) | product(inverse(inverse(inverse(b))), identity, inverse(b)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(identity, inverse(b), inverse(b))) | (~product(inverse(inverse(b)), inverse(b), identity)) | (~product(inverse(inverse(inverse(b))), inverse(inverse(b)), identity)) | product(inverse(inverse(inverse(b))), identity, inverse(b)))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(34,plain,
% 8.31/8.48 ((product(inverse(inverse(inverse(b))), identity, inverse(b)) | (~product(identity, inverse(b), inverse(b))) | (~product(inverse(inverse(b)), inverse(b), identity)) | (~product(inverse(inverse(inverse(b))), inverse(inverse(b)), identity))) <=> ((~product(identity, inverse(b), inverse(b))) | (~product(inverse(inverse(b)), inverse(b), identity)) | (~product(inverse(inverse(inverse(b))), inverse(inverse(b)), identity)) | product(inverse(inverse(inverse(b))), identity, inverse(b)))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(35,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(inverse(inverse(b))), identity, inverse(b)) | (~product(identity, inverse(b), inverse(b))) | (~product(inverse(inverse(b)), inverse(b), identity)) | (~product(inverse(inverse(inverse(b))), inverse(inverse(b)), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(identity, inverse(b), inverse(b))) | (~product(inverse(inverse(b)), inverse(b), identity)) | (~product(inverse(inverse(inverse(b))), inverse(inverse(b)), identity)) | product(inverse(inverse(inverse(b))), identity, inverse(b))))),
% 8.31/8.48 inference(monotonicity,[status(thm)],[34])).
% 8.31/8.48 tff(36,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(inverse(inverse(b))), identity, inverse(b)) | (~product(identity, inverse(b), inverse(b))) | (~product(inverse(inverse(b)), inverse(b), identity)) | (~product(inverse(inverse(inverse(b))), inverse(inverse(b)), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(identity, inverse(b), inverse(b))) | (~product(inverse(inverse(b)), inverse(b), identity)) | (~product(inverse(inverse(inverse(b))), inverse(inverse(b)), identity)) | product(inverse(inverse(inverse(b))), identity, inverse(b)))),
% 8.31/8.48 inference(transitivity,[status(thm)],[35, 33])).
% 8.31/8.48 tff(37,plain,
% 8.31/8.48 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(inverse(inverse(b))), identity, inverse(b)) | (~product(identity, inverse(b), inverse(b))) | (~product(inverse(inverse(b)), inverse(b), identity)) | (~product(inverse(inverse(inverse(b))), inverse(inverse(b)), identity)))),
% 8.31/8.48 inference(quant_inst,[status(thm)],[])).
% 8.31/8.48 tff(38,plain,
% 8.31/8.48 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(identity, inverse(b), inverse(b))) | (~product(inverse(inverse(b)), inverse(b), identity)) | (~product(inverse(inverse(inverse(b))), inverse(inverse(b)), identity)) | product(inverse(inverse(inverse(b))), identity, inverse(b))),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[37, 36])).
% 8.31/8.48 tff(39,plain,
% 8.31/8.48 (product(inverse(inverse(inverse(b))), identity, inverse(b))),
% 8.31/8.48 inference(unit_resolution,[status(thm)],[38, 32, 22, 13, 11])).
% 8.31/8.48 tff(40,plain,
% 8.31/8.48 ((~![X: $i] : product(identity, X, X)) | product(identity, inverse(a), inverse(a))),
% 8.31/8.48 inference(quant_inst,[status(thm)],[])).
% 8.31/8.48 tff(41,plain,
% 8.31/8.48 (product(identity, inverse(a), inverse(a))),
% 8.31/8.48 inference(unit_resolution,[status(thm)],[40, 20])).
% 8.31/8.48 tff(42,plain,
% 8.31/8.48 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(a), a, identity)),
% 8.31/8.48 inference(quant_inst,[status(thm)],[])).
