%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP039-3 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.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 05:06:55 AM UTC 2025
% Result : Unsatisfiable 2.14s 2.33s
% Output : Proof 2.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.13 % Problem : GRP039-3 : TPTP v9.0.0. Released v1.0.0.
% 0.03/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n003.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Jun 19 15:30:09 EDT 2025
% 0.13/0.35 % CPUTime :
% 2.14/2.33 % SZS status Unsatisfiable
% 2.14/2.33 % SZS output start Proof
% 2.14/2.33 tff(product_type, type, (
% 2.14/2.33 product: ( $i * $i * $i ) > $o)).
% 2.14/2.33 tff(inverse_type, type, (
% 2.14/2.33 inverse: $i > $i)).
% 2.14/2.33 tff(a_type, type, (
% 2.14/2.33 a: $i)).
% 2.14/2.33 tff(multiply_type, type, (
% 2.14/2.33 multiply: ( $i * $i ) > $i)).
% 2.14/2.33 tff(b_type, type, (
% 2.14/2.33 b: $i)).
% 2.14/2.33 tff(identity_type, type, (
% 2.14/2.33 identity: $i)).
% 2.14/2.33 tff(subgroup_member_type, type, (
% 2.14/2.33 subgroup_member: $i > $o)).
% 2.14/2.33 tff(element_in_O2_type, type, (
% 2.14/2.33 element_in_O2: ( $i * $i ) > $i)).
% 2.14/2.33 tff(d_type, type, (
% 2.14/2.33 d: $i)).
% 2.14/2.33 tff(c_type, type, (
% 2.14/2.33 c: $i)).
% 2.14/2.33 tff(1,plain,
% 2.14/2.33 (^[Y: $i, X: $i] : refl(product(X, Y, multiply(X, Y)) <=> product(X, Y, multiply(X, Y)))),
% 2.14/2.33 inference(bind,[status(th)],[])).
% 2.14/2.33 tff(2,plain,
% 2.14/2.33 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 2.14/2.33 inference(quant_intro,[status(thm)],[1])).
% 2.14/2.33 tff(3,plain,
% 2.14/2.33 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 2.14/2.33 inference(rewrite,[status(thm)],[])).
% 2.14/2.33 tff(4,axiom,(![Y: $i, X: $i] : product(X, Y, multiply(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','total_function1')).
% 2.14/2.33 tff(5,plain,
% 2.14/2.33 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 2.14/2.33 inference(modus_ponens,[status(thm)],[4, 3])).
% 2.14/2.33 tff(6,plain,(
% 2.14/2.33 ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 2.14/2.33 inference(skolemize,[status(sab)],[5])).
% 2.14/2.33 tff(7,plain,
% 2.14/2.33 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 2.14/2.33 inference(modus_ponens,[status(thm)],[6, 2])).
% 2.14/2.33 tff(8,plain,
% 2.14/2.33 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))),
% 2.14/2.33 inference(quant_inst,[status(thm)],[])).
% 2.14/2.33 tff(9,plain,
% 2.14/2.33 (product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))),
% 2.14/2.33 inference(unit_resolution,[status(thm)],[8, 7])).
% 2.14/2.33 tff(10,plain,
% 2.14/2.33 (^[X: $i] : refl(product(X, inverse(X), identity) <=> product(X, inverse(X), identity))),
% 2.14/2.33 inference(bind,[status(th)],[])).
% 2.14/2.33 tff(11,plain,
% 2.14/2.33 (![X: $i] : product(X, inverse(X), identity) <=> ![X: $i] : product(X, inverse(X), identity)),
% 2.14/2.33 inference(quant_intro,[status(thm)],[10])).
% 2.14/2.33 tff(12,plain,
% 2.14/2.33 (![X: $i] : product(X, inverse(X), identity) <=> ![X: $i] : product(X, inverse(X), identity)),
% 2.14/2.33 inference(rewrite,[status(thm)],[])).
% 2.14/2.33 tff(13,axiom,(![X: $i] : product(X, inverse(X), identity)), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','right_inverse')).
% 2.14/2.33 tff(14,plain,
% 2.14/2.33 (![X: $i] : product(X, inverse(X), identity)),
% 2.14/2.33 inference(modus_ponens,[status(thm)],[13, 12])).
% 2.14/2.33 tff(15,plain,(
% 2.14/2.33 ![X: $i] : product(X, inverse(X), identity)),
% 2.14/2.33 inference(skolemize,[status(sab)],[14])).
% 2.14/2.33 tff(16,plain,
% 2.14/2.33 (![X: $i] : product(X, inverse(X), identity)),
% 2.14/2.33 inference(modus_ponens,[status(thm)],[15, 11])).
% 2.14/2.33 tff(17,plain,
% 2.14/2.33 ((~![X: $i] : product(X, inverse(X), identity)) | product(multiply(a, b), inverse(multiply(a, b)), identity)),
% 2.14/2.33 inference(quant_inst,[status(thm)],[])).
% 2.14/2.33 tff(18,plain,
% 2.14/2.33 (product(multiply(a, b), inverse(multiply(a, b)), identity)),
% 2.14/2.33 inference(unit_resolution,[status(thm)],[17, 16])).
% 2.14/2.33 tff(19,plain,
% 2.14/2.33 (^[X: $i] : refl(product(X, identity, X) <=> product(X, identity, X))),
% 2.14/2.33 inference(bind,[status(th)],[])).
% 2.14/2.33 tff(20,plain,
% 2.14/2.33 (![X: $i] : product(X, identity, X) <=> ![X: $i] : product(X, identity, X)),
% 2.14/2.33 inference(quant_intro,[status(thm)],[19])).
% 2.14/2.33 tff(21,plain,
% 2.14/2.33 (![X: $i] : product(X, identity, X) <=> ![X: $i] : product(X, identity, X)),
% 2.14/2.33 inference(rewrite,[status(thm)],[])).
% 2.14/2.33 tff(22,axiom,(![X: $i] : product(X, identity, X)), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','right_identity')).
% 2.14/2.33 tff(23,plain,
% 2.14/2.33 (![X: $i] : product(X, identity, X)),
% 2.14/2.33 inference(modus_ponens,[status(thm)],[22, 21])).
% 2.14/2.33 tff(24,plain,(
% 2.14/2.33 ![X: $i] : product(X, identity, X)),
% 2.14/2.33 inference(skolemize,[status(sab)],[23])).
% 2.14/2.33 tff(25,plain,
% 2.14/2.33 (![X: $i] : product(X, identity, X)),
% 2.14/2.33 inference(modus_ponens,[status(thm)],[24, 20])).
% 2.14/2.33 tff(26,plain,
% 2.14/2.33 ((~![X: $i] : product(X, identity, X)) | product(inverse(a), identity, inverse(a))),
% 2.14/2.33 inference(quant_inst,[status(thm)],[])).
% 2.14/2.33 tff(27,plain,
% 2.14/2.33 (product(inverse(a), identity, inverse(a))),
% 2.14/2.33 inference(unit_resolution,[status(thm)],[26, 25])).
% 2.14/2.33 tff(28,plain,
% 2.14/2.33 (^[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))))),
% 2.14/2.33 inference(bind,[status(th)],[])).
% 2.14/2.33 tff(29,plain,
% 2.14/2.33 (![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)))),
% 2.14/2.33 inference(quant_intro,[status(thm)],[28])).
% 2.14/2.33 tff(30,plain,
% 2.14/2.33 (![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)))),
% 2.14/2.33 inference(rewrite,[status(thm)],[])).
% 2.14/2.33 tff(31,plain,
% 2.14/2.33 (^[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)))))),
% 2.14/2.33 inference(bind,[status(th)],[])).
% 2.14/2.33 tff(32,plain,
% 2.14/2.33 (![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)))),
% 2.14/2.33 inference(quant_intro,[status(thm)],[31])).
% 2.14/2.33 tff(33,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/sandbox2/benchmark/Axioms/GRP003-0.ax','associativity2')).
% 2.14/2.33 tff(34,plain,
% 2.14/2.33 (![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)))),
% 2.14/2.33 inference(modus_ponens,[status(thm)],[33, 32])).
% 2.14/2.33 tff(35,plain,
% 2.14/2.33 (![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)))),
% 2.14/2.33 inference(modus_ponens,[status(thm)],[34, 30])).
% 2.14/2.33 tff(36,plain,(
% 2.14/2.33 ![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)))),
% 2.14/2.33 inference(skolemize,[status(sab)],[35])).
% 2.14/2.33 tff(37,plain,
% 2.14/2.33 (![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)))),
% 2.14/2.33 inference(modus_ponens,[status(thm)],[36, 29])).
% 2.14/2.33 tff(38,plain,
% 2.14/2.33 (((~![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(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), identity, inverse(a))) | (~product(multiply(a, b), inverse(multiply(a, b)), identity)) | product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(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(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), identity, inverse(a))) | (~product(multiply(a, b), inverse(multiply(a, b)), identity)) | product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a)))),
% 2.14/2.33 inference(rewrite,[status(thm)],[])).
