%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP133-2.003 : TPTP v9.0.0. Released v1.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:07:13 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.38s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GRP133-2.003 : TPTP v9.0.0. Released v1.2.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.32 % Computer : n009.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 300
% 0.12/0.32 % DateTime : Thu Jun 19 15:43:48 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.38 % SZS status Unsatisfiable
% 0.19/0.38 % SZS output start Proof
% 0.19/0.38 tff(product_type, type, (
% 0.19/0.38 product: ( $i * $i * $i ) > $o)).
% 0.19/0.38 tff(e_1_type, type, (
% 0.19/0.38 e_1: $i)).
% 0.19/0.38 tff(e_3_type, type, (
% 0.19/0.38 e_3: $i)).
% 0.19/0.38 tff(e_2_type, type, (
% 0.19/0.38 e_2: $i)).
% 0.19/0.38 tff(equalish_type, type, (
% 0.19/0.38 equalish: ( $i * $i ) > $o)).
% 0.19/0.38 tff(next_type, type, (
% 0.19/0.38 next: ( $i * $i ) > $o)).
% 0.19/0.38 tff(greater_type, type, (
% 0.19/0.38 greater: ( $i * $i ) > $o)).
% 0.19/0.38 tff(group_element_type, type, (
% 0.19/0.38 group_element: $i > $o)).
% 0.19/0.38 tff(1,assumption,(product(e_3, e_1, e_2)), introduced(assumption)).
% 0.19/0.38 tff(2,assumption,(product(e_1, e_1, e_1)), introduced(assumption)).
% 0.19/0.38 tff(3,assumption,(product(e_1, e_3, e_1)), introduced(assumption)).
% 0.19/0.38 tff(4,plain,
% 0.19/0.38 (^[W: $i, Z: $i, Y: $i, X: $i] : refl((equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y))) <=> (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(5,plain,
% 0.19/0.38 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[4])).
% 0.19/0.38 tff(6,plain,
% 0.19/0.38 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(7,plain,
% 0.19/0.38 (^[W: $i, Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~product(X, W, Y)) | (~product(X, Z, Y))) <=> ((~product(X, Z, Y)) | (~product(X, W, Y)))), ((((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z)) <=> (((~product(X, Z, Y)) | (~product(X, W, Y))) | equalish(W, Z)))), rewrite((((~product(X, Z, Y)) | (~product(X, W, Y))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))), ((((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(8,plain,
% 0.19/0.38 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[7])).
% 0.19/0.38 tff(9,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_right_cancellation')).
% 0.19/0.38 tff(10,plain,
% 0.19/0.38 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[9, 8])).
% 0.19/0.38 tff(11,plain,
% 0.19/0.38 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[10, 6])).
% 0.19/0.38 tff(12,plain,(
% 0.19/0.38 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[11])).
% 0.19/0.38 tff(13,plain,
% 0.19/0.38 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[12, 5])).
% 0.19/0.38 tff(14,plain,
% 0.19/0.38 ((~equalish(e_3, e_1)) <=> (~equalish(e_3, e_1))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(15,axiom,(~equalish(e_3, e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_3_is_not_e_1')).
% 0.19/0.38 tff(16,plain,
% 0.19/0.38 (~equalish(e_3, e_1)),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.19/0.38 tff(17,plain,
% 0.19/0.38 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_3, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_3, e_1)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(18,plain,
% 0.19/0.38 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_3, e_1)))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(19,plain,
% 0.19/0.39 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_3, e_1))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.19/0.39 tff(20,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[19, 16, 13, 2, 3])).
% 0.19/0.39 tff(21,plain,((~product(e_1, e_3, e_1)) | (~product(e_1, e_1, e_1))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.39 tff(22,plain,
% 0.19/0.39 (~product(e_1, e_3, e_1)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[21, 2])).
% 0.19/0.39 tff(23,assumption,(product(e_3, e_1, e_3)), introduced(assumption)).
% 0.19/0.39 tff(24,assumption,(product(e_3, e_3, e_1)), introduced(assumption)).
% 0.19/0.39 tff(25,plain,
% 0.19/0.39 (^[Y: $i, X: $i, X1: $i] : refl(((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y))) <=> ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(26,plain,
% 0.19/0.39 (![Y: $i, X: $i, X1: $i] : ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y))) <=> ![Y: $i, X: $i, X1: $i] : ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[25])).