% 8.31/8.48 tff(43,plain,
% 8.31/8.48 (product(inverse(a), a, identity)),
% 8.31/8.48 inference(unit_resolution,[status(thm)],[42, 9])).
% 8.31/8.48 tff(44,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(inverse(a), a, identity)) | (~product(identity, inverse(a), inverse(a))) | product(identity, identity, identity))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(inverse(a), a, identity)) | (~product(identity, inverse(a), inverse(a))) | product(identity, identity, identity))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(45,plain,
% 8.31/8.48 ((product(identity, identity, identity) | (~product(inverse(a), a, identity)) | (~product(inverse(a), a, identity)) | (~product(identity, inverse(a), inverse(a)))) <=> ((~product(inverse(a), a, identity)) | (~product(identity, inverse(a), inverse(a))) | product(identity, identity, identity))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(46,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(identity, identity, identity) | (~product(inverse(a), a, identity)) | (~product(inverse(a), a, identity)) | (~product(identity, inverse(a), inverse(a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(inverse(a), a, identity)) | (~product(identity, inverse(a), inverse(a))) | product(identity, identity, identity)))),
% 8.31/8.48 inference(monotonicity,[status(thm)],[45])).
% 8.31/8.48 tff(47,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(identity, identity, identity) | (~product(inverse(a), a, identity)) | (~product(inverse(a), a, identity)) | (~product(identity, inverse(a), inverse(a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(inverse(a), a, identity)) | (~product(identity, inverse(a), inverse(a))) | product(identity, identity, identity))),
% 8.31/8.48 inference(transitivity,[status(thm)],[46, 44])).
% 8.31/8.48 tff(48,plain,
% 8.31/8.48 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(identity, identity, identity) | (~product(inverse(a), a, identity)) | (~product(inverse(a), a, identity)) | (~product(identity, inverse(a), inverse(a))))),
% 8.31/8.48 inference(quant_inst,[status(thm)],[])).
% 8.31/8.48 tff(49,plain,
% 8.31/8.48 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(inverse(a), a, identity)) | (~product(identity, inverse(a), inverse(a))) | product(identity, identity, identity)),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[48, 47])).
% 8.31/8.48 tff(50,plain,
% 8.31/8.48 (product(identity, identity, identity)),
% 8.31/8.48 inference(unit_resolution,[status(thm)],[49, 32, 43, 41])).
% 8.31/8.48 tff(51,plain,
% 8.31/8.48 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : refl((product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) <=> (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))))),
% 8.31/8.48 inference(bind,[status(th)],[])).
% 8.31/8.48 tff(52,plain,
% 8.31/8.48 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 8.31/8.48 inference(quant_intro,[status(thm)],[51])).
% 8.31/8.48 tff(53,plain,
% 8.31/8.48 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(54,plain,
% 8.31/8.48 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : trans(monotonicity(rewrite((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) <=> ((~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))), (((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W)) <=> (((~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) | product(U, Z, W)))), rewrite((((~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) | product(U, Z, W)) <=> (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))), (((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W)) <=> (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))))),
% 8.31/8.48 inference(bind,[status(th)],[])).
% 8.31/8.48 tff(55,plain,
% 8.31/8.48 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W)) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 8.31/8.48 inference(quant_intro,[status(thm)],[54])).
% 8.31/8.48 tff(56,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W))), file('/export/starexec/sandbox/benchmark/Axioms/GRP005-0.ax','associativity2')).
% 8.31/8.48 tff(57,plain,
% 8.31/8.48 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[56, 55])).
% 8.31/8.48 tff(58,plain,
% 8.31/8.48 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[57, 53])).
% 8.31/8.48 tff(59,plain,(
% 8.31/8.48 ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 8.31/8.48 inference(skolemize,[status(sab)],[58])).
% 8.31/8.48 tff(60,plain,
% 8.31/8.48 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[59, 52])).