% 2.14/2.33 tff(39,plain,
% 2.14/2.33 ((product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a)) | (~product(multiply(a, b), inverse(multiply(a, b)), identity)) | (~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), identity, inverse(a)))) <=> ((~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), identity, inverse(a))) | (~product(multiply(a, b), inverse(multiply(a, b)), identity)) | product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a)))),
% 2.14/2.33 inference(rewrite,[status(thm)],[])).
% 2.14/2.33 tff(40,plain,
% 2.14/2.33 (((~![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(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a)) | (~product(multiply(a, b), inverse(multiply(a, b)), identity)) | (~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), identity, inverse(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(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), identity, inverse(a))) | (~product(multiply(a, b), inverse(multiply(a, b)), identity)) | product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a))))),
% 2.14/2.34 inference(monotonicity,[status(thm)],[39])).
% 2.14/2.34 tff(41,plain,
% 2.14/2.34 (((~![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(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a)) | (~product(multiply(a, b), inverse(multiply(a, b)), identity)) | (~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), identity, inverse(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(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), identity, inverse(a))) | (~product(multiply(a, b), inverse(multiply(a, b)), identity)) | product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a)))),
% 2.14/2.34 inference(transitivity,[status(thm)],[40, 38])).
% 2.14/2.34 tff(42,plain,
% 2.14/2.34 ((~![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(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a)) | (~product(multiply(a, b), inverse(multiply(a, b)), identity)) | (~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), identity, inverse(a))))),
% 2.14/2.34 inference(quant_inst,[status(thm)],[])).
% 2.14/2.34 tff(43,plain,
% 2.14/2.34 ((~![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(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), identity, inverse(a))) | (~product(multiply(a, b), inverse(multiply(a, b)), identity)) | product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[42, 41])).
% 2.14/2.34 tff(44,plain,
% 2.14/2.34 (product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a))),
% 2.14/2.34 inference(unit_resolution,[status(thm)],[43, 37, 27, 18, 9])).
% 2.14/2.34 tff(45,plain,
% 2.14/2.34 (^[X: $i] : refl(product(identity, X, X) <=> product(identity, X, X))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(46,plain,
% 2.14/2.34 (![X: $i] : product(identity, X, X) <=> ![X: $i] : product(identity, X, X)),
% 2.14/2.34 inference(quant_intro,[status(thm)],[45])).
% 2.14/2.34 tff(47,plain,
% 2.14/2.34 (![X: $i] : product(identity, X, X) <=> ![X: $i] : product(identity, X, X)),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(48,axiom,(![X: $i] : product(identity, X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','left_identity')).
% 2.14/2.34 tff(49,plain,
% 2.14/2.34 (![X: $i] : product(identity, X, X)),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[48, 47])).
% 2.14/2.34 tff(50,plain,(
% 2.14/2.34 ![X: $i] : product(identity, X, X)),
% 2.14/2.34 inference(skolemize,[status(sab)],[49])).
% 2.14/2.34 tff(51,plain,
% 2.14/2.34 (![X: $i] : product(identity, X, X)),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[50, 46])).
% 2.14/2.34 tff(52,plain,
% 2.14/2.34 ((~![X: $i] : product(identity, X, X)) | product(identity, b, b)),
% 2.14/2.34 inference(quant_inst,[status(thm)],[])).
% 2.14/2.34 tff(53,plain,
% 2.14/2.34 (product(identity, b, b)),
% 2.14/2.34 inference(unit_resolution,[status(thm)],[52, 51])).
% 2.14/2.34 tff(54,plain,
% 2.14/2.34 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(a, b, multiply(a, b))),
% 2.14/2.34 inference(quant_inst,[status(thm)],[])).
% 2.14/2.34 tff(55,plain,
% 2.14/2.34 (product(a, b, multiply(a, b))),
% 2.14/2.34 inference(unit_resolution,[status(thm)],[54, 7])).
% 2.14/2.34 tff(56,plain,
% 2.14/2.34 (^[X: $i] : refl(product(inverse(X), X, identity) <=> product(inverse(X), X, identity))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(57,plain,
% 2.14/2.34 (![X: $i] : product(inverse(X), X, identity) <=> ![X: $i] : product(inverse(X), X, identity)),
% 2.14/2.34 inference(quant_intro,[status(thm)],[56])).
% 2.14/2.34 tff(58,plain,
% 2.14/2.34 (![X: $i] : product(inverse(X), X, identity) <=> ![X: $i] : product(inverse(X), X, identity)),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(59,axiom,(![X: $i] : product(inverse(X), X, identity)), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','left_inverse')).
% 2.14/2.34 tff(60,plain,
% 2.14/2.34 (![X: $i] : product(inverse(X), X, identity)),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[59, 58])).
% 2.14/2.34 tff(61,plain,(
% 2.14/2.34 ![X: $i] : product(inverse(X), X, identity)),
% 2.14/2.34 inference(skolemize,[status(sab)],[60])).
% 2.14/2.34 tff(62,plain,
% 2.14/2.34 (![X: $i] : product(inverse(X), X, identity)),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[61, 57])).
% 2.14/2.34 tff(63,plain,
% 2.14/2.34 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(a), a, identity)),
% 2.14/2.34 inference(quant_inst,[status(thm)],[])).
% 2.14/2.34 tff(64,plain,
% 2.14/2.34 (product(inverse(a), a, identity)),
% 2.14/2.34 inference(unit_resolution,[status(thm)],[63, 62])).
% 2.14/2.34 tff(65,plain,
% 2.14/2.34 (^[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))))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(66,plain,
% 2.14/2.34 (![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)))),
% 2.14/2.34 inference(quant_intro,[status(thm)],[65])).
% 2.14/2.34 tff(67,plain,
% 2.14/2.34 (![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)))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(68,plain,
% 2.14/2.34 (^[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)))))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(69,plain,
% 2.14/2.34 (![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)))),
% 2.14/2.34 inference(quant_intro,[status(thm)],[68])).
% 2.14/2.34 tff(70,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/sandbox2/benchmark/Axioms/GRP003-0.ax','associativity1')).
% 2.14/2.34 tff(71,plain,
% 2.14/2.34 (![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)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[70, 69])).
% 2.14/2.34 tff(72,plain,
% 2.14/2.34 (![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)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[71, 67])).
% 2.14/2.34 tff(73,plain,(
% 2.14/2.34 ![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)))),
% 2.14/2.34 inference(skolemize,[status(sab)],[72])).
% 2.14/2.34 tff(74,plain,
% 2.14/2.34 (![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)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[73, 66])).
% 2.14/2.34 tff(75,plain,
% 2.14/2.34 (((~![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(a, b, multiply(a, b))) | (~product(identity, b, b)) | product(inverse(a), multiply(a, b), 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(inverse(a), a, identity)) | (~product(a, b, multiply(a, b))) | (~product(identity, b, b)) | product(inverse(a), multiply(a, b), b))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(76,plain,
% 2.14/2.34 ((product(inverse(a), multiply(a, b), b) | (~product(identity, b, b)) | (~product(a, b, multiply(a, b))) | (~product(inverse(a), a, identity))) <=> ((~product(inverse(a), a, identity)) | (~product(a, b, multiply(a, b))) | (~product(identity, b, b)) | product(inverse(a), multiply(a, b), b))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(77,plain,
% 2.14/2.34 (((~![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), multiply(a, b), b) | (~product(identity, b, b)) | (~product(a, b, multiply(a, b))) | (~product(inverse(a), 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(a, b, multiply(a, b))) | (~product(identity, b, b)) | product(inverse(a), multiply(a, b), b)))),
% 2.14/2.34 inference(monotonicity,[status(thm)],[76])).
% 2.14/2.34 tff(78,plain,
% 2.14/2.34 (((~![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), multiply(a, b), b) | (~product(identity, b, b)) | (~product(a, b, multiply(a, b))) | (~product(inverse(a), 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(a, b, multiply(a, b))) | (~product(identity, b, b)) | product(inverse(a), multiply(a, b), b))),
% 2.14/2.34 inference(transitivity,[status(thm)],[77, 75])).
% 2.14/2.34 tff(79,plain,
% 2.14/2.34 ((~![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), multiply(a, b), b) | (~product(identity, b, b)) | (~product(a, b, multiply(a, b))) | (~product(inverse(a), a, identity)))),
% 2.14/2.34 inference(quant_inst,[status(thm)],[])).
% 2.14/2.34 tff(80,plain,
% 2.14/2.34 ((~![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(a, b, multiply(a, b))) | (~product(identity, b, b)) | product(inverse(a), multiply(a, b), b)),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[79, 78])).
% 2.14/2.34 tff(81,plain,
% 2.14/2.34 (product(inverse(a), multiply(a, b), b)),
% 2.14/2.34 inference(unit_resolution,[status(thm)],[80, 74, 64, 55, 53])).
% 2.14/2.34 tff(82,plain,
% 2.14/2.34 (^[W: $i, Z: $i, Y: $i, X: $i] : refl(((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z))) <=> ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z))))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(83,plain,
% 2.14/2.34 (![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 2.14/2.34 inference(quant_intro,[status(thm)],[82])).
% 2.14/2.34 tff(84,plain,
% 2.14/2.34 (![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(85,plain,
% 2.14/2.34 (^[W: $i, Z: $i, Y: $i, X: $i] : rewrite((((~product(X, Y, Z)) | (~product(X, Y, W))) | (Z = W)) <=> ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z))))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(86,plain,
% 2.14/2.34 (![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] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 2.14/2.34 inference(quant_intro,[status(thm)],[85])).