% 0.19/0.39 tff(27,plain,
% 0.19/0.39 (![Y: $i, X: $i, X1: $i] : ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y))) <=> ![Y: $i, X: $i, X1: $i] : ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(28,plain,
% 0.19/0.39 (^[Y: $i, X: $i, X1: $i] : trans(monotonicity(rewrite(((~product(X, e_1, Y)) | (~next(X, X1))) <=> ((~next(X, X1)) | (~product(X, e_1, Y)))), ((((~product(X, e_1, Y)) | (~next(X, X1))) | (~greater(Y, X1))) <=> (((~next(X, X1)) | (~product(X, e_1, Y))) | (~greater(Y, X1))))), rewrite((((~next(X, X1)) | (~product(X, e_1, Y))) | (~greater(Y, X1))) <=> ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y)))), ((((~product(X, e_1, Y)) | (~next(X, X1))) | (~greater(Y, X1))) <=> ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(29,plain,
% 0.19/0.39 (![Y: $i, X: $i, X1: $i] : (((~product(X, e_1, Y)) | (~next(X, X1))) | (~greater(Y, X1))) <=> ![Y: $i, X: $i, X1: $i] : ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[28])).
% 0.19/0.39 tff(30,axiom,(![Y: $i, X: $i, X1: $i] : (((~product(X, e_1, Y)) | (~next(X, X1))) | (~greater(Y, X1)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','no_redundancy')).
% 0.19/0.39 tff(31,plain,
% 0.19/0.39 (![Y: $i, X: $i, X1: $i] : ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[30, 29])).
% 0.19/0.39 tff(32,plain,
% 0.19/0.39 (![Y: $i, X: $i, X1: $i] : ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[31, 27])).
% 0.19/0.39 tff(33,plain,(
% 0.19/0.39 ![Y: $i, X: $i, X1: $i] : ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[32])).
% 0.19/0.39 tff(34,plain,
% 0.19/0.39 (![Y: $i, X: $i, X1: $i] : ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[33, 26])).
% 0.19/0.39 tff(35,plain,
% 0.19/0.39 (greater(e_3, e_2) <=> greater(e_3, e_2)),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(36,axiom,(greater(e_3, e_2)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_3_greater_e_2')).
% 0.19/0.39 tff(37,plain,
% 0.19/0.39 (greater(e_3, e_2)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.19/0.39 tff(38,plain,
% 0.19/0.39 (next(e_1, e_2) <=> next(e_1, e_2)),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(39,axiom,(next(e_1, e_2)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_1_then_e_2')).
% 0.19/0.39 tff(40,plain,
% 0.19/0.39 (next(e_1, e_2)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[39, 38])).
% 0.19/0.39 tff(41,plain,
% 0.19/0.39 (((~![Y: $i, X: $i, X1: $i] : ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y)))) | ((~greater(e_3, e_2)) | (~next(e_1, e_2)) | (~product(e_1, e_1, e_3)))) <=> ((~![Y: $i, X: $i, X1: $i] : ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y)))) | (~greater(e_3, e_2)) | (~next(e_1, e_2)) | (~product(e_1, e_1, e_3)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(42,plain,
% 0.19/0.39 ((~![Y: $i, X: $i, X1: $i] : ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y)))) | ((~greater(e_3, e_2)) | (~next(e_1, e_2)) | (~product(e_1, e_1, e_3)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(43,plain,
% 0.19/0.39 ((~![Y: $i, X: $i, X1: $i] : ((~greater(Y, X1)) | (~next(X, X1)) | (~product(X, e_1, Y)))) | (~greater(e_3, e_2)) | (~next(e_1, e_2)) | (~product(e_1, e_1, e_3))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.19/0.39 tff(44,plain,
% 0.19/0.39 (~product(e_1, e_1, e_3)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[43, 40, 37, 34])).
% 0.19/0.39 tff(45,plain,
% 0.19/0.39 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : refl(((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1))) <=> ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(46,plain,
% 0.19/0.39 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[45])).
% 0.19/0.39 tff(47,plain,
% 0.19/0.39 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(48,plain,
% 0.19/0.39 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : rewrite((((~product(X, Y, Z1)) | (~product(Y, X, Z2))) | product(Z1, Z2, X)) <=> ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(49,plain,
% 0.19/0.39 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (((~product(X, Y, Z1)) | (~product(Y, X, Z2))) | product(Z1, Z2, X)) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[48])).
% 0.19/0.39 tff(50,axiom,(![Y: $i, Z1: $i, X: $i, Z2: $i] : (((~product(X, Y, Z1)) | (~product(Y, X, Z2))) | product(Z1, Z2, X))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','qg3')).
% 0.19/0.39 tff(51,plain,
% 0.19/0.39 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.19/0.39 tff(52,plain,
% 0.19/0.39 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[51, 47])).
% 0.19/0.39 tff(53,plain,(
% 0.19/0.39 ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[52])).
% 0.19/0.39 tff(54,plain,
% 0.19/0.39 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[53, 46])).