% 8.31/8.48 tff(61,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(inverse(inverse(inverse(b))), identity, inverse(b))) | (~product(identity, identity, identity)) | product(inverse(b), identity, inverse(b)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(inverse(inverse(inverse(b))), identity, inverse(b))) | (~product(identity, identity, identity)) | product(inverse(b), identity, inverse(b)))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(62,plain,
% 8.31/8.48 ((product(inverse(b), identity, inverse(b)) | (~product(identity, identity, identity)) | (~product(inverse(inverse(inverse(b))), identity, inverse(b))) | (~product(inverse(inverse(inverse(b))), identity, inverse(b)))) <=> ((~product(inverse(inverse(inverse(b))), identity, inverse(b))) | (~product(identity, identity, identity)) | product(inverse(b), identity, inverse(b)))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(63,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(inverse(b), identity, inverse(b)) | (~product(identity, identity, identity)) | (~product(inverse(inverse(inverse(b))), identity, inverse(b))) | (~product(inverse(inverse(inverse(b))), identity, inverse(b))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(inverse(inverse(inverse(b))), identity, inverse(b))) | (~product(identity, identity, identity)) | product(inverse(b), identity, inverse(b))))),
% 8.31/8.48 inference(monotonicity,[status(thm)],[62])).
% 8.31/8.48 tff(64,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(inverse(b), identity, inverse(b)) | (~product(identity, identity, identity)) | (~product(inverse(inverse(inverse(b))), identity, inverse(b))) | (~product(inverse(inverse(inverse(b))), identity, inverse(b))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(inverse(inverse(inverse(b))), identity, inverse(b))) | (~product(identity, identity, identity)) | product(inverse(b), identity, inverse(b)))),
% 8.31/8.48 inference(transitivity,[status(thm)],[63, 61])).
% 8.31/8.48 tff(65,plain,
% 8.31/8.48 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(inverse(b), identity, inverse(b)) | (~product(identity, identity, identity)) | (~product(inverse(inverse(inverse(b))), identity, inverse(b))) | (~product(inverse(inverse(inverse(b))), identity, inverse(b))))),
% 8.31/8.48 inference(quant_inst,[status(thm)],[])).
% 8.31/8.48 tff(66,plain,
% 8.31/8.48 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(inverse(inverse(inverse(b))), identity, inverse(b))) | (~product(identity, identity, identity)) | product(inverse(b), identity, inverse(b))),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[65, 64])).
% 8.31/8.48 tff(67,plain,
% 8.31/8.48 ($false),
% 8.31/8.48 inference(unit_resolution,[status(thm)],[66, 60, 50, 39, 2])).
% 8.31/8.48 tff(68,plain,(product(inverse(b), identity, inverse(b))), inference(lemma,lemma(discharge,[]))).
% 8.31/8.48 tff(69,plain,
% 8.31/8.48 ((~![X: $i] : product(identity, X, X)) | product(identity, a, a)),
% 8.31/8.48 inference(quant_inst,[status(thm)],[])).
% 8.31/8.48 tff(70,plain,
% 8.31/8.48 (product(identity, a, a)),
% 8.31/8.48 inference(unit_resolution,[status(thm)],[69, 20])).
% 8.31/8.48 tff(71,plain,
% 8.31/8.48 (equalish(a, b) <=> equalish(a, b)),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(72,axiom,(equalish(a, b)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','a_equals_b')).
% 8.31/8.48 tff(73,plain,
% 8.31/8.48 (equalish(a, b)),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[72, 71])).
% 8.31/8.48 tff(74,plain,
% 8.31/8.48 (^[W: $i, Z: $i, Y: $i, X: $i] : refl(((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y))) <=> ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y))))),
% 8.31/8.48 inference(bind,[status(th)],[])).
% 8.31/8.48 tff(75,plain,
% 8.31/8.48 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y)))),
% 8.31/8.48 inference(quant_intro,[status(thm)],[74])).
