%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP033-3 : TPTP v9.0.0. Bugfixed v4.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n010.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 05:06:54 AM UTC 2025
% Result : Unsatisfiable 81.05s 81.23s
% Output : Proof 81.05s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : GRP033-3 : TPTP v9.0.0. Bugfixed v4.0.0.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n010.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Jun 19 15:29:56 EDT 2025
% 0.12/0.33 % CPUTime :
% 81.05/81.23 % SZS status Unsatisfiable
% 81.05/81.23 % SZS output start Proof
% 81.05/81.23 tff(product_type, type, (
% 81.05/81.23 product: ( $i * $i * $i ) > $o)).
% 81.05/81.23 tff(j_type, type, (
% 81.05/81.23 j: $i > $i)).
% 81.05/81.23 tff(inverse_type, type, (
% 81.05/81.23 inverse: $i > $i)).
% 81.05/81.23 tff(identity_type, type, (
% 81.05/81.23 identity: $i)).
% 81.05/81.23 tff(equalish_type, type, (
% 81.05/81.23 equalish: ( $i * $i ) > $o)).
% 81.05/81.23 tff(subgroup_member_type, type, (
% 81.05/81.23 subgroup_member: $i > $o)).
% 81.05/81.23 tff(a_type, type, (
% 81.05/81.23 a: $i)).
% 81.05/81.23 tff(1,plain,
% 81.05/81.23 (^[X: $i] : refl(product(X, identity, X) <=> product(X, identity, X))),
% 81.05/81.23 inference(bind,[status(th)],[])).
% 81.05/81.23 tff(2,plain,
% 81.05/81.23 (![X: $i] : product(X, identity, X) <=> ![X: $i] : product(X, identity, X)),
% 81.05/81.23 inference(quant_intro,[status(thm)],[1])).
% 81.05/81.23 tff(3,plain,
% 81.05/81.23 (![X: $i] : product(X, identity, X) <=> ![X: $i] : product(X, identity, X)),
% 81.05/81.23 inference(rewrite,[status(thm)],[])).
% 81.05/81.23 tff(4,axiom,(![X: $i] : product(X, identity, X)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','right_identity')).
% 81.05/81.23 tff(5,plain,
% 81.05/81.23 (![X: $i] : product(X, identity, X)),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[4, 3])).
% 81.05/81.23 tff(6,plain,(
% 81.05/81.23 ![X: $i] : product(X, identity, X)),
% 81.05/81.23 inference(skolemize,[status(sab)],[5])).
% 81.05/81.23 tff(7,plain,
% 81.05/81.23 (![X: $i] : product(X, identity, X)),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[6, 2])).
% 81.05/81.23 tff(8,plain,
% 81.05/81.23 ((~![X: $i] : product(X, identity, X)) | product(j(inverse(identity)), identity, j(inverse(identity)))),
% 81.05/81.23 inference(quant_inst,[status(thm)],[])).
% 81.05/81.23 tff(9,plain,
% 81.05/81.23 (product(j(inverse(identity)), identity, j(inverse(identity)))),
% 81.05/81.23 inference(unit_resolution,[status(thm)],[8, 7])).
% 81.05/81.23 tff(10,plain,
% 81.05/81.23 (^[X: $i] : refl(product(identity, X, X) <=> product(identity, X, X))),
% 81.05/81.23 inference(bind,[status(th)],[])).
% 81.05/81.23 tff(11,plain,
% 81.05/81.23 (![X: $i] : product(identity, X, X) <=> ![X: $i] : product(identity, X, X)),
% 81.05/81.23 inference(quant_intro,[status(thm)],[10])).
% 81.05/81.23 tff(12,plain,
% 81.05/81.23 (![X: $i] : product(identity, X, X) <=> ![X: $i] : product(identity, X, X)),
% 81.05/81.23 inference(rewrite,[status(thm)],[])).
% 81.05/81.23 tff(13,axiom,(![X: $i] : product(identity, X, X)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','left_identity')).
% 81.05/81.23 tff(14,plain,
% 81.05/81.23 (![X: $i] : product(identity, X, X)),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[13, 12])).
% 81.05/81.23 tff(15,plain,(
% 81.05/81.23 ![X: $i] : product(identity, X, X)),
% 81.05/81.23 inference(skolemize,[status(sab)],[14])).
% 81.05/81.23 tff(16,plain,
% 81.05/81.23 (![X: $i] : product(identity, X, X)),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[15, 11])).
% 81.05/81.23 tff(17,plain,
% 81.05/81.23 ((~![X: $i] : product(identity, X, X)) | product(identity, j(inverse(identity)), j(inverse(identity)))),
% 81.05/81.23 inference(quant_inst,[status(thm)],[])).
% 81.05/81.23 tff(18,plain,
% 81.05/81.23 (product(identity, j(inverse(identity)), j(inverse(identity)))),
% 81.05/81.23 inference(unit_resolution,[status(thm)],[17, 16])).