% 2.14/2.34 tff(87,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, Z)) | (~product(X, Y, W))) | (Z = W))), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','total_function2')).
% 2.14/2.34 tff(88,plain,
% 2.14/2.34 (![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[87, 86])).
% 2.14/2.34 tff(89,plain,
% 2.14/2.34 (![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[88, 84])).
% 2.14/2.34 tff(90,plain,(
% 2.14/2.34 ![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 2.14/2.34 inference(skolemize,[status(sab)],[89])).
% 2.14/2.34 tff(91,plain,
% 2.14/2.34 (![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[90, 83])).
% 2.14/2.34 tff(92,plain,
% 2.14/2.34 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), multiply(a, b), b)) | (b = multiply(inverse(a), multiply(a, b))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), multiply(a, b), b)) | (b = multiply(inverse(a), multiply(a, b))))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(93,plain,
% 2.14/2.34 (((b = multiply(inverse(a), multiply(a, b))) | (~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), multiply(a, b), b))) <=> ((~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), multiply(a, b), b)) | (b = multiply(inverse(a), multiply(a, b))))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(94,plain,
% 2.14/2.34 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((b = multiply(inverse(a), multiply(a, b))) | (~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), multiply(a, b), b)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), multiply(a, b), b)) | (b = multiply(inverse(a), multiply(a, b)))))),
% 2.14/2.34 inference(monotonicity,[status(thm)],[93])).
% 2.14/2.34 tff(95,plain,
% 2.14/2.34 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((b = multiply(inverse(a), multiply(a, b))) | (~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), multiply(a, b), b)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), multiply(a, b), b)) | (b = multiply(inverse(a), multiply(a, b))))),
% 2.14/2.34 inference(transitivity,[status(thm)],[94, 92])).
% 2.14/2.34 tff(96,plain,
% 2.14/2.34 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((b = multiply(inverse(a), multiply(a, b))) | (~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), multiply(a, b), b)))),
% 2.14/2.34 inference(quant_inst,[status(thm)],[])).
% 2.14/2.34 tff(97,plain,
% 2.14/2.34 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (~product(inverse(a), multiply(a, b), multiply(inverse(a), multiply(a, b)))) | (~product(inverse(a), multiply(a, b), b)) | (b = multiply(inverse(a), multiply(a, b)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[96, 95])).
% 2.14/2.34 tff(98,plain,
% 2.14/2.34 (b = multiply(inverse(a), multiply(a, b))),
% 2.14/2.34 inference(unit_resolution,[status(thm)],[97, 91, 9, 81])).
% 2.14/2.34 tff(99,plain,
% 2.14/2.34 (subgroup_member(b) <=> subgroup_member(multiply(inverse(a), multiply(a, b)))),
% 2.14/2.34 inference(monotonicity,[status(thm)],[98])).
% 2.14/2.34 tff(100,plain,
% 2.14/2.34 (subgroup_member(b) <=> subgroup_member(b)),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(101,axiom,(subgroup_member(b)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','b_is_in_subgroup')).
% 2.14/2.34 tff(102,plain,
% 2.14/2.34 (subgroup_member(b)),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[101, 100])).
% 2.14/2.34 tff(103,plain,
% 2.14/2.34 (subgroup_member(multiply(inverse(a), multiply(a, b)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[102, 99])).
% 2.14/2.34 tff(104,assumption,(~subgroup_member(inverse(multiply(a, b)))), introduced(assumption)).
% 2.14/2.34 tff(105,plain,
% 2.14/2.34 (![A: $i] : (inverse(inverse(A)) = A) <=> ![A: $i] : (inverse(inverse(A)) = A)),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(106,plain,
% 2.14/2.34 (![A: $i] : (inverse(inverse(A)) = A) <=> ![A: $i] : (inverse(inverse(A)) = A)),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(107,axiom,(![A: $i] : (inverse(inverse(A)) = A)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','inverse_is_self_cancelling')).
% 2.14/2.34 tff(108,plain,
% 2.14/2.34 (![A: $i] : (inverse(inverse(A)) = A)),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[107, 106])).
% 2.14/2.34 tff(109,plain,(
% 2.14/2.34 ![A: $i] : (inverse(inverse(A)) = A)),
% 2.14/2.34 inference(skolemize,[status(sab)],[108])).
% 2.14/2.34 tff(110,plain,
% 2.14/2.34 (![A: $i] : (inverse(inverse(A)) = A)),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[109, 105])).
% 2.14/2.34 tff(111,plain,
% 2.14/2.34 ((~![A: $i] : (inverse(inverse(A)) = A)) | (inverse(inverse(a)) = a)),
% 2.14/2.34 inference(quant_inst,[status(thm)],[])).
% 2.14/2.34 tff(112,plain,
% 2.14/2.34 (inverse(inverse(a)) = a),
% 2.14/2.34 inference(unit_resolution,[status(thm)],[111, 110])).
% 2.14/2.34 tff(113,plain,
% 2.14/2.34 (subgroup_member(inverse(inverse(a))) <=> subgroup_member(a)),
% 2.14/2.34 inference(monotonicity,[status(thm)],[112])).
% 2.14/2.34 tff(114,plain,
% 2.14/2.34 (subgroup_member(a) <=> subgroup_member(inverse(inverse(a)))),
% 2.14/2.34 inference(symmetry,[status(thm)],[113])).
% 2.14/2.34 tff(115,plain,
% 2.14/2.34 ((~subgroup_member(a)) <=> (~subgroup_member(inverse(inverse(a))))),
% 2.14/2.34 inference(monotonicity,[status(thm)],[114])).
% 2.14/2.34 tff(116,assumption,(~subgroup_member(c)), introduced(assumption)).
% 2.14/2.34 tff(117,plain,
% 2.14/2.34 (product(b, inverse(a), c) <=> product(b, inverse(a), c)),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(118,axiom,(product(b, inverse(a), c)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','b_times_a_inverse_is_c')).
% 2.14/2.34 tff(119,plain,
% 2.14/2.34 (product(b, inverse(a), c)),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[118, 117])).
% 2.14/2.34 tff(120,plain,
% 2.14/2.34 (^[B: $i, A: $i, C: $i] : refl((subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A))) <=> (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A))))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(121,plain,
% 2.14/2.34 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A))) <=> ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.14/2.34 inference(quant_intro,[status(thm)],[120])).
% 2.14/2.34 tff(122,plain,
% 2.14/2.34 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A))) <=> ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(123,plain,
% 2.14/2.34 (^[B: $i, A: $i, C: $i] : trans(monotonicity(rewrite((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, inverse(B), C))) <=> ((~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))), (((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, inverse(B), C))) | subgroup_member(C)) <=> (((~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A))) | subgroup_member(C)))), rewrite((((~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A))) | subgroup_member(C)) <=> (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))), (((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, inverse(B), C))) | subgroup_member(C)) <=> (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(124,plain,
% 2.14/2.34 (![B: $i, A: $i, C: $i] : ((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, inverse(B), C))) | subgroup_member(C)) <=> ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.14/2.34 inference(quant_intro,[status(thm)],[123])).
% 2.14/2.34 tff(125,axiom,(![B: $i, A: $i, C: $i] : ((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, inverse(B), C))) | subgroup_member(C))), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-2.ax','closure_of_product_and_inverse')).
% 2.14/2.34 tff(126,plain,
% 2.14/2.34 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[125, 124])).
% 2.14/2.34 tff(127,plain,
% 2.14/2.34 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[126, 122])).
% 2.14/2.34 tff(128,plain,(
% 2.14/2.34 ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.14/2.34 inference(skolemize,[status(sab)],[127])).
% 2.14/2.34 tff(129,plain,
% 2.14/2.34 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[128, 121])).
% 2.14/2.34 tff(130,plain,
% 2.14/2.34 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(a)) | (~subgroup_member(b)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(a)) | (~subgroup_member(b)))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(131,plain,
% 2.14/2.34 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(a)) | (~subgroup_member(b)))),
% 2.14/2.34 inference(quant_inst,[status(thm)],[])).
% 2.14/2.34 tff(132,plain,
% 2.14/2.34 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(a)) | (~subgroup_member(b))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[131, 130])).
% 2.14/2.34 tff(133,plain,
% 2.14/2.34 (subgroup_member(c) | (~subgroup_member(a))),
% 2.14/2.34 inference(unit_resolution,[status(thm)],[132, 129, 102, 119])).
% 2.14/2.34 tff(134,plain,
% 2.14/2.34 (~subgroup_member(a)),
% 2.14/2.34 inference(unit_resolution,[status(thm)],[133, 116])).
% 2.14/2.34 tff(135,plain,
% 2.14/2.34 ((~subgroup_member(d)) <=> (~subgroup_member(d))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(136,axiom,(~subgroup_member(d)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_d_is_in_subgroup')).
% 2.14/2.34 tff(137,plain,
% 2.14/2.34 (~subgroup_member(d)),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[136, 135])).