% 8.31/8.48 tff(76,plain,
% 8.31/8.48 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y)))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(77,plain,
% 8.31/8.48 (^[W: $i, Z: $i, Y: $i, X: $i] : rewrite((((~equalish(X, Y)) | (~product(W, Z, X))) | product(W, Z, Y)) <=> ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y))))),
% 8.31/8.48 inference(bind,[status(th)],[])).
% 8.31/8.48 tff(78,plain,
% 8.31/8.48 (![W: $i, Z: $i, Y: $i, X: $i] : (((~equalish(X, Y)) | (~product(W, Z, X))) | product(W, Z, Y)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y)))),
% 8.31/8.48 inference(quant_intro,[status(thm)],[77])).
% 8.31/8.48 tff(79,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~equalish(X, Y)) | (~product(W, Z, X))) | product(W, Z, Y))), file('/export/starexec/sandbox/benchmark/Axioms/GRP005-0.ax','product_substitution3')).
% 8.31/8.48 tff(80,plain,
% 8.31/8.48 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y)))),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[79, 78])).
% 8.31/8.48 tff(81,plain,
% 8.31/8.48 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y)))),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[80, 76])).
% 8.31/8.48 tff(82,plain,(
% 8.31/8.48 ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y)))),
% 8.31/8.48 inference(skolemize,[status(sab)],[81])).
% 8.31/8.48 tff(83,plain,
% 8.31/8.48 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y)))),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[82, 75])).
% 8.31/8.48 tff(84,plain,
% 8.31/8.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y)))) | ((~product(identity, a, a)) | product(identity, a, b) | (~equalish(a, b)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y)))) | (~product(identity, a, a)) | product(identity, a, b) | (~equalish(a, b)))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(85,plain,
% 8.31/8.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y)))) | ((~product(identity, a, a)) | product(identity, a, b) | (~equalish(a, b)))),
% 8.31/8.48 inference(quant_inst,[status(thm)],[])).
% 8.31/8.48 tff(86,plain,
% 8.31/8.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(W, Z, X)) | product(W, Z, Y) | (~equalish(X, Y)))) | (~product(identity, a, a)) | product(identity, a, b) | (~equalish(a, b))),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[85, 84])).
% 8.31/8.48 tff(87,plain,
% 8.31/8.48 (product(identity, a, b)),
% 8.31/8.48 inference(unit_resolution,[status(thm)],[86, 83, 73, 70])).
% 8.31/8.48 tff(88,assumption,(~product(inverse(inverse(a)), identity, a)), introduced(assumption)).
% 8.31/8.48 tff(89,plain,
% 8.31/8.48 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(inverse(a)), inverse(a), identity)),
% 8.31/8.48 inference(quant_inst,[status(thm)],[])).
% 8.31/8.48 tff(90,plain,
% 8.31/8.48 (product(inverse(inverse(a)), inverse(a), identity)),
% 8.31/8.48 inference(unit_resolution,[status(thm)],[89, 9])).
% 8.31/8.48 tff(91,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(inverse(a), a, identity)) | (~product(identity, a, a)) | (~product(inverse(inverse(a)), inverse(a), identity)) | product(inverse(inverse(a)), identity, a))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(inverse(a), a, identity)) | (~product(identity, a, a)) | (~product(inverse(inverse(a)), inverse(a), identity)) | product(inverse(inverse(a)), identity, a))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(92,plain,
% 8.31/8.48 ((product(inverse(inverse(a)), identity, a) | (~product(identity, a, a)) | (~product(inverse(a), a, identity)) | (~product(inverse(inverse(a)), inverse(a), identity))) <=> ((~product(inverse(a), a, identity)) | (~product(identity, a, a)) | (~product(inverse(inverse(a)), inverse(a), identity)) | product(inverse(inverse(a)), identity, a))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(93,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(inverse(a)), identity, a) | (~product(identity, a, a)) | (~product(inverse(a), a, identity)) | (~product(inverse(inverse(a)), inverse(a), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(inverse(a), a, identity)) | (~product(identity, a, a)) | (~product(inverse(inverse(a)), inverse(a), identity)) | product(inverse(inverse(a)), identity, a)))),
% 8.31/8.48 inference(monotonicity,[status(thm)],[92])).