% 0.19/0.39 tff(55,plain,
% 0.19/0.39 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (product(e_1, e_1, e_3) | (~product(e_3, e_3, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_1, e_1, e_3) | (~product(e_3, e_3, e_1)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(56,plain,
% 0.19/0.39 (((~product(e_3, e_3, e_1)) | product(e_1, e_1, e_3) | (~product(e_3, e_3, e_1))) <=> (product(e_1, e_1, e_3) | (~product(e_3, e_3, e_1)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(57,plain,
% 0.19/0.39 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_3, e_3, e_1)) | product(e_1, e_1, e_3) | (~product(e_3, e_3, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (product(e_1, e_1, e_3) | (~product(e_3, e_3, e_1))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[56])).
% 0.19/0.39 tff(58,plain,
% 0.19/0.39 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_3, e_3, e_1)) | product(e_1, e_1, e_3) | (~product(e_3, e_3, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_1, e_1, e_3) | (~product(e_3, e_3, e_1)))),
% 0.19/0.39 inference(transitivity,[status(thm)],[57, 55])).
% 0.19/0.39 tff(59,plain,
% 0.19/0.39 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_3, e_3, e_1)) | product(e_1, e_1, e_3) | (~product(e_3, e_3, e_1)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(60,plain,
% 0.19/0.39 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_1, e_1, e_3) | (~product(e_3, e_3, e_1))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.19/0.39 tff(61,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[60, 54, 44, 24])).
% 0.19/0.39 tff(62,plain,(~product(e_3, e_3, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.39 tff(63,plain,
% 0.19/0.39 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (product(e_3, e_3, e_1) | (~product(e_3, e_1, e_3)) | (~product(e_1, e_3, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_3, e_3, e_1) | (~product(e_3, e_1, e_3)) | (~product(e_1, e_3, e_3)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(64,plain,
% 0.19/0.39 (((~product(e_3, e_1, e_3)) | product(e_3, e_3, e_1) | (~product(e_1, e_3, e_3))) <=> (product(e_3, e_3, e_1) | (~product(e_3, e_1, e_3)) | (~product(e_1, e_3, e_3)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(65,plain,
% 0.19/0.39 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_3, e_1, e_3)) | product(e_3, e_3, e_1) | (~product(e_1, e_3, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (product(e_3, e_3, e_1) | (~product(e_3, e_1, e_3)) | (~product(e_1, e_3, e_3))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[64])).
% 0.19/0.39 tff(66,plain,
% 0.19/0.39 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_3, e_1, e_3)) | product(e_3, e_3, e_1) | (~product(e_1, e_3, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_3, e_3, e_1) | (~product(e_3, e_1, e_3)) | (~product(e_1, e_3, e_3)))),
% 0.19/0.39 inference(transitivity,[status(thm)],[65, 63])).
% 0.19/0.39 tff(67,plain,
% 0.19/0.39 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_3, e_1, e_3)) | product(e_3, e_3, e_1) | (~product(e_1, e_3, e_3)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(68,plain,
% 0.19/0.39 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_3, e_3, e_1) | (~product(e_3, e_1, e_3)) | (~product(e_1, e_3, e_3))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[67, 66])).
% 0.19/0.39 tff(69,plain,
% 0.19/0.39 (~product(e_1, e_3, e_3)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[68, 54, 62, 23])).
% 0.19/0.39 tff(70,assumption,(~product(e_1, e_3, e_1)), introduced(assumption)).
% 0.19/0.39 tff(71,plain,
% 0.19/0.39 (^[Y: $i, X: $i] : refl((product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))) <=> (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(72,plain,
% 0.19/0.39 (![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))) <=> ![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[71])).
% 0.19/0.39 tff(73,plain,
% 0.19/0.39 (![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))) <=> ![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(74,plain,
% 0.19/0.39 (^[Y: $i, X: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) <=> (product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))), (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) <=> ((product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))) | product(X, Y, e_2)))), rewrite(((product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))) | product(X, Y, e_2)) <=> (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))), (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) <=> (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3)) <=> ((product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))) | product(X, Y, e_3)))), rewrite(((product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))) | product(X, Y, e_3)) <=> (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3)) <=> (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(75,plain,
% 0.19/0.40 (![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3)) <=> ![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[74])).
% 0.19/0.40 tff(76,axiom,(![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_total_function1')).
% 0.19/0.40 tff(77,plain,
% 0.19/0.40 (![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.19/0.40 tff(78,plain,
% 0.19/0.40 (![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[77, 73])).
% 0.19/0.40 tff(79,plain,(
% 0.19/0.40 ![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.40 inference(skolemize,[status(sab)],[78])).
% 0.19/0.40 tff(80,plain,
% 0.19/0.40 (![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[79, 72])).
% 0.19/0.40 tff(81,plain,
% 0.19/0.40 (group_element(e_3) <=> group_element(e_3)),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(82,axiom,(group_element(e_3)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_3')).