% 2.14/2.34 tff(138,plain,
% 2.14/2.34 (^[B: $i, A: $i] : refl((subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B)) <=> (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B)))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(139,plain,
% 2.14/2.34 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B)) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 2.14/2.34 inference(quant_intro,[status(thm)],[138])).
% 2.14/2.34 tff(140,plain,
% 2.14/2.34 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B)) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(141,plain,
% 2.14/2.34 (^[B: $i, A: $i] : trans(monotonicity(rewrite((product(A, element_in_O2(A, B), B) | subgroup_member(B)) <=> (subgroup_member(B) | product(A, element_in_O2(A, B), B))), (((product(A, element_in_O2(A, B), B) | subgroup_member(B)) | subgroup_member(A)) <=> ((subgroup_member(B) | product(A, element_in_O2(A, B), B)) | subgroup_member(A)))), rewrite(((subgroup_member(B) | product(A, element_in_O2(A, B), B)) | subgroup_member(A)) <=> (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))), (((product(A, element_in_O2(A, B), B) | subgroup_member(B)) | subgroup_member(A)) <=> (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(142,plain,
% 2.14/2.34 (![B: $i, A: $i] : ((product(A, element_in_O2(A, B), B) | subgroup_member(B)) | subgroup_member(A)) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 2.14/2.34 inference(quant_intro,[status(thm)],[141])).
% 2.14/2.34 tff(143,axiom,(![B: $i, A: $i] : ((product(A, element_in_O2(A, B), B) | subgroup_member(B)) | subgroup_member(A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','property_of_O2')).
% 2.14/2.34 tff(144,plain,
% 2.14/2.34 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[143, 142])).
% 2.14/2.34 tff(145,plain,
% 2.14/2.34 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[144, 140])).
% 2.14/2.34 tff(146,plain,(
% 2.14/2.34 ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 2.14/2.34 inference(skolemize,[status(sab)],[145])).
% 2.14/2.34 tff(147,plain,
% 2.14/2.34 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[146, 139])).
% 2.14/2.34 tff(148,plain,
% 2.14/2.34 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | (subgroup_member(a) | subgroup_member(d) | product(a, element_in_O2(a, d), d))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | subgroup_member(a) | subgroup_member(d) | product(a, element_in_O2(a, d), d))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(149,plain,
% 2.14/2.34 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | (subgroup_member(a) | subgroup_member(d) | product(a, element_in_O2(a, d), d))),
% 2.14/2.34 inference(quant_inst,[status(thm)],[])).
% 2.14/2.34 tff(150,plain,
% 2.14/2.34 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | subgroup_member(a) | subgroup_member(d) | product(a, element_in_O2(a, d), d)),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[149, 148])).
% 2.14/2.34 tff(151,plain,
% 2.14/2.34 (subgroup_member(a) | product(a, element_in_O2(a, d), d)),
% 2.14/2.34 inference(unit_resolution,[status(thm)],[150, 147, 137])).
% 2.14/2.34 tff(152,plain,
% 2.14/2.34 (product(a, element_in_O2(a, d), d)),
% 2.14/2.34 inference(unit_resolution,[status(thm)],[151, 134])).
% 2.14/2.34 tff(153,plain,
% 2.14/2.34 (product(a, c, d) <=> product(a, c, d)),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(154,axiom,(product(a, c, d)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','a_times_c_is_d')).
% 2.14/2.34 tff(155,plain,
% 2.14/2.34 (product(a, c, d)),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[154, 153])).
% 2.14/2.34 tff(156,plain,
% 2.14/2.34 (^[B: $i, D: $i, A: $i, C: $i] : refl(((D = B) | (~product(A, D, C)) | (~product(A, B, C))) <=> ((D = B) | (~product(A, D, C)) | (~product(A, B, C))))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(157,plain,
% 2.14/2.34 (![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C))) <=> ![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))),
% 2.14/2.34 inference(quant_intro,[status(thm)],[156])).
% 2.14/2.34 tff(158,plain,
% 2.14/2.34 (![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C))) <=> ![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(159,plain,
% 2.14/2.34 (^[B: $i, D: $i, A: $i, C: $i] : trans(monotonicity(rewrite(((~product(A, B, C)) | (~product(A, D, C))) <=> ((~product(A, D, C)) | (~product(A, B, C)))), ((((~product(A, B, C)) | (~product(A, D, C))) | (D = B)) <=> (((~product(A, D, C)) | (~product(A, B, C))) | (D = B)))), rewrite((((~product(A, D, C)) | (~product(A, B, C))) | (D = B)) <=> ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))), ((((~product(A, B, C)) | (~product(A, D, C))) | (D = B)) <=> ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(160,plain,
% 2.14/2.34 (![B: $i, D: $i, A: $i, C: $i] : (((~product(A, B, C)) | (~product(A, D, C))) | (D = B)) <=> ![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))),
% 2.14/2.34 inference(quant_intro,[status(thm)],[159])).
% 2.14/2.34 tff(161,axiom,(![B: $i, D: $i, A: $i, C: $i] : (((~product(A, B, C)) | (~product(A, D, C))) | (D = B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_right_cancellation')).
% 2.14/2.34 tff(162,plain,
% 2.14/2.34 (![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[161, 160])).
% 2.14/2.34 tff(163,plain,
% 2.14/2.34 (![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[162, 158])).
% 2.14/2.34 tff(164,plain,(
% 2.14/2.34 ![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))),
% 2.14/2.34 inference(skolemize,[status(sab)],[163])).
% 2.14/2.34 tff(165,plain,
% 2.14/2.34 (![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[164, 157])).
% 2.14/2.34 tff(166,plain,
% 2.14/2.34 (((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((~product(a, element_in_O2(a, d), d)) | (~product(a, c, d)) | (c = element_in_O2(a, d)))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | (~product(a, element_in_O2(a, d), d)) | (~product(a, c, d)) | (c = element_in_O2(a, d)))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(167,plain,
% 2.14/2.34 (((c = element_in_O2(a, d)) | (~product(a, c, d)) | (~product(a, element_in_O2(a, d), d))) <=> ((~product(a, element_in_O2(a, d), d)) | (~product(a, c, d)) | (c = element_in_O2(a, d)))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(168,plain,
% 2.14/2.34 (((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((c = element_in_O2(a, d)) | (~product(a, c, d)) | (~product(a, element_in_O2(a, d), d)))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((~product(a, element_in_O2(a, d), d)) | (~product(a, c, d)) | (c = element_in_O2(a, d))))),
% 2.14/2.34 inference(monotonicity,[status(thm)],[167])).
% 2.14/2.34 tff(169,plain,
% 2.14/2.34 (((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((c = element_in_O2(a, d)) | (~product(a, c, d)) | (~product(a, element_in_O2(a, d), d)))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | (~product(a, element_in_O2(a, d), d)) | (~product(a, c, d)) | (c = element_in_O2(a, d)))),
% 2.14/2.34 inference(transitivity,[status(thm)],[168, 166])).
% 2.14/2.34 tff(170,plain,
% 2.14/2.34 ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((c = element_in_O2(a, d)) | (~product(a, c, d)) | (~product(a, element_in_O2(a, d), d)))),
% 2.14/2.34 inference(quant_inst,[status(thm)],[])).
% 2.14/2.34 tff(171,plain,
% 2.14/2.34 ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | (~product(a, element_in_O2(a, d), d)) | (~product(a, c, d)) | (c = element_in_O2(a, d))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[170, 169])).
% 2.14/2.34 tff(172,plain,
% 2.14/2.34 (c = element_in_O2(a, d)),
% 2.14/2.34 inference(unit_resolution,[status(thm)],[171, 165, 155, 152])).
% 2.14/2.34 tff(173,plain,
% 2.14/2.34 (element_in_O2(a, d) = c),
% 2.14/2.34 inference(symmetry,[status(thm)],[172])).
% 2.14/2.34 tff(174,plain,
% 2.14/2.34 (subgroup_member(element_in_O2(a, d)) <=> subgroup_member(c)),
% 2.14/2.34 inference(monotonicity,[status(thm)],[173])).
% 2.14/2.34 tff(175,plain,
% 2.14/2.34 (subgroup_member(c) <=> subgroup_member(element_in_O2(a, d))),
% 2.14/2.34 inference(symmetry,[status(thm)],[174])).
% 2.14/2.34 tff(176,plain,
% 2.14/2.34 ((~subgroup_member(c)) <=> (~subgroup_member(element_in_O2(a, d)))),
% 2.14/2.34 inference(monotonicity,[status(thm)],[175])).
% 2.14/2.34 tff(177,plain,
% 2.14/2.34 (~subgroup_member(element_in_O2(a, d))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[116, 176])).
% 2.14/2.34 tff(178,plain,
% 2.14/2.34 (^[B: $i, A: $i] : refl((subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B))) <=> (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B))))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(179,plain,
% 2.14/2.34 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B))) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))),
% 2.14/2.34 inference(quant_intro,[status(thm)],[178])).