% 8.31/8.48 tff(94,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(inverse(a)), identity, a) | (~product(identity, a, a)) | (~product(inverse(a), a, identity)) | (~product(inverse(inverse(a)), inverse(a), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(inverse(a), a, identity)) | (~product(identity, a, a)) | (~product(inverse(inverse(a)), inverse(a), identity)) | product(inverse(inverse(a)), identity, a))),
% 8.31/8.48 inference(transitivity,[status(thm)],[93, 91])).
% 8.31/8.48 tff(95,plain,
% 8.31/8.48 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(inverse(a)), identity, a) | (~product(identity, a, a)) | (~product(inverse(a), a, identity)) | (~product(inverse(inverse(a)), inverse(a), identity)))),
% 8.31/8.48 inference(quant_inst,[status(thm)],[])).
% 8.31/8.48 tff(96,plain,
% 8.31/8.48 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(inverse(a), a, identity)) | (~product(identity, a, a)) | (~product(inverse(inverse(a)), inverse(a), identity)) | product(inverse(inverse(a)), identity, a)),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[95, 94])).
% 8.31/8.48 tff(97,plain,
% 8.31/8.48 ($false),
% 8.31/8.48 inference(unit_resolution,[status(thm)],[96, 32, 43, 70, 90, 88])).
% 8.31/8.48 tff(98,plain,(product(inverse(inverse(a)), identity, a)), inference(lemma,lemma(discharge,[]))).
% 8.31/8.48 tff(99,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(identity, inverse(a), inverse(a))) | (~product(inverse(inverse(a)), inverse(a), identity)) | product(a, inverse(a), identity) | (~product(inverse(inverse(a)), identity, a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(identity, inverse(a), inverse(a))) | (~product(inverse(inverse(a)), inverse(a), identity)) | product(a, inverse(a), identity) | (~product(inverse(inverse(a)), identity, a)))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(100,plain,
% 8.31/8.48 ((product(a, inverse(a), identity) | (~product(identity, inverse(a), inverse(a))) | (~product(inverse(inverse(a)), identity, a)) | (~product(inverse(inverse(a)), inverse(a), identity))) <=> ((~product(identity, inverse(a), inverse(a))) | (~product(inverse(inverse(a)), inverse(a), identity)) | product(a, inverse(a), identity) | (~product(inverse(inverse(a)), identity, a)))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(101,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(a, inverse(a), identity) | (~product(identity, inverse(a), inverse(a))) | (~product(inverse(inverse(a)), identity, a)) | (~product(inverse(inverse(a)), inverse(a), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(identity, inverse(a), inverse(a))) | (~product(inverse(inverse(a)), inverse(a), identity)) | product(a, inverse(a), identity) | (~product(inverse(inverse(a)), identity, a))))),
% 8.31/8.48 inference(monotonicity,[status(thm)],[100])).
% 8.31/8.48 tff(102,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(a, inverse(a), identity) | (~product(identity, inverse(a), inverse(a))) | (~product(inverse(inverse(a)), identity, a)) | (~product(inverse(inverse(a)), inverse(a), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(identity, inverse(a), inverse(a))) | (~product(inverse(inverse(a)), inverse(a), identity)) | product(a, inverse(a), identity) | (~product(inverse(inverse(a)), identity, a)))),
% 8.31/8.48 inference(transitivity,[status(thm)],[101, 99])).
% 8.31/8.48 tff(103,plain,
% 8.31/8.48 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(a, inverse(a), identity) | (~product(identity, inverse(a), inverse(a))) | (~product(inverse(inverse(a)), identity, a)) | (~product(inverse(inverse(a)), inverse(a), identity)))),
% 8.31/8.48 inference(quant_inst,[status(thm)],[])).