% 81.05/81.23 tff(19,plain,
% 81.05/81.23 ((~![X: $i] : product(X, identity, X)) | product(inverse(identity), identity, inverse(identity))),
% 81.05/81.23 inference(quant_inst,[status(thm)],[])).
% 81.05/81.23 tff(20,plain,
% 81.05/81.23 (product(inverse(identity), identity, inverse(identity))),
% 81.05/81.23 inference(unit_resolution,[status(thm)],[19, 7])).
% 81.05/81.23 tff(21,plain,
% 81.05/81.23 (^[X: $i] : refl(product(inverse(X), X, identity) <=> product(inverse(X), X, identity))),
% 81.05/81.23 inference(bind,[status(th)],[])).
% 81.05/81.23 tff(22,plain,
% 81.05/81.23 (![X: $i] : product(inverse(X), X, identity) <=> ![X: $i] : product(inverse(X), X, identity)),
% 81.05/81.23 inference(quant_intro,[status(thm)],[21])).
% 81.05/81.23 tff(23,plain,
% 81.05/81.23 (![X: $i] : product(inverse(X), X, identity) <=> ![X: $i] : product(inverse(X), X, identity)),
% 81.05/81.23 inference(rewrite,[status(thm)],[])).
% 81.05/81.23 tff(24,axiom,(![X: $i] : product(inverse(X), X, identity)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','left_inverse')).
% 81.05/81.23 tff(25,plain,
% 81.05/81.23 (![X: $i] : product(inverse(X), X, identity)),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[24, 23])).
% 81.05/81.23 tff(26,plain,(
% 81.05/81.23 ![X: $i] : product(inverse(X), X, identity)),
% 81.05/81.23 inference(skolemize,[status(sab)],[25])).
% 81.05/81.23 tff(27,plain,
% 81.05/81.23 (![X: $i] : product(inverse(X), X, identity)),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[26, 22])).
% 81.05/81.23 tff(28,plain,
% 81.05/81.23 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(identity), identity, identity)),
% 81.05/81.23 inference(quant_inst,[status(thm)],[])).
% 81.05/81.23 tff(29,plain,
% 81.05/81.23 (product(inverse(identity), identity, identity)),
% 81.05/81.23 inference(unit_resolution,[status(thm)],[28, 27])).
% 81.05/81.23 tff(30,plain,
% 81.05/81.23 (^[W: $i, Z: $i, Y: $i, X: $i] : refl((equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z))) <=> (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z))))),
% 81.05/81.23 inference(bind,[status(th)],[])).
% 81.05/81.23 tff(31,plain,
% 81.05/81.23 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 81.05/81.23 inference(quant_intro,[status(thm)],[30])).
% 81.05/81.23 tff(32,plain,
% 81.05/81.23 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 81.05/81.23 inference(rewrite,[status(thm)],[])).
% 81.05/81.23 tff(33,plain,
% 81.05/81.23 (^[W: $i, Z: $i, Y: $i, X: $i] : rewrite((((~product(X, Y, Z)) | (~product(X, Y, W))) | equalish(Z, W)) <=> (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z))))),
% 81.05/81.23 inference(bind,[status(th)],[])).
% 81.05/81.23 tff(34,plain,
% 81.05/81.23 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, Z)) | (~product(X, Y, W))) | equalish(Z, W)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 81.05/81.23 inference(quant_intro,[status(thm)],[33])).
% 81.05/81.23 tff(35,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, Z)) | (~product(X, Y, W))) | equalish(Z, W))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','total_function2')).
% 81.05/81.23 tff(36,plain,
% 81.05/81.23 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[35, 34])).
% 81.05/81.23 tff(37,plain,
% 81.05/81.23 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[36, 32])).
% 81.05/81.23 tff(38,plain,(
% 81.05/81.23 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 81.05/81.23 inference(skolemize,[status(sab)],[37])).
% 81.05/81.23 tff(39,plain,
% 81.05/81.23 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[38, 31])).
% 81.05/81.23 tff(40,plain,
% 81.05/81.23 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (equalish(identity, inverse(identity)) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | equalish(identity, inverse(identity)) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity))))),
% 81.05/81.23 inference(rewrite,[status(thm)],[])).
% 81.05/81.23 tff(41,plain,
% 81.05/81.23 ((equalish(identity, inverse(identity)) | (~product(inverse(identity), identity, inverse(identity))) | (~product(inverse(identity), identity, identity))) <=> (equalish(identity, inverse(identity)) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity))))),
% 81.05/81.23 inference(rewrite,[status(thm)],[])).
% 81.05/81.23 tff(42,plain,
% 81.05/81.23 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (equalish(identity, inverse(identity)) | (~product(inverse(identity), identity, inverse(identity))) | (~product(inverse(identity), identity, identity)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (equalish(identity, inverse(identity)) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity)))))),
% 81.05/81.23 inference(monotonicity,[status(thm)],[41])).