% 2.14/2.34 tff(180,plain,
% 2.14/2.34 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B))) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(181,plain,
% 2.14/2.34 (^[B: $i, A: $i] : trans(monotonicity(rewrite((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) <=> (subgroup_member(B) | subgroup_member(element_in_O2(A, B)))), (((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A)) <=> ((subgroup_member(B) | subgroup_member(element_in_O2(A, B))) | subgroup_member(A)))), rewrite(((subgroup_member(B) | subgroup_member(element_in_O2(A, B))) | subgroup_member(A)) <=> (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))), (((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A)) <=> (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(182,plain,
% 2.14/2.34 (![B: $i, A: $i] : ((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A)) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))),
% 2.14/2.34 inference(quant_intro,[status(thm)],[181])).
% 2.14/2.34 tff(183,axiom,(![B: $i, A: $i] : ((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','an_element_in_O2')).
% 2.14/2.34 tff(184,plain,
% 2.14/2.34 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[183, 182])).
% 2.14/2.34 tff(185,plain,
% 2.14/2.34 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[184, 180])).
% 2.14/2.34 tff(186,plain,(
% 2.14/2.34 ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))),
% 2.14/2.34 inference(skolemize,[status(sab)],[185])).
% 2.14/2.34 tff(187,plain,
% 2.14/2.34 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[186, 179])).
% 2.14/2.34 tff(188,plain,
% 2.14/2.34 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | (subgroup_member(a) | subgroup_member(d) | subgroup_member(element_in_O2(a, d)))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | subgroup_member(a) | subgroup_member(d) | subgroup_member(element_in_O2(a, d)))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(189,plain,
% 2.14/2.34 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | (subgroup_member(a) | subgroup_member(d) | subgroup_member(element_in_O2(a, d)))),
% 2.14/2.34 inference(quant_inst,[status(thm)],[])).
% 2.14/2.34 tff(190,plain,
% 2.14/2.34 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | subgroup_member(a) | subgroup_member(d) | subgroup_member(element_in_O2(a, d))),
% 2.14/2.34 inference(modus_ponens,[status(thm)],[189, 188])).
% 2.14/2.34 tff(191,plain,
% 2.14/2.34 (subgroup_member(element_in_O2(a, d))),
% 2.14/2.34 inference(unit_resolution,[status(thm)],[190, 187, 137, 134])).
% 2.14/2.34 tff(192,plain,
% 2.14/2.34 ($false),
% 2.14/2.34 inference(unit_resolution,[status(thm)],[191, 177])).
% 2.14/2.34 tff(193,plain,(subgroup_member(c)), inference(lemma,lemma(discharge,[]))).
% 2.14/2.34 tff(194,plain,
% 2.14/2.34 (^[A: $i] : refl(((~subgroup_member(A)) | subgroup_member(inverse(A))) <=> ((~subgroup_member(A)) | subgroup_member(inverse(A))))),
% 2.14/2.34 inference(bind,[status(th)],[])).
% 2.14/2.34 tff(195,plain,
% 2.14/2.34 (![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A))) <=> ![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))),
% 2.14/2.34 inference(quant_intro,[status(thm)],[194])).
% 2.14/2.34 tff(196,plain,
% 2.14/2.34 (![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A))) <=> ![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))),
% 2.14/2.34 inference(rewrite,[status(thm)],[])).
% 2.14/2.34 tff(197,axiom,(![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','subgroup_member_inverse_are_in_subgroup')).
% 2.14/2.34 tff(198,plain,
% 2.14/2.34 (![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))),
% 2.14/2.35 inference(modus_ponens,[status(thm)],[197, 196])).
% 2.14/2.35 tff(199,plain,(
% 2.14/2.35 ![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))),
% 2.14/2.35 inference(skolemize,[status(sab)],[198])).
% 2.14/2.35 tff(200,plain,
% 2.14/2.35 (![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))),
% 2.14/2.35 inference(modus_ponens,[status(thm)],[199, 195])).
% 2.14/2.35 tff(201,plain,
% 2.14/2.35 (((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | ((~subgroup_member(c)) | subgroup_member(inverse(c)))) <=> ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | (~subgroup_member(c)) | subgroup_member(inverse(c)))),
% 2.14/2.35 inference(rewrite,[status(thm)],[])).
% 2.14/2.35 tff(202,plain,
% 2.14/2.35 ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | ((~subgroup_member(c)) | subgroup_member(inverse(c)))),
% 2.14/2.35 inference(quant_inst,[status(thm)],[])).
% 2.14/2.35 tff(203,plain,
% 2.14/2.35 ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | (~subgroup_member(c)) | subgroup_member(inverse(c))),
% 2.14/2.35 inference(modus_ponens,[status(thm)],[202, 201])).
% 2.14/2.35 tff(204,plain,
% 2.14/2.35 ((~subgroup_member(c)) | subgroup_member(inverse(c))),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[203, 200])).
% 2.14/2.35 tff(205,plain,
% 2.14/2.35 (subgroup_member(inverse(c))),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[204, 193])).
% 2.14/2.35 tff(206,plain,
% 2.14/2.35 ((~![A: $i] : (inverse(inverse(A)) = A)) | (inverse(inverse(c)) = c)),
% 2.14/2.35 inference(quant_inst,[status(thm)],[])).
% 2.14/2.35 tff(207,plain,
% 2.14/2.35 (inverse(inverse(c)) = c),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[206, 110])).
% 2.14/2.35 tff(208,plain,
% 2.14/2.35 (product(a, inverse(inverse(c)), d) <=> product(a, c, d)),
% 2.14/2.35 inference(monotonicity,[status(thm)],[207])).
% 2.14/2.35 tff(209,plain,
% 2.14/2.35 (product(a, c, d) <=> product(a, inverse(inverse(c)), d)),
% 2.14/2.35 inference(symmetry,[status(thm)],[208])).
% 2.14/2.35 tff(210,plain,
% 2.14/2.35 (product(a, inverse(inverse(c)), d)),
% 2.14/2.35 inference(modus_ponens,[status(thm)],[155, 209])).
% 2.14/2.35 tff(211,plain,
% 2.14/2.35 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(d) | (~subgroup_member(a)) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(d) | (~subgroup_member(a)) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c))))),
% 2.14/2.35 inference(rewrite,[status(thm)],[])).
% 2.14/2.35 tff(212,plain,
% 2.14/2.35 ((subgroup_member(d) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c))) | (~subgroup_member(a))) <=> (subgroup_member(d) | (~subgroup_member(a)) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c))))),
% 2.14/2.35 inference(rewrite,[status(thm)],[])).
% 2.14/2.35 tff(213,plain,
% 2.14/2.35 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(d) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c))) | (~subgroup_member(a)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(d) | (~subgroup_member(a)) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c)))))),
% 2.14/2.35 inference(monotonicity,[status(thm)],[212])).
% 2.14/2.35 tff(214,plain,
% 2.14/2.35 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(d) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c))) | (~subgroup_member(a)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(d) | (~subgroup_member(a)) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c))))),
% 2.14/2.35 inference(transitivity,[status(thm)],[213, 211])).
% 2.14/2.35 tff(215,plain,
% 2.14/2.35 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(d) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c))) | (~subgroup_member(a)))),
% 2.14/2.35 inference(quant_inst,[status(thm)],[])).
% 2.14/2.35 tff(216,plain,
% 2.14/2.35 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(d) | (~subgroup_member(a)) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c)))),
% 2.14/2.35 inference(modus_ponens,[status(thm)],[215, 214])).
% 2.14/2.35 tff(217,plain,
% 2.14/2.35 ((~subgroup_member(a)) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c)))),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[216, 129, 137])).
% 2.14/2.35 tff(218,plain,
% 2.14/2.35 ((~subgroup_member(a)) | (~subgroup_member(inverse(c)))),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[217, 210])).
% 2.14/2.35 tff(219,plain,
% 2.14/2.35 (~subgroup_member(a)),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[218, 205])).
% 2.14/2.35 tff(220,plain,
% 2.14/2.35 (~subgroup_member(inverse(inverse(a)))),
% 2.14/2.35 inference(modus_ponens,[status(thm)],[219, 115])).
% 2.14/2.35 tff(221,plain,
% 2.14/2.35 (((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | ((~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a))))) <=> ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | (~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a))))),
% 2.14/2.35 inference(rewrite,[status(thm)],[])).
% 2.14/2.35 tff(222,plain,
% 2.14/2.35 ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | ((~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a))))),
% 2.14/2.35 inference(quant_inst,[status(thm)],[])).
% 2.14/2.35 tff(223,plain,
% 2.14/2.35 ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | (~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a)))),
% 2.14/2.35 inference(modus_ponens,[status(thm)],[222, 221])).
% 2.14/2.35 tff(224,plain,
% 2.14/2.35 ((~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a)))),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[223, 200])).
% 2.14/2.35 tff(225,plain,
% 2.14/2.35 (~subgroup_member(inverse(a))),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[224, 220])).
% 2.14/2.35 tff(226,plain,
% 2.14/2.35 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | (subgroup_member(inverse(a)) | subgroup_member(inverse(multiply(a, b))) | product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a)))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | subgroup_member(inverse(a)) | subgroup_member(inverse(multiply(a, b))) | product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a)))),
% 2.14/2.35 inference(rewrite,[status(thm)],[])).
% 2.14/2.35 tff(227,plain,
% 2.14/2.35 ((subgroup_member(inverse(multiply(a, b))) | subgroup_member(inverse(a)) | product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))) <=> (subgroup_member(inverse(a)) | subgroup_member(inverse(multiply(a, b))) | product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a)))),
% 2.14/2.35 inference(rewrite,[status(thm)],[])).