% 8.31/8.48 tff(104,plain,
% 8.31/8.48 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(identity, inverse(a), inverse(a))) | (~product(inverse(inverse(a)), inverse(a), identity)) | product(a, inverse(a), identity) | (~product(inverse(inverse(a)), identity, a))),
% 8.31/8.48 inference(modus_ponens,[status(thm)],[103, 102])).
% 8.31/8.48 tff(105,plain,
% 8.31/8.48 (product(a, inverse(a), identity)),
% 8.31/8.48 inference(unit_resolution,[status(thm)],[104, 60, 41, 90, 98])).
% 8.31/8.48 tff(106,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(identity, a, b)) | (~product(a, inverse(a), identity)) | product(b, inverse(a), identity) | (~product(identity, identity, identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(identity, a, b)) | (~product(a, inverse(a), identity)) | product(b, inverse(a), identity) | (~product(identity, identity, identity)))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(107,plain,
% 8.31/8.48 ((product(b, inverse(a), identity) | (~product(a, inverse(a), identity)) | (~product(identity, a, b)) | (~product(identity, identity, identity))) <=> ((~product(identity, a, b)) | (~product(a, inverse(a), identity)) | product(b, inverse(a), identity) | (~product(identity, identity, identity)))),
% 8.31/8.48 inference(rewrite,[status(thm)],[])).
% 8.31/8.48 tff(108,plain,
% 8.31/8.48 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(b, inverse(a), identity) | (~product(a, inverse(a), identity)) | (~product(identity, a, b)) | (~product(identity, identity, identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(identity, a, b)) | (~product(a, inverse(a), identity)) | product(b, inverse(a), identity) | (~product(identity, identity, identity))))),
% 8.31/8.49 inference(monotonicity,[status(thm)],[107])).
% 8.31/8.49 tff(109,plain,
% 8.31/8.49 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(b, inverse(a), identity) | (~product(a, inverse(a), identity)) | (~product(identity, a, b)) | (~product(identity, identity, identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(identity, a, b)) | (~product(a, inverse(a), identity)) | product(b, inverse(a), identity) | (~product(identity, identity, identity)))),
% 8.31/8.49 inference(transitivity,[status(thm)],[108, 106])).
% 8.31/8.49 tff(110,plain,
% 8.31/8.49 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(b, inverse(a), identity) | (~product(a, inverse(a), identity)) | (~product(identity, a, b)) | (~product(identity, identity, identity)))),
% 8.31/8.49 inference(quant_inst,[status(thm)],[])).
% 8.31/8.49 tff(111,plain,
% 8.31/8.49 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(identity, a, b)) | (~product(a, inverse(a), identity)) | product(b, inverse(a), identity) | (~product(identity, identity, identity))),
% 8.31/8.49 inference(modus_ponens,[status(thm)],[110, 109])).
% 8.31/8.49 tff(112,plain,
% 8.31/8.49 (product(b, inverse(a), identity)),
% 8.31/8.49 inference(unit_resolution,[status(thm)],[111, 60, 50, 105, 87])).
% 8.31/8.49 tff(113,plain,
% 8.31/8.49 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(b), b, identity)),
% 8.31/8.49 inference(quant_inst,[status(thm)],[])).
% 8.31/8.49 tff(114,plain,
% 8.31/8.49 (product(inverse(b), b, identity)),
% 8.31/8.49 inference(unit_resolution,[status(thm)],[113, 9])).
% 8.31/8.49 tff(115,plain,
% 8.31/8.49 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(inverse(b), b, identity)) | (~product(b, inverse(a), identity)) | product(identity, inverse(a), inverse(b)) | (~product(inverse(b), identity, inverse(b))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(inverse(b), b, identity)) | (~product(b, inverse(a), identity)) | product(identity, inverse(a), inverse(b)) | (~product(inverse(b), identity, inverse(b))))),
% 8.31/8.49 inference(rewrite,[status(thm)],[])).