% 81.05/81.23 tff(43,plain,
% 81.05/81.23 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (equalish(identity, inverse(identity)) | (~product(inverse(identity), identity, inverse(identity))) | (~product(inverse(identity), identity, identity)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | equalish(identity, inverse(identity)) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity))))),
% 81.05/81.23 inference(transitivity,[status(thm)],[42, 40])).
% 81.05/81.23 tff(44,plain,
% 81.05/81.23 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (equalish(identity, inverse(identity)) | (~product(inverse(identity), identity, inverse(identity))) | (~product(inverse(identity), identity, identity)))),
% 81.05/81.23 inference(quant_inst,[status(thm)],[])).
% 81.05/81.23 tff(45,plain,
% 81.05/81.23 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(Z, W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | equalish(identity, inverse(identity)) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity)))),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[44, 43])).
% 81.05/81.23 tff(46,plain,
% 81.05/81.23 (equalish(identity, inverse(identity))),
% 81.05/81.23 inference(unit_resolution,[status(thm)],[45, 39, 29, 20])).
% 81.05/81.23 tff(47,plain,
% 81.05/81.23 ((~![X: $i] : product(identity, X, X)) | product(identity, identity, identity)),
% 81.05/81.23 inference(quant_inst,[status(thm)],[])).
% 81.05/81.23 tff(48,plain,
% 81.05/81.23 (product(identity, identity, identity)),
% 81.05/81.23 inference(unit_resolution,[status(thm)],[47, 16])).
% 81.05/81.23 tff(49,plain,
% 81.05/81.23 (^[W: $i, Z: $i, Y: $i, X: $i] : refl(((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X))) <=> ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X))))),
% 81.05/81.23 inference(bind,[status(th)],[])).
% 81.05/81.23 tff(50,plain,
% 81.05/81.23 (![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))),
% 81.05/81.23 inference(quant_intro,[status(thm)],[49])).
% 81.05/81.23 tff(51,plain,
% 81.05/81.23 (![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))),
% 81.05/81.23 inference(rewrite,[status(thm)],[])).
% 81.05/81.23 tff(52,plain,
% 81.05/81.23 (^[W: $i, Z: $i, Y: $i, X: $i] : rewrite((((~equalish(X, Y)) | (~product(W, Z, X))) | product(W, Z, Y)) <=> ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X))))),
% 81.05/81.23 inference(bind,[status(th)],[])).
% 81.05/81.23 tff(53,plain,
% 81.05/81.23 (![W: $i, Z: $i, Y: $i, X: $i] : (((~equalish(X, Y)) | (~product(W, Z, X))) | product(W, Z, Y)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))),
% 81.05/81.23 inference(quant_intro,[status(thm)],[52])).
% 81.05/81.23 tff(54,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~equalish(X, Y)) | (~product(W, Z, X))) | product(W, Z, Y))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_substitution3')).
% 81.05/81.23 tff(55,plain,
% 81.05/81.23 (![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[54, 53])).
% 81.05/81.23 tff(56,plain,
% 81.05/81.23 (![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[55, 51])).
% 81.05/81.23 tff(57,plain,(
% 81.05/81.23 ![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))),
% 81.05/81.23 inference(skolemize,[status(sab)],[56])).
% 81.05/81.23 tff(58,plain,
% 81.05/81.23 (![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[57, 50])).
% 81.05/81.23 tff(59,plain,
% 81.05/81.23 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))) | ((~product(identity, identity, identity)) | product(identity, identity, inverse(identity)) | (~equalish(identity, inverse(identity))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))) | (~product(identity, identity, identity)) | product(identity, identity, inverse(identity)) | (~equalish(identity, inverse(identity))))),
% 81.05/81.23 inference(rewrite,[status(thm)],[])).
% 81.05/81.23 tff(60,plain,
% 81.05/81.23 (((~equalish(identity, inverse(identity))) | product(identity, identity, inverse(identity)) | (~product(identity, identity, identity))) <=> ((~product(identity, identity, identity)) | product(identity, identity, inverse(identity)) | (~equalish(identity, inverse(identity))))),
% 81.05/81.23 inference(rewrite,[status(thm)],[])).
% 81.05/81.23 tff(61,plain,
% 81.05/81.23 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))) | ((~equalish(identity, inverse(identity))) | product(identity, identity, inverse(identity)) | (~product(identity, identity, identity)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))) | ((~product(identity, identity, identity)) | product(identity, identity, inverse(identity)) | (~equalish(identity, inverse(identity)))))),
% 81.05/81.23 inference(monotonicity,[status(thm)],[60])).
% 81.05/81.23 tff(62,plain,
% 81.05/81.23 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))) | ((~equalish(identity, inverse(identity))) | product(identity, identity, inverse(identity)) | (~product(identity, identity, identity)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))) | (~product(identity, identity, identity)) | product(identity, identity, inverse(identity)) | (~equalish(identity, inverse(identity))))),
% 81.05/81.23 inference(transitivity,[status(thm)],[61, 59])).