% 2.14/2.35 tff(228,plain,
% 2.14/2.35 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | (subgroup_member(inverse(multiply(a, b))) | subgroup_member(inverse(a)) | product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a)))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | (subgroup_member(inverse(a)) | subgroup_member(inverse(multiply(a, b))) | product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))))),
% 2.14/2.35 inference(monotonicity,[status(thm)],[227])).
% 2.14/2.35 tff(229,plain,
% 2.14/2.35 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | (subgroup_member(inverse(multiply(a, b))) | subgroup_member(inverse(a)) | product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a)))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | subgroup_member(inverse(a)) | subgroup_member(inverse(multiply(a, b))) | product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a)))),
% 2.14/2.35 inference(transitivity,[status(thm)],[228, 226])).
% 2.14/2.35 tff(230,plain,
% 2.14/2.35 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | (subgroup_member(inverse(multiply(a, b))) | subgroup_member(inverse(a)) | product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a)))),
% 2.14/2.35 inference(quant_inst,[status(thm)],[])).
% 2.14/2.35 tff(231,plain,
% 2.14/2.35 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | subgroup_member(inverse(a)) | subgroup_member(inverse(multiply(a, b))) | product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))),
% 2.14/2.35 inference(modus_ponens,[status(thm)],[230, 229])).
% 2.14/2.35 tff(232,plain,
% 2.14/2.35 (subgroup_member(inverse(multiply(a, b))) | product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[231, 147, 225])).
% 2.14/2.35 tff(233,plain,
% 2.14/2.35 (product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[232, 104])).
% 2.14/2.35 tff(234,plain,
% 2.14/2.35 ((~![X: $i] : product(identity, X, X)) | product(identity, multiply(identity, d), multiply(identity, d))),
% 2.14/2.35 inference(quant_inst,[status(thm)],[])).
% 2.14/2.35 tff(235,plain,
% 2.14/2.35 (product(identity, multiply(identity, d), multiply(identity, d))),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[234, 51])).
% 2.14/2.35 tff(236,plain,
% 2.14/2.35 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(identity, d, multiply(identity, d))),
% 2.14/2.35 inference(quant_inst,[status(thm)],[])).
% 2.14/2.35 tff(237,plain,
% 2.14/2.35 (product(identity, d, multiply(identity, d))),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[236, 7])).
% 2.14/2.35 tff(238,plain,
% 2.14/2.35 (((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((d = multiply(identity, d)) | (~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, d, multiply(identity, d))))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | (d = multiply(identity, d)) | (~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, d, multiply(identity, d))))),
% 2.14/2.35 inference(rewrite,[status(thm)],[])).
% 2.14/2.35 tff(239,plain,
% 2.14/2.35 (((d = multiply(identity, d)) | (~product(identity, d, multiply(identity, d))) | (~product(identity, multiply(identity, d), multiply(identity, d)))) <=> ((d = multiply(identity, d)) | (~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, d, multiply(identity, d))))),
% 2.14/2.35 inference(rewrite,[status(thm)],[])).
% 2.14/2.35 tff(240,plain,
% 2.14/2.35 (((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((d = multiply(identity, d)) | (~product(identity, d, multiply(identity, d))) | (~product(identity, multiply(identity, d), multiply(identity, d))))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((d = multiply(identity, d)) | (~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, d, multiply(identity, d)))))),
% 2.14/2.35 inference(monotonicity,[status(thm)],[239])).
% 2.14/2.35 tff(241,plain,
% 2.14/2.35 (((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((d = multiply(identity, d)) | (~product(identity, d, multiply(identity, d))) | (~product(identity, multiply(identity, d), multiply(identity, d))))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | (d = multiply(identity, d)) | (~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, d, multiply(identity, d))))),
% 2.14/2.35 inference(transitivity,[status(thm)],[240, 238])).
% 2.14/2.35 tff(242,plain,
% 2.14/2.35 ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((d = multiply(identity, d)) | (~product(identity, d, multiply(identity, d))) | (~product(identity, multiply(identity, d), multiply(identity, d))))),
% 2.14/2.35 inference(quant_inst,[status(thm)],[])).
% 2.14/2.35 tff(243,plain,
% 2.14/2.35 ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | (d = multiply(identity, d)) | (~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, d, multiply(identity, d)))),
% 2.14/2.35 inference(modus_ponens,[status(thm)],[242, 241])).
% 2.14/2.35 tff(244,plain,
% 2.14/2.35 (d = multiply(identity, d)),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[243, 165, 237, 235])).
% 2.14/2.35 tff(245,plain,
% 2.14/2.35 (multiply(identity, d) = d),
% 2.14/2.35 inference(symmetry,[status(thm)],[244])).
% 2.14/2.35 tff(246,plain,
% 2.14/2.35 (product(multiply(a, b), inverse(a), multiply(identity, d)) <=> product(multiply(a, b), inverse(a), d)),
% 2.14/2.35 inference(monotonicity,[status(thm)],[245])).
% 2.14/2.35 tff(247,plain,
% 2.14/2.35 (product(multiply(a, b), inverse(a), d) <=> product(multiply(a, b), inverse(a), multiply(identity, d))),
% 2.14/2.35 inference(symmetry,[status(thm)],[246])).
% 2.14/2.35 tff(248,plain,
% 2.14/2.35 (((~![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), c)) | (~product(a, b, multiply(a, b))) | product(multiply(a, b), inverse(a), d) | (~product(a, c, d)))) <=> ((~![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), c)) | (~product(a, b, multiply(a, b))) | product(multiply(a, b), inverse(a), d) | (~product(a, c, d)))),
% 2.14/2.35 inference(rewrite,[status(thm)],[])).
% 2.14/2.35 tff(249,plain,
% 2.14/2.35 ((product(multiply(a, b), inverse(a), d) | (~product(b, inverse(a), c)) | (~product(a, b, multiply(a, b))) | (~product(a, c, d))) <=> ((~product(b, inverse(a), c)) | (~product(a, b, multiply(a, b))) | product(multiply(a, b), inverse(a), d) | (~product(a, c, d)))),
% 2.14/2.35 inference(rewrite,[status(thm)],[])).
% 2.14/2.35 tff(250,plain,
% 2.14/2.35 (((~![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(multiply(a, b), inverse(a), d) | (~product(b, inverse(a), c)) | (~product(a, b, multiply(a, b))) | (~product(a, c, d)))) <=> ((~![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), c)) | (~product(a, b, multiply(a, b))) | product(multiply(a, b), inverse(a), d) | (~product(a, c, d))))),
% 2.14/2.35 inference(monotonicity,[status(thm)],[249])).
% 2.14/2.35 tff(251,plain,
% 2.14/2.35 (((~![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(multiply(a, b), inverse(a), d) | (~product(b, inverse(a), c)) | (~product(a, b, multiply(a, b))) | (~product(a, c, d)))) <=> ((~![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), c)) | (~product(a, b, multiply(a, b))) | product(multiply(a, b), inverse(a), d) | (~product(a, c, d)))),
% 2.14/2.35 inference(transitivity,[status(thm)],[250, 248])).
% 2.14/2.35 tff(252,plain,
% 2.14/2.35 ((~![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(multiply(a, b), inverse(a), d) | (~product(b, inverse(a), c)) | (~product(a, b, multiply(a, b))) | (~product(a, c, d)))),
% 2.14/2.35 inference(quant_inst,[status(thm)],[])).
% 2.14/2.35 tff(253,plain,
% 2.14/2.35 ((~![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), c)) | (~product(a, b, multiply(a, b))) | product(multiply(a, b), inverse(a), d) | (~product(a, c, d))),
% 2.14/2.35 inference(modus_ponens,[status(thm)],[252, 251])).
% 2.14/2.35 tff(254,plain,
% 2.14/2.35 (product(multiply(a, b), inverse(a), d)),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[253, 37, 119, 155, 55])).
% 2.14/2.35 tff(255,plain,
% 2.14/2.35 (product(multiply(a, b), inverse(a), multiply(identity, d))),
% 2.14/2.35 inference(modus_ponens,[status(thm)],[254, 247])).
% 2.14/2.35 tff(256,plain,
% 2.14/2.35 (((~![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(multiply(a, b), inverse(multiply(a, b)), identity)) | (~product(multiply(a, b), inverse(a), multiply(identity, d))) | (~product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))) | product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d)))) <=> ((~![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(multiply(a, b), inverse(multiply(a, b)), identity)) | (~product(multiply(a, b), inverse(a), multiply(identity, d))) | (~product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))) | product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d)))),
% 2.14/2.35 inference(rewrite,[status(thm)],[])).
% 2.14/2.35 tff(257,plain,
% 2.14/2.35 ((product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d)) | (~product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))) | (~product(multiply(a, b), inverse(multiply(a, b)), identity)) | (~product(multiply(a, b), inverse(a), multiply(identity, d)))) <=> ((~product(multiply(a, b), inverse(multiply(a, b)), identity)) | (~product(multiply(a, b), inverse(a), multiply(identity, d))) | (~product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))) | product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d)))),
% 2.14/2.35 inference(rewrite,[status(thm)],[])).