% 8.31/8.49 tff(116,plain,
% 8.31/8.49 ((product(identity, inverse(a), inverse(b)) | (~product(b, inverse(a), identity)) | (~product(inverse(b), b, identity)) | (~product(inverse(b), identity, inverse(b)))) <=> ((~product(inverse(b), b, identity)) | (~product(b, inverse(a), identity)) | product(identity, inverse(a), inverse(b)) | (~product(inverse(b), identity, inverse(b))))),
% 8.31/8.49 inference(rewrite,[status(thm)],[])).
% 8.31/8.49 tff(117,plain,
% 8.31/8.49 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(identity, inverse(a), inverse(b)) | (~product(b, inverse(a), identity)) | (~product(inverse(b), b, identity)) | (~product(inverse(b), identity, inverse(b))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(inverse(b), b, identity)) | (~product(b, inverse(a), identity)) | product(identity, inverse(a), inverse(b)) | (~product(inverse(b), identity, inverse(b)))))),
% 8.31/8.49 inference(monotonicity,[status(thm)],[116])).
% 8.31/8.49 tff(118,plain,
% 8.31/8.49 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(identity, inverse(a), inverse(b)) | (~product(b, inverse(a), identity)) | (~product(inverse(b), b, identity)) | (~product(inverse(b), identity, inverse(b))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(inverse(b), b, identity)) | (~product(b, inverse(a), identity)) | product(identity, inverse(a), inverse(b)) | (~product(inverse(b), identity, inverse(b))))),
% 8.31/8.49 inference(transitivity,[status(thm)],[117, 115])).
% 8.31/8.49 tff(119,plain,
% 8.31/8.49 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(identity, inverse(a), inverse(b)) | (~product(b, inverse(a), identity)) | (~product(inverse(b), b, identity)) | (~product(inverse(b), identity, inverse(b))))),
% 8.31/8.49 inference(quant_inst,[status(thm)],[])).
% 8.31/8.49 tff(120,plain,
% 8.31/8.49 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(inverse(b), b, identity)) | (~product(b, inverse(a), identity)) | product(identity, inverse(a), inverse(b)) | (~product(inverse(b), identity, inverse(b)))),
% 8.31/8.49 inference(modus_ponens,[status(thm)],[119, 118])).
% 8.31/8.49 tff(121,plain,
% 8.31/8.49 ($false),
% 8.31/8.49 inference(unit_resolution,[status(thm)],[120, 60, 114, 112, 68, 1])).
% 8.31/8.49 tff(122,plain,(product(identity, inverse(a), inverse(b))), inference(lemma,lemma(discharge,[]))).
% 8.31/8.49 tff(123,plain,
% 8.31/8.49 ((~equalish(inverse(a), inverse(b))) <=> (~equalish(inverse(a), inverse(b)))),
% 8.31/8.49 inference(rewrite,[status(thm)],[])).
% 8.31/8.49 tff(124,axiom,(~equalish(inverse(a), inverse(b))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_inverse_substitution')).
% 8.31/8.49 tff(125,plain,
% 8.31/8.49 (~equalish(inverse(a), inverse(b))),
% 8.31/8.49 inference(modus_ponens,[status(thm)],[124, 123])).
% 8.31/8.49 tff(126,plain,
% 8.31/8.49 (^[W: $i, Z: $i, Y: $i, X: $i] : refl((equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z))) <=> (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z))))),
% 8.31/8.49 inference(bind,[status(th)],[])).
% 8.31/8.49 tff(127,plain,
% 8.31/8.49 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 8.31/8.49 inference(quant_intro,[status(thm)],[126])).
% 8.31/8.49 tff(128,plain,
% 8.31/8.49 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 8.31/8.49 inference(rewrite,[status(thm)],[])).
% 8.31/8.49 tff(129,plain,
% 8.31/8.49 (^[W: $i, Z: $i, Y: $i, X: $i] : rewrite((((~product(X, Y, Z)) | (~product(X, Y, W))) | equalish(Z, W)) <=> (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z))))),
% 8.31/8.49 inference(bind,[status(th)],[])).