% 81.05/81.23 tff(63,plain,
% 81.05/81.23 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))) | ((~equalish(identity, inverse(identity))) | product(identity, identity, inverse(identity)) | (~product(identity, identity, identity)))),
% 81.05/81.23 inference(quant_inst,[status(thm)],[])).
% 81.05/81.23 tff(64,plain,
% 81.05/81.23 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | product(W, Z, Y) | (~product(W, Z, X)))) | (~product(identity, identity, identity)) | product(identity, identity, inverse(identity)) | (~equalish(identity, inverse(identity)))),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[63, 62])).
% 81.05/81.23 tff(65,plain,
% 81.05/81.23 (product(identity, identity, inverse(identity))),
% 81.05/81.23 inference(unit_resolution,[status(thm)],[64, 58, 48, 46])).
% 81.05/81.23 tff(66,assumption,(~product(inverse(identity), j(inverse(identity)), j(inverse(identity)))), introduced(assumption)).
% 81.05/81.23 tff(67,plain,
% 81.05/81.23 (^[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))))),
% 81.05/81.23 inference(bind,[status(th)],[])).
% 81.05/81.23 tff(68,plain,
% 81.05/81.23 (![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)))),
% 81.05/81.23 inference(quant_intro,[status(thm)],[67])).
% 81.05/81.23 tff(69,plain,
% 81.05/81.23 (![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)))),
% 81.05/81.23 inference(rewrite,[status(thm)],[])).
% 81.05/81.23 tff(70,plain,
% 81.05/81.23 (^[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)))))),
% 81.05/81.23 inference(bind,[status(th)],[])).
% 81.05/81.23 tff(71,plain,
% 81.05/81.23 (![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)))),
% 81.05/81.23 inference(quant_intro,[status(thm)],[70])).
% 81.05/81.23 tff(72,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/theBenchmark.p','associativity2')).
% 81.05/81.23 tff(73,plain,
% 81.05/81.23 (![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)))),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[72, 71])).
% 81.05/81.23 tff(74,plain,
% 81.05/81.23 (![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)))),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[73, 69])).
% 81.05/81.23 tff(75,plain,(
% 81.05/81.23 ![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)))),
% 81.05/81.23 inference(skolemize,[status(sab)],[74])).
% 81.05/81.23 tff(76,plain,
% 81.05/81.23 (![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)))),
% 81.05/81.23 inference(modus_ponens,[status(thm)],[75, 68])).
% 81.05/81.23 tff(77,plain,
% 81.05/81.23 (((~![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(identity), j(inverse(identity)), j(inverse(identity))) | (~product(identity, identity, inverse(identity))) | (~product(identity, j(inverse(identity)), j(inverse(identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | product(inverse(identity), j(inverse(identity)), j(inverse(identity))) | (~product(identity, identity, inverse(identity))) | (~product(identity, j(inverse(identity)), j(inverse(identity)))))),
% 81.05/81.23 inference(rewrite,[status(thm)],[])).
% 81.05/81.23 tff(78,plain,
% 81.05/81.23 ((product(inverse(identity), j(inverse(identity)), j(inverse(identity))) | (~product(identity, j(inverse(identity)), j(inverse(identity)))) | (~product(identity, identity, inverse(identity))) | (~product(identity, j(inverse(identity)), j(inverse(identity))))) <=> (product(inverse(identity), j(inverse(identity)), j(inverse(identity))) | (~product(identity, identity, inverse(identity))) | (~product(identity, j(inverse(identity)), j(inverse(identity)))))),
% 81.05/81.23 inference(rewrite,[status(thm)],[])).
% 81.05/81.23 tff(79,plain,
% 81.05/81.23 (((~![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(identity), j(inverse(identity)), j(inverse(identity))) | (~product(identity, j(inverse(identity)), j(inverse(identity)))) | (~product(identity, identity, inverse(identity))) | (~product(identity, j(inverse(identity)), j(inverse(identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(inverse(identity), j(inverse(identity)), j(inverse(identity))) | (~product(identity, identity, inverse(identity))) | (~product(identity, j(inverse(identity)), j(inverse(identity))))))),
% 81.05/81.23 inference(monotonicity,[status(thm)],[78])).
% 81.05/81.23 tff(80,plain,
% 81.05/81.23 (((~![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(identity), j(inverse(identity)), j(inverse(identity))) | (~product(identity, j(inverse(identity)), j(inverse(identity)))) | (~product(identity, identity, inverse(identity))) | (~product(identity, j(inverse(identity)), j(inverse(identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | product(inverse(identity), j(inverse(identity)), j(inverse(identity))) | (~product(identity, identity, inverse(identity))) | (~product(identity, j(inverse(identity)), j(inverse(identity)))))),
% 81.05/81.23 inference(transitivity,[status(thm)],[79, 77])).