% 2.14/2.35 tff(258,plain,
% 2.14/2.35 (((~![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, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d)) | (~product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))) | (~product(multiply(a, b), inverse(multiply(a, b)), identity)) | (~product(multiply(a, b), inverse(a), multiply(identity, d))))) <=> ((~![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(multiply(a, b), inverse(multiply(a, b)), identity)) | (~product(multiply(a, b), inverse(a), multiply(identity, d))) | (~product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))) | product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d))))),
% 2.14/2.35 inference(monotonicity,[status(thm)],[257])).
% 2.14/2.35 tff(259,plain,
% 2.14/2.35 (((~![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, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d)) | (~product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))) | (~product(multiply(a, b), inverse(multiply(a, b)), identity)) | (~product(multiply(a, b), inverse(a), multiply(identity, d))))) <=> ((~![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(multiply(a, b), inverse(multiply(a, b)), identity)) | (~product(multiply(a, b), inverse(a), multiply(identity, d))) | (~product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))) | product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d)))),
% 2.14/2.35 inference(transitivity,[status(thm)],[258, 256])).
% 2.14/2.35 tff(260,plain,
% 2.14/2.35 ((~![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, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d)) | (~product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))) | (~product(multiply(a, b), inverse(multiply(a, b)), identity)) | (~product(multiply(a, b), inverse(a), multiply(identity, d))))),
% 2.14/2.35 inference(quant_inst,[status(thm)],[])).
% 2.14/2.35 tff(261,plain,
% 2.14/2.35 ((~![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(multiply(a, b), inverse(multiply(a, b)), identity)) | (~product(multiply(a, b), inverse(a), multiply(identity, d))) | (~product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))) | product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d))),
% 2.14/2.35 inference(modus_ponens,[status(thm)],[260, 259])).
% 2.14/2.35 tff(262,plain,
% 2.14/2.35 ((~product(inverse(multiply(a, b)), element_in_O2(inverse(multiply(a, b)), inverse(a)), inverse(a))) | product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d))),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[261, 37, 18, 255])).
% 2.14/2.35 tff(263,plain,
% 2.14/2.35 (product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d))),
% 2.14/2.35 inference(unit_resolution,[status(thm)],[262, 233])).
% 2.14/2.35 tff(264,plain,
% 2.14/2.35 (((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d))) | (multiply(identity, d) = element_in_O2(inverse(multiply(a, b)), inverse(a))))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | (~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d))) | (multiply(identity, d) = element_in_O2(inverse(multiply(a, b)), inverse(a))))),
% 2.14/2.35 inference(rewrite,[status(thm)],[])).
% 2.14/2.35 tff(265,plain,
% 2.14/2.35 (((multiply(identity, d) = element_in_O2(inverse(multiply(a, b)), inverse(a))) | (~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d)))) <=> ((~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d))) | (multiply(identity, d) = element_in_O2(inverse(multiply(a, b)), inverse(a))))),
% 2.20/2.35 inference(rewrite,[status(thm)],[])).
% 2.20/2.35 tff(266,plain,
% 2.20/2.35 (((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((multiply(identity, d) = element_in_O2(inverse(multiply(a, b)), inverse(a))) | (~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d))))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d))) | (multiply(identity, d) = element_in_O2(inverse(multiply(a, b)), inverse(a)))))),
% 2.20/2.35 inference(monotonicity,[status(thm)],[265])).
% 2.20/2.35 tff(267,plain,
% 2.20/2.35 (((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((multiply(identity, d) = element_in_O2(inverse(multiply(a, b)), inverse(a))) | (~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d))))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | (~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d))) | (multiply(identity, d) = element_in_O2(inverse(multiply(a, b)), inverse(a))))),
% 2.20/2.35 inference(transitivity,[status(thm)],[266, 264])).
% 2.20/2.35 tff(268,plain,
% 2.20/2.35 ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((multiply(identity, d) = element_in_O2(inverse(multiply(a, b)), inverse(a))) | (~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d))))),
% 2.20/2.35 inference(quant_inst,[status(thm)],[])).
% 2.20/2.35 tff(269,plain,
% 2.20/2.35 ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | (~product(identity, multiply(identity, d), multiply(identity, d))) | (~product(identity, element_in_O2(inverse(multiply(a, b)), inverse(a)), multiply(identity, d))) | (multiply(identity, d) = element_in_O2(inverse(multiply(a, b)), inverse(a)))),
% 2.20/2.35 inference(modus_ponens,[status(thm)],[268, 267])).
% 2.20/2.35 tff(270,plain,
% 2.20/2.35 (multiply(identity, d) = element_in_O2(inverse(multiply(a, b)), inverse(a))),
% 2.20/2.35 inference(unit_resolution,[status(thm)],[269, 165, 235, 263])).
% 2.20/2.35 tff(271,plain,
% 2.20/2.35 (element_in_O2(inverse(multiply(a, b)), inverse(a)) = multiply(identity, d)),
% 2.20/2.35 inference(symmetry,[status(thm)],[270])).
% 2.20/2.35 tff(272,plain,
% 2.20/2.35 (subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a))) <=> subgroup_member(multiply(identity, d))),
% 2.20/2.35 inference(monotonicity,[status(thm)],[271])).
% 2.20/2.35 tff(273,plain,
% 2.20/2.35 (subgroup_member(multiply(identity, d)) <=> subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a)))),
% 2.20/2.35 inference(symmetry,[status(thm)],[272])).
% 2.20/2.35 tff(274,plain,
% 2.20/2.35 ((~subgroup_member(multiply(identity, d))) <=> (~subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a))))),
% 2.20/2.35 inference(monotonicity,[status(thm)],[273])).
% 2.20/2.35 tff(275,plain,
% 2.20/2.35 (subgroup_member(d) <=> subgroup_member(multiply(identity, d))),
% 2.20/2.35 inference(monotonicity,[status(thm)],[244])).
% 2.20/2.35 tff(276,plain,
% 2.20/2.35 ((~subgroup_member(d)) <=> (~subgroup_member(multiply(identity, d)))),
% 2.20/2.35 inference(monotonicity,[status(thm)],[275])).
% 2.20/2.35 tff(277,plain,
% 2.20/2.35 (~subgroup_member(multiply(identity, d))),
% 2.20/2.35 inference(modus_ponens,[status(thm)],[137, 276])).
% 2.20/2.35 tff(278,plain,
% 2.20/2.35 (~subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a)))),
% 2.20/2.35 inference(modus_ponens,[status(thm)],[277, 274])).
% 2.20/2.36 tff(279,plain,
% 2.20/2.36 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | (subgroup_member(inverse(a)) | subgroup_member(inverse(multiply(a, b))) | subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a))))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | subgroup_member(inverse(a)) | subgroup_member(inverse(multiply(a, b))) | subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a))))),
% 2.20/2.36 inference(rewrite,[status(thm)],[])).
% 2.20/2.36 tff(280,plain,
% 2.20/2.36 ((subgroup_member(inverse(multiply(a, b))) | subgroup_member(inverse(a)) | subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a)))) <=> (subgroup_member(inverse(a)) | subgroup_member(inverse(multiply(a, b))) | subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a))))),
% 2.20/2.36 inference(rewrite,[status(thm)],[])).
% 2.20/2.36 tff(281,plain,
% 2.20/2.36 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | (subgroup_member(inverse(multiply(a, b))) | subgroup_member(inverse(a)) | subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a))))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | (subgroup_member(inverse(a)) | subgroup_member(inverse(multiply(a, b))) | subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a)))))),
% 2.20/2.36 inference(monotonicity,[status(thm)],[280])).
% 2.20/2.36 tff(282,plain,
% 2.20/2.36 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | (subgroup_member(inverse(multiply(a, b))) | subgroup_member(inverse(a)) | subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a))))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | subgroup_member(inverse(a)) | subgroup_member(inverse(multiply(a, b))) | subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a))))),
% 2.20/2.36 inference(transitivity,[status(thm)],[281, 279])).
% 2.20/2.36 tff(283,plain,
% 2.20/2.36 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | (subgroup_member(inverse(multiply(a, b))) | subgroup_member(inverse(a)) | subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a))))),
% 2.20/2.36 inference(quant_inst,[status(thm)],[])).
% 2.20/2.36 tff(284,plain,
% 2.20/2.36 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | subgroup_member(inverse(a)) | subgroup_member(inverse(multiply(a, b))) | subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a)))),
% 2.20/2.36 inference(modus_ponens,[status(thm)],[283, 282])).
% 2.20/2.36 tff(285,plain,
% 2.20/2.36 (subgroup_member(inverse(multiply(a, b))) | subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a)))),
% 2.20/2.36 inference(unit_resolution,[status(thm)],[284, 187, 225])).
% 2.20/2.36 tff(286,plain,
% 2.20/2.36 (subgroup_member(element_in_O2(inverse(multiply(a, b)), inverse(a)))),
% 2.20/2.36 inference(unit_resolution,[status(thm)],[285, 104])).
% 2.20/2.36 tff(287,plain,
% 2.20/2.36 ($false),
% 2.20/2.36 inference(unit_resolution,[status(thm)],[286, 278])).