% 8.31/8.49 tff(130,plain,
% 8.31/8.49 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, Z)) | (~product(X, Y, W))) | equalish(Z, W)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 8.31/8.49 inference(quant_intro,[status(thm)],[129])).
% 8.31/8.49 tff(131,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, Z)) | (~product(X, Y, W))) | equalish(Z, W))), file('/export/starexec/sandbox/benchmark/Axioms/GRP005-0.ax','total_function2')).
% 8.31/8.49 tff(132,plain,
% 8.31/8.49 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 8.31/8.49 inference(modus_ponens,[status(thm)],[131, 130])).
% 8.31/8.49 tff(133,plain,
% 8.31/8.49 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 8.31/8.49 inference(modus_ponens,[status(thm)],[132, 128])).
% 8.31/8.49 tff(134,plain,(
% 8.31/8.49 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 8.31/8.49 inference(skolemize,[status(sab)],[133])).
% 8.31/8.49 tff(135,plain,
% 8.31/8.49 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 8.31/8.49 inference(modus_ponens,[status(thm)],[134, 127])).
% 8.31/8.49 tff(136,plain,
% 8.31/8.49 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (equalish(inverse(a), inverse(b)) | (~product(identity, inverse(a), inverse(a))) | (~product(identity, inverse(a), inverse(b))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | equalish(inverse(a), inverse(b)) | (~product(identity, inverse(a), inverse(a))) | (~product(identity, inverse(a), inverse(b))))),
% 8.31/8.50 inference(rewrite,[status(thm)],[])).
% 8.31/8.50 tff(137,plain,
% 8.31/8.50 ((equalish(inverse(a), inverse(b)) | (~product(identity, inverse(a), inverse(b))) | (~product(identity, inverse(a), inverse(a)))) <=> (equalish(inverse(a), inverse(b)) | (~product(identity, inverse(a), inverse(a))) | (~product(identity, inverse(a), inverse(b))))),
% 8.31/8.50 inference(rewrite,[status(thm)],[])).
% 8.31/8.50 tff(138,plain,
% 8.31/8.50 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (equalish(inverse(a), inverse(b)) | (~product(identity, inverse(a), inverse(b))) | (~product(identity, inverse(a), inverse(a))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (equalish(inverse(a), inverse(b)) | (~product(identity, inverse(a), inverse(a))) | (~product(identity, inverse(a), inverse(b)))))),
% 8.31/8.50 inference(monotonicity,[status(thm)],[137])).
% 8.31/8.50 tff(139,plain,
% 8.31/8.50 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (equalish(inverse(a), inverse(b)) | (~product(identity, inverse(a), inverse(b))) | (~product(identity, inverse(a), inverse(a))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | equalish(inverse(a), inverse(b)) | (~product(identity, inverse(a), inverse(a))) | (~product(identity, inverse(a), inverse(b))))),
% 8.31/8.50 inference(transitivity,[status(thm)],[138, 136])).
% 8.31/8.50 tff(140,plain,
% 8.31/8.50 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (equalish(inverse(a), inverse(b)) | (~product(identity, inverse(a), inverse(b))) | (~product(identity, inverse(a), inverse(a))))),
% 8.31/8.50 inference(quant_inst,[status(thm)],[])).
% 8.31/8.50 tff(141,plain,
% 8.31/8.50 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | equalish(inverse(a), inverse(b)) | (~product(identity, inverse(a), inverse(a))) | (~product(identity, inverse(a), inverse(b)))),
% 8.31/8.50 inference(modus_ponens,[status(thm)],[140, 139])).
% 8.31/8.50 tff(142,plain,
% 8.31/8.50 ($false),
% 8.31/8.50 inference(unit_resolution,[status(thm)],[141, 135, 125, 41, 122])).
% 8.31/8.50 % SZS output end Proof
% 8.31/8.50 % E exiting
%------------------------------------------------------------------------------