% 81.05/81.23 tff(81,plain,
% 81.05/81.23 ((~![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(identity), j(inverse(identity)), j(inverse(identity))) | (~product(identity, j(inverse(identity)), j(inverse(identity)))) | (~product(identity, identity, inverse(identity))) | (~product(identity, j(inverse(identity)), j(inverse(identity)))))),
% 81.05/81.24 inference(quant_inst,[status(thm)],[])).
% 81.05/81.24 tff(82,plain,
% 81.05/81.24 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | product(inverse(identity), j(inverse(identity)), j(inverse(identity))) | (~product(identity, identity, inverse(identity))) | (~product(identity, j(inverse(identity)), j(inverse(identity))))),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[81, 80])).
% 81.05/81.24 tff(83,plain,
% 81.05/81.24 ($false),
% 81.05/81.24 inference(unit_resolution,[status(thm)],[82, 76, 66, 65, 18])).
% 81.05/81.24 tff(84,plain,(product(inverse(identity), j(inverse(identity)), j(inverse(identity)))), inference(lemma,lemma(discharge,[]))).
% 81.05/81.24 tff(85,plain,
% 81.05/81.24 (^[X: $i] : refl(product(X, inverse(X), identity) <=> product(X, inverse(X), identity))),
% 81.05/81.24 inference(bind,[status(th)],[])).
% 81.05/81.24 tff(86,plain,
% 81.05/81.24 (![X: $i] : product(X, inverse(X), identity) <=> ![X: $i] : product(X, inverse(X), identity)),
% 81.05/81.24 inference(quant_intro,[status(thm)],[85])).
% 81.05/81.24 tff(87,plain,
% 81.05/81.24 (![X: $i] : product(X, inverse(X), identity) <=> ![X: $i] : product(X, inverse(X), identity)),
% 81.05/81.24 inference(rewrite,[status(thm)],[])).
% 81.05/81.24 tff(88,axiom,(![X: $i] : product(X, inverse(X), identity)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','right_inverse')).
% 81.05/81.24 tff(89,plain,
% 81.05/81.24 (![X: $i] : product(X, inverse(X), identity)),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[88, 87])).
% 81.05/81.24 tff(90,plain,(
% 81.05/81.24 ![X: $i] : product(X, inverse(X), identity)),
% 81.05/81.24 inference(skolemize,[status(sab)],[89])).
% 81.05/81.24 tff(91,plain,
% 81.05/81.24 (![X: $i] : product(X, inverse(X), identity)),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[90, 86])).
% 81.05/81.24 tff(92,plain,
% 81.05/81.24 ((~![X: $i] : product(X, inverse(X), identity)) | product(a, inverse(a), identity)),
% 81.05/81.24 inference(quant_inst,[status(thm)],[])).
% 81.05/81.24 tff(93,plain,
% 81.05/81.24 (product(a, inverse(a), identity)),
% 81.05/81.24 inference(unit_resolution,[status(thm)],[92, 91])).
% 81.05/81.24 tff(94,plain,
% 81.05/81.24 (subgroup_member(a) <=> subgroup_member(a)),
% 81.05/81.24 inference(rewrite,[status(thm)],[])).
% 81.05/81.24 tff(95,axiom,(subgroup_member(a)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','a_is_in_subgroup')).
% 81.05/81.24 tff(96,plain,
% 81.05/81.24 (subgroup_member(a)),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[95, 94])).
% 81.05/81.24 tff(97,plain,
% 81.05/81.24 (^[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))))),
% 81.05/81.24 inference(bind,[status(th)],[])).
% 81.05/81.24 tff(98,plain,
% 81.05/81.24 (![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)))),
% 81.05/81.24 inference(quant_intro,[status(thm)],[97])).
% 81.05/81.24 tff(99,plain,
% 81.05/81.24 (![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)))),
% 81.05/81.24 inference(rewrite,[status(thm)],[])).
% 81.05/81.24 tff(100,plain,
% 81.05/81.24 (^[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)))))),
% 81.05/81.24 inference(bind,[status(th)],[])).
% 81.05/81.24 tff(101,plain,
% 81.05/81.24 (![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)))),
% 81.05/81.24 inference(quant_intro,[status(thm)],[100])).
% 81.05/81.24 tff(102,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/theBenchmark.p','closure_of_product_and_inverse')).
% 81.05/81.24 tff(103,plain,
% 81.05/81.24 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[102, 101])).
% 81.05/81.24 tff(104,plain,
% 81.05/81.24 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[103, 99])).
% 81.05/81.24 tff(105,plain,(
% 81.05/81.24 ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 81.05/81.24 inference(skolemize,[status(sab)],[104])).
% 81.05/81.24 tff(106,plain,
% 81.05/81.24 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[105, 98])).
% 81.05/81.24 tff(107,plain,
% 81.05/81.24 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | ((~subgroup_member(a)) | subgroup_member(identity) | (~product(a, inverse(a), identity)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(a)) | subgroup_member(identity) | (~product(a, inverse(a), identity)))),
% 81.05/81.24 inference(rewrite,[status(thm)],[])).