% 2.20/2.36 tff(288,plain,(subgroup_member(inverse(multiply(a, b)))), inference(lemma,lemma(discharge,[]))).
% 2.20/2.36 tff(289,plain,
% 2.20/2.36 ((~![A: $i] : (inverse(inverse(A)) = A)) | (inverse(inverse(multiply(a, b))) = multiply(a, b))),
% 2.20/2.36 inference(quant_inst,[status(thm)],[])).
% 2.20/2.36 tff(290,plain,
% 2.20/2.36 (inverse(inverse(multiply(a, b))) = multiply(a, b)),
% 2.20/2.36 inference(unit_resolution,[status(thm)],[289, 110])).
% 2.20/2.36 tff(291,plain,
% 2.20/2.36 (product(identity, inverse(inverse(multiply(a, b))), multiply(a, b)) <=> product(identity, multiply(a, b), multiply(a, b))),
% 2.20/2.36 inference(monotonicity,[status(thm)],[290])).
% 2.20/2.36 tff(292,plain,
% 2.20/2.36 (product(identity, multiply(a, b), multiply(a, b)) <=> product(identity, inverse(inverse(multiply(a, b))), multiply(a, b))),
% 2.20/2.36 inference(symmetry,[status(thm)],[291])).
% 2.20/2.36 tff(293,plain,
% 2.20/2.36 ((~![X: $i] : product(identity, X, X)) | product(identity, multiply(a, b), multiply(a, b))),
% 2.20/2.36 inference(quant_inst,[status(thm)],[])).
% 2.20/2.36 tff(294,plain,
% 2.20/2.36 (product(identity, multiply(a, b), multiply(a, b))),
% 2.20/2.36 inference(unit_resolution,[status(thm)],[293, 51])).
% 2.20/2.36 tff(295,plain,
% 2.20/2.36 (product(identity, inverse(inverse(multiply(a, b))), multiply(a, b))),
% 2.20/2.36 inference(modus_ponens,[status(thm)],[294, 292])).
% 2.20/2.36 tff(296,plain,
% 2.20/2.36 (subgroup_member(identity) <=> subgroup_member(identity)),
% 2.20/2.36 inference(rewrite,[status(thm)],[])).
% 2.20/2.36 tff(297,axiom,(subgroup_member(identity)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','identity_is_in_subgroup')).
% 2.20/2.36 tff(298,plain,
% 2.20/2.36 (subgroup_member(identity)),
% 2.20/2.36 inference(modus_ponens,[status(thm)],[297, 296])).
% 2.20/2.36 tff(299,plain,
% 2.20/2.36 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(multiply(a, b)) | (~subgroup_member(identity)) | (~subgroup_member(inverse(multiply(a, b)))) | (~product(identity, inverse(inverse(multiply(a, b))), multiply(a, b))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(multiply(a, b)) | (~subgroup_member(identity)) | (~subgroup_member(inverse(multiply(a, b)))) | (~product(identity, inverse(inverse(multiply(a, b))), multiply(a, b))))),
% 2.20/2.36 inference(rewrite,[status(thm)],[])).
% 2.20/2.36 tff(300,plain,
% 2.20/2.36 ((subgroup_member(multiply(a, b)) | (~product(identity, inverse(inverse(multiply(a, b))), multiply(a, b))) | (~subgroup_member(inverse(multiply(a, b)))) | (~subgroup_member(identity))) <=> (subgroup_member(multiply(a, b)) | (~subgroup_member(identity)) | (~subgroup_member(inverse(multiply(a, b)))) | (~product(identity, inverse(inverse(multiply(a, b))), multiply(a, b))))),
% 2.20/2.36 inference(rewrite,[status(thm)],[])).
% 2.20/2.36 tff(301,plain,
% 2.20/2.36 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(multiply(a, b)) | (~product(identity, inverse(inverse(multiply(a, b))), multiply(a, b))) | (~subgroup_member(inverse(multiply(a, b)))) | (~subgroup_member(identity)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(multiply(a, b)) | (~subgroup_member(identity)) | (~subgroup_member(inverse(multiply(a, b)))) | (~product(identity, inverse(inverse(multiply(a, b))), multiply(a, b)))))),
% 2.20/2.36 inference(monotonicity,[status(thm)],[300])).
% 2.20/2.36 tff(302,plain,
% 2.20/2.36 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(multiply(a, b)) | (~product(identity, inverse(inverse(multiply(a, b))), multiply(a, b))) | (~subgroup_member(inverse(multiply(a, b)))) | (~subgroup_member(identity)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(multiply(a, b)) | (~subgroup_member(identity)) | (~subgroup_member(inverse(multiply(a, b)))) | (~product(identity, inverse(inverse(multiply(a, b))), multiply(a, b))))),
% 2.20/2.36 inference(transitivity,[status(thm)],[301, 299])).
% 2.20/2.36 tff(303,plain,
% 2.20/2.36 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(multiply(a, b)) | (~product(identity, inverse(inverse(multiply(a, b))), multiply(a, b))) | (~subgroup_member(inverse(multiply(a, b)))) | (~subgroup_member(identity)))),
% 2.20/2.36 inference(quant_inst,[status(thm)],[])).
% 2.20/2.36 tff(304,plain,
% 2.20/2.36 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(multiply(a, b)) | (~subgroup_member(identity)) | (~subgroup_member(inverse(multiply(a, b)))) | (~product(identity, inverse(inverse(multiply(a, b))), multiply(a, b)))),
% 2.20/2.36 inference(modus_ponens,[status(thm)],[303, 302])).
% 2.20/2.36 tff(305,plain,
% 2.20/2.36 (subgroup_member(multiply(a, b)) | (~subgroup_member(inverse(multiply(a, b)))) | (~product(identity, inverse(inverse(multiply(a, b))), multiply(a, b)))),
% 2.20/2.36 inference(unit_resolution,[status(thm)],[304, 129, 298])).
% 2.20/2.36 tff(306,plain,
% 2.20/2.36 (subgroup_member(multiply(a, b)) | (~subgroup_member(inverse(multiply(a, b))))),
% 2.20/2.36 inference(unit_resolution,[status(thm)],[305, 295])).
% 2.20/2.36 tff(307,plain,
% 2.20/2.36 (subgroup_member(multiply(a, b))),
% 2.20/2.36 inference(unit_resolution,[status(thm)],[306, 288])).
% 2.20/2.36 tff(308,plain,
% 2.20/2.36 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~subgroup_member(multiply(a, b))) | (~subgroup_member(multiply(inverse(a), multiply(a, b)))) | (~product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(inverse(a)) | (~subgroup_member(multiply(a, b))) | (~subgroup_member(multiply(inverse(a), multiply(a, b)))) | (~product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a))))),
% 2.20/2.36 inference(rewrite,[status(thm)],[])).
% 2.20/2.36 tff(309,plain,
% 2.20/2.36 ((subgroup_member(inverse(a)) | (~product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a))) | (~subgroup_member(multiply(a, b))) | (~subgroup_member(multiply(inverse(a), multiply(a, b))))) <=> (subgroup_member(inverse(a)) | (~subgroup_member(multiply(a, b))) | (~subgroup_member(multiply(inverse(a), multiply(a, b)))) | (~product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a))))),
% 2.20/2.36 inference(rewrite,[status(thm)],[])).
% 2.20/2.36 tff(310,plain,
% 2.20/2.36 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a))) | (~subgroup_member(multiply(a, b))) | (~subgroup_member(multiply(inverse(a), multiply(a, b)))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~subgroup_member(multiply(a, b))) | (~subgroup_member(multiply(inverse(a), multiply(a, b)))) | (~product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a)))))),
% 2.20/2.36 inference(monotonicity,[status(thm)],[309])).
% 2.20/2.36 tff(311,plain,
% 2.20/2.36 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a))) | (~subgroup_member(multiply(a, b))) | (~subgroup_member(multiply(inverse(a), multiply(a, b)))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(inverse(a)) | (~subgroup_member(multiply(a, b))) | (~subgroup_member(multiply(inverse(a), multiply(a, b)))) | (~product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a))))),
% 2.20/2.36 inference(transitivity,[status(thm)],[310, 308])).
% 2.20/2.36 tff(312,plain,
% 2.20/2.36 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a))) | (~subgroup_member(multiply(a, b))) | (~subgroup_member(multiply(inverse(a), multiply(a, b)))))),
% 2.20/2.36 inference(quant_inst,[status(thm)],[])).
% 2.20/2.36 tff(313,plain,
% 2.20/2.36 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(inverse(a)) | (~subgroup_member(multiply(a, b))) | (~subgroup_member(multiply(inverse(a), multiply(a, b)))) | (~product(multiply(inverse(a), multiply(a, b)), inverse(multiply(a, b)), inverse(a)))),
% 2.20/2.36 inference(modus_ponens,[status(thm)],[312, 311])).
% 2.20/2.36 tff(314,plain,
% 2.20/2.36 ($false),
% 2.20/2.36 inference(unit_resolution,[status(thm)],[313, 129, 225, 307, 103, 44])).
% 2.20/2.36 % SZS output end Proof
% 2.20/2.36 % E exiting
%------------------------------------------------------------------------------