% 81.05/81.24 tff(108,plain,
% 81.05/81.24 ((subgroup_member(identity) | (~product(a, inverse(a), identity)) | (~subgroup_member(a)) | (~subgroup_member(a))) <=> ((~subgroup_member(a)) | subgroup_member(identity) | (~product(a, inverse(a), identity)))),
% 81.05/81.24 inference(rewrite,[status(thm)],[])).
% 81.05/81.24 tff(109,plain,
% 81.05/81.24 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(identity) | (~product(a, inverse(a), identity)) | (~subgroup_member(a)) | (~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(a)) | subgroup_member(identity) | (~product(a, inverse(a), identity))))),
% 81.05/81.24 inference(monotonicity,[status(thm)],[108])).
% 81.05/81.24 tff(110,plain,
% 81.05/81.24 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(identity) | (~product(a, inverse(a), identity)) | (~subgroup_member(a)) | (~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(a)) | subgroup_member(identity) | (~product(a, inverse(a), identity)))),
% 81.05/81.24 inference(transitivity,[status(thm)],[109, 107])).
% 81.05/81.24 tff(111,plain,
% 81.05/81.24 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(identity) | (~product(a, inverse(a), identity)) | (~subgroup_member(a)) | (~subgroup_member(a)))),
% 81.05/81.24 inference(quant_inst,[status(thm)],[])).
% 81.05/81.24 tff(112,plain,
% 81.05/81.24 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(a)) | subgroup_member(identity) | (~product(a, inverse(a), identity))),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[111, 110])).
% 81.05/81.24 tff(113,plain,
% 81.05/81.24 (subgroup_member(identity)),
% 81.05/81.24 inference(unit_resolution,[status(thm)],[112, 106, 96, 93])).
% 81.05/81.24 tff(114,plain,
% 81.05/81.24 (^[B: $i, A: $i] : refl(((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A))) <=> ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A))))),
% 81.05/81.24 inference(bind,[status(th)],[])).
% 81.05/81.24 tff(115,plain,
% 81.05/81.24 (![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A))) <=> ![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))),
% 81.05/81.24 inference(quant_intro,[status(thm)],[114])).
% 81.05/81.24 tff(116,plain,
% 81.05/81.24 (![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A))) <=> ![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))),
% 81.05/81.24 inference(rewrite,[status(thm)],[])).
% 81.05/81.24 tff(117,plain,
% 81.05/81.24 (^[B: $i, A: $i] : trans(monotonicity(rewrite(((~equalish(A, B)) | (~subgroup_member(A))) <=> ((~equalish(A, B)) | (~subgroup_member(A)))), ((((~equalish(A, B)) | (~subgroup_member(A))) | subgroup_member(B)) <=> (((~equalish(A, B)) | (~subgroup_member(A))) | subgroup_member(B)))), rewrite((((~equalish(A, B)) | (~subgroup_member(A))) | subgroup_member(B)) <=> ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))), ((((~equalish(A, B)) | (~subgroup_member(A))) | subgroup_member(B)) <=> ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))))),
% 81.05/81.24 inference(bind,[status(th)],[])).
% 81.05/81.24 tff(118,plain,
% 81.05/81.24 (![B: $i, A: $i] : (((~equalish(A, B)) | (~subgroup_member(A))) | subgroup_member(B)) <=> ![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))),
% 81.05/81.24 inference(quant_intro,[status(thm)],[117])).
% 81.05/81.24 tff(119,axiom,(![B: $i, A: $i] : (((~equalish(A, B)) | (~subgroup_member(A))) | subgroup_member(B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','subgroup_member_substitution')).
% 81.05/81.24 tff(120,plain,
% 81.05/81.24 (![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[119, 118])).
% 81.05/81.24 tff(121,plain,
% 81.05/81.24 (![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[120, 116])).
% 81.05/81.24 tff(122,plain,(
% 81.05/81.24 ![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))),
% 81.05/81.24 inference(skolemize,[status(sab)],[121])).
% 81.05/81.24 tff(123,plain,
% 81.05/81.24 (![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[122, 115])).
% 81.05/81.24 tff(124,plain,
% 81.05/81.24 (((~![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))) | ((~subgroup_member(identity)) | subgroup_member(inverse(identity)) | (~equalish(identity, inverse(identity))))) <=> ((~![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))) | (~subgroup_member(identity)) | subgroup_member(inverse(identity)) | (~equalish(identity, inverse(identity))))),
% 81.05/81.24 inference(rewrite,[status(thm)],[])).
% 81.05/81.24 tff(125,plain,
% 81.05/81.24 (((~equalish(identity, inverse(identity))) | subgroup_member(inverse(identity)) | (~subgroup_member(identity))) <=> ((~subgroup_member(identity)) | subgroup_member(inverse(identity)) | (~equalish(identity, inverse(identity))))),
% 81.05/81.24 inference(rewrite,[status(thm)],[])).
% 81.05/81.24 tff(126,plain,
% 81.05/81.24 (((~![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))) | ((~equalish(identity, inverse(identity))) | subgroup_member(inverse(identity)) | (~subgroup_member(identity)))) <=> ((~![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))) | ((~subgroup_member(identity)) | subgroup_member(inverse(identity)) | (~equalish(identity, inverse(identity)))))),
% 81.05/81.24 inference(monotonicity,[status(thm)],[125])).
% 81.05/81.24 tff(127,plain,
% 81.05/81.24 (((~![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))) | ((~equalish(identity, inverse(identity))) | subgroup_member(inverse(identity)) | (~subgroup_member(identity)))) <=> ((~![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))) | (~subgroup_member(identity)) | subgroup_member(inverse(identity)) | (~equalish(identity, inverse(identity))))),
% 81.05/81.24 inference(transitivity,[status(thm)],[126, 124])).
% 81.05/81.24 tff(128,plain,
% 81.05/81.24 ((~![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))) | ((~equalish(identity, inverse(identity))) | subgroup_member(inverse(identity)) | (~subgroup_member(identity)))),
% 81.05/81.24 inference(quant_inst,[status(thm)],[])).
% 81.05/81.24 tff(129,plain,
% 81.05/81.24 ((~![B: $i, A: $i] : ((~equalish(A, B)) | subgroup_member(B) | (~subgroup_member(A)))) | (~subgroup_member(identity)) | subgroup_member(inverse(identity)) | (~equalish(identity, inverse(identity)))),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[128, 127])).
% 81.05/81.24 tff(130,plain,
% 81.05/81.24 (subgroup_member(inverse(identity))),
% 81.05/81.24 inference(unit_resolution,[status(thm)],[129, 123, 113, 46])).
% 81.05/81.24 tff(131,plain,
% 81.05/81.24 (^[A: $i] : refl(((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A)))) <=> ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A)))))),
% 81.05/81.24 inference(bind,[status(th)],[])).
% 81.05/81.24 tff(132,plain,
% 81.05/81.24 (![A: $i] : ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A)))) <=> ![A: $i] : ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A))))),
% 81.05/81.24 inference(quant_intro,[status(thm)],[131])).
% 81.05/81.24 tff(133,plain,
% 81.05/81.24 (![A: $i] : ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A)))) <=> ![A: $i] : ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A))))),
% 81.05/81.24 inference(rewrite,[status(thm)],[])).
% 81.05/81.24 tff(134,plain,
% 81.05/81.24 (^[A: $i] : rewrite((((~product(j(A), A, j(A))) | (~product(A, j(A), j(A)))) | (~subgroup_member(A))) <=> ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A)))))),
% 81.05/81.24 inference(bind,[status(th)],[])).
% 81.05/81.24 tff(135,plain,
% 81.05/81.24 (![A: $i] : (((~product(j(A), A, j(A))) | (~product(A, j(A), j(A)))) | (~subgroup_member(A))) <=> ![A: $i] : ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A))))),
% 81.05/81.24 inference(quant_intro,[status(thm)],[134])).
% 81.05/81.24 tff(136,axiom,(![A: $i] : (((~product(j(A), A, j(A))) | (~product(A, j(A), j(A)))) | (~subgroup_member(A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_subgr2')).
% 81.05/81.24 tff(137,plain,
% 81.05/81.24 (![A: $i] : ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A))))),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[136, 135])).
% 81.05/81.24 tff(138,plain,
% 81.05/81.24 (![A: $i] : ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A))))),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[137, 133])).
% 81.05/81.24 tff(139,plain,(
% 81.05/81.24 ![A: $i] : ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A))))),
% 81.05/81.24 inference(skolemize,[status(sab)],[138])).
% 81.05/81.24 tff(140,plain,
% 81.05/81.24 (![A: $i] : ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A))))),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[139, 132])).
% 81.05/81.24 tff(141,plain,
% 81.05/81.24 (((~![A: $i] : ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A))))) | ((~subgroup_member(inverse(identity))) | (~product(inverse(identity), j(inverse(identity)), j(inverse(identity)))) | (~product(j(inverse(identity)), inverse(identity), j(inverse(identity)))))) <=> ((~![A: $i] : ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A))))) | (~subgroup_member(inverse(identity))) | (~product(inverse(identity), j(inverse(identity)), j(inverse(identity)))) | (~product(j(inverse(identity)), inverse(identity), j(inverse(identity)))))),
% 81.05/81.24 inference(rewrite,[status(thm)],[])).
% 81.05/81.24 tff(142,plain,
% 81.05/81.24 ((~![A: $i] : ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A))))) | ((~subgroup_member(inverse(identity))) | (~product(inverse(identity), j(inverse(identity)), j(inverse(identity)))) | (~product(j(inverse(identity)), inverse(identity), j(inverse(identity)))))),
% 81.05/81.24 inference(quant_inst,[status(thm)],[])).
% 81.05/81.24 tff(143,plain,
% 81.05/81.24 ((~![A: $i] : ((~subgroup_member(A)) | (~product(A, j(A), j(A))) | (~product(j(A), A, j(A))))) | (~subgroup_member(inverse(identity))) | (~product(inverse(identity), j(inverse(identity)), j(inverse(identity)))) | (~product(j(inverse(identity)), inverse(identity), j(inverse(identity))))),
% 81.05/81.24 inference(modus_ponens,[status(thm)],[142, 141])).
% 81.05/81.24 tff(144,plain,
% 81.05/81.24 ((~product(inverse(identity), j(inverse(identity)), j(inverse(identity)))) | (~product(j(inverse(identity)), inverse(identity), j(inverse(identity))))),
% 81.05/81.24 inference(unit_resolution,[status(thm)],[143, 140, 130])).
% 81.05/81.24 tff(145,plain,
% 81.05/81.24 (~product(j(inverse(identity)), inverse(identity), j(inverse(identity)))),
% 81.05/81.24 inference(unit_resolution,[status(thm)],[144, 84])).
% 81.05/81.24 tff(146,plain,
% 81.05/81.24 ((~![X: $i] : product(X, inverse(X), identity)) | product(identity, inverse(identity), identity)),
% 81.05/81.24 inference(quant_inst,[status(thm)],[])).
% 81.05/81.24 tff(147,plain,
% 81.05/81.24 (product(identity, inverse(identity), identity)),
% 81.05/81.24 inference(unit_resolution,[status(thm)],[146, 91])).
% 81.05/81.24 tff(148,plain,
% 81.05/81.24 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(identity, inverse(identity), identity)) | product(j(inverse(identity)), inverse(identity), j(inverse(identity))) | (~product(j(inverse(identity)), identity, j(inverse(identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(identity, inverse(identity), identity)) | product(j(inverse(identity)), inverse(identity), j(inverse(identity))) | (~product(j(inverse(identity)), identity, j(inverse(identity)))))),
% 81.05/81.24 inference(rewrite,[status(thm)],[])).
% 81.05/81.24 tff(149,plain,
% 81.05/81.24 ((product(j(inverse(identity)), inverse(identity), j(inverse(identity))) | (~product(identity, inverse(identity), identity)) | (~product(j(inverse(identity)), identity, j(inverse(identity)))) | (~product(j(inverse(identity)), identity, j(inverse(identity))))) <=> ((~product(identity, inverse(identity), identity)) | product(j(inverse(identity)), inverse(identity), j(inverse(identity))) | (~product(j(inverse(identity)), identity, j(inverse(identity)))))),
% 81.05/81.24 inference(rewrite,[status(thm)],[])).
% 81.05/81.24 tff(150,plain,
% 81.05/81.24 (((~![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(j(inverse(identity)), inverse(identity), j(inverse(identity))) | (~product(identity, inverse(identity), identity)) | (~product(j(inverse(identity)), identity, j(inverse(identity)))) | (~product(j(inverse(identity)), identity, j(inverse(identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(identity, inverse(identity), identity)) | product(j(inverse(identity)), inverse(identity), j(inverse(identity))) | (~product(j(inverse(identity)), identity, j(inverse(identity))))))),
% 81.05/81.24 inference(monotonicity,[status(thm)],[149])).
% 81.05/81.24 tff(151,plain,
% 81.05/81.24 (((~![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(j(inverse(identity)), inverse(identity), j(inverse(identity))) | (~product(identity, inverse(identity), identity)) | (~product(j(inverse(identity)), identity, j(inverse(identity)))) | (~product(j(inverse(identity)), identity, j(inverse(identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(identity, inverse(identity), identity)) | product(j(inverse(identity)), inverse(identity), j(inverse(identity))) | (~product(j(inverse(identity)), identity, j(inverse(identity)))))),
% 81.05/81.24 inference(transitivity,[status(thm)],[150, 148])).
% 81.05/81.24 tff(152,plain,
% 81.05/81.24 ((~![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(j(inverse(identity)), inverse(identity), j(inverse(identity))) | (~product(identity, inverse(identity), identity)) | (~product(j(inverse(identity)), identity, j(inverse(identity)))) | (~product(j(inverse(identity)), identity, j(inverse(identity)))))),
% 81.05/81.24 inference(quant_inst,[status(thm)],[])).
% 81.05/81.26 tff(153,plain,
% 81.05/81.26 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(identity, inverse(identity), identity)) | product(j(inverse(identity)), inverse(identity), j(inverse(identity))) | (~product(j(inverse(identity)), identity, j(inverse(identity))))),
% 81.05/81.26 inference(modus_ponens,[status(thm)],[152, 151])).
% 81.05/81.26 tff(154,plain,
% 81.05/81.26 ($false),
% 81.05/81.26 inference(unit_resolution,[status(thm)],[153, 76, 147, 145, 9])).
% 81.05/81.26 % SZS output end Proof
% 81.12/81.26 % E exiting
%------------------------------------------------------------------------------