% 0.19/0.40 tff(83,plain,
% 0.19/0.40 (group_element(e_3)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.19/0.40 tff(84,plain,
% 0.19/0.40 (group_element(e_1) <=> group_element(e_1)),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(85,axiom,(group_element(e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_1')).
% 0.19/0.40 tff(86,plain,
% 0.19/0.40 (group_element(e_1)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.19/0.40 tff(87,plain,
% 0.19/0.40 (((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | ((~group_element(e_1)) | (~group_element(e_3)) | product(e_1, e_3, e_3) | product(e_1, e_3, e_1) | product(e_1, e_3, e_2))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (~group_element(e_1)) | (~group_element(e_3)) | product(e_1, e_3, e_3) | product(e_1, e_3, e_1) | product(e_1, e_3, e_2))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(88,plain,
% 0.19/0.40 ((product(e_1, e_3, e_3) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_1))) <=> ((~group_element(e_1)) | (~group_element(e_3)) | product(e_1, e_3, e_3) | product(e_1, e_3, e_1) | product(e_1, e_3, e_2))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(89,plain,
% 0.19/0.40 (((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_3, e_3) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | ((~group_element(e_1)) | (~group_element(e_3)) | product(e_1, e_3, e_3) | product(e_1, e_3, e_1) | product(e_1, e_3, e_2)))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[88])).
% 0.19/0.40 tff(90,plain,
% 0.19/0.40 (((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_3, e_3) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (~group_element(e_1)) | (~group_element(e_3)) | product(e_1, e_3, e_3) | product(e_1, e_3, e_1) | product(e_1, e_3, e_2))),
% 0.19/0.40 inference(transitivity,[status(thm)],[89, 87])).
% 0.19/0.40 tff(91,plain,
% 0.19/0.40 ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_3, e_3) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_1)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(92,plain,
% 0.19/0.40 ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (~group_element(e_1)) | (~group_element(e_3)) | product(e_1, e_3, e_3) | product(e_1, e_3, e_1) | product(e_1, e_3, e_2)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.19/0.40 tff(93,plain,
% 0.19/0.40 (product(e_1, e_3, e_3) | product(e_1, e_3, e_2)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[92, 86, 83, 80, 70])).
% 0.19/0.40 tff(94,plain,
% 0.19/0.40 (product(e_1, e_3, e_2)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[93, 69])).
% 0.19/0.40 tff(95,assumption,(product(e_1, e_3, e_2)), introduced(assumption)).
% 0.19/0.40 tff(96,assumption,(product(e_3, e_3, e_2)), introduced(assumption)).
% 0.19/0.40 tff(97,plain,
% 0.19/0.40 (^[W: $i, Z: $i, Y: $i, X: $i] : refl((equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X))) <=> (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(98,plain,
% 0.19/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[97])).
% 0.19/0.40 tff(99,plain,
% 0.19/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(100,plain,
% 0.19/0.40 (^[W: $i, Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~product(W, Y, X)) | (~product(Z, Y, X))) <=> ((~product(Z, Y, X)) | (~product(W, Y, X)))), ((((~product(W, Y, X)) | (~product(Z, Y, X))) | equalish(W, Z)) <=> (((~product(Z, Y, X)) | (~product(W, Y, X))) | equalish(W, Z)))), rewrite((((~product(Z, Y, X)) | (~product(W, Y, X))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))), ((((~product(W, Y, X)) | (~product(Z, Y, X))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(101,plain,
% 0.19/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product(W, Y, X)) | (~product(Z, Y, X))) | equalish(W, Z)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[100])).
% 0.19/0.40 tff(102,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(W, Y, X)) | (~product(Z, Y, X))) | equalish(W, Z))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_left_cancellation')).
% 0.19/0.40 tff(103,plain,
% 0.19/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.19/0.40 tff(104,plain,
% 0.19/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[103, 99])).
% 0.19/0.40 tff(105,plain,(
% 0.19/0.40 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.19/0.40 inference(skolemize,[status(sab)],[104])).
% 0.19/0.40 tff(106,plain,
% 0.19/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[105, 98])).
% 0.19/0.40 tff(107,plain,
% 0.19/0.40 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_3, e_3, e_2)) | (~product(e_1, e_3, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_3, e_3, e_2)) | (~product(e_1, e_3, e_2)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(108,plain,
% 0.19/0.40 ((equalish(e_3, e_1) | (~product(e_1, e_3, e_2)) | (~product(e_3, e_3, e_2))) <=> (equalish(e_3, e_1) | (~product(e_3, e_3, e_2)) | (~product(e_1, e_3, e_2)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(109,plain,
% 0.19/0.40 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_3, e_2)) | (~product(e_3, e_3, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_3, e_3, e_2)) | (~product(e_1, e_3, e_2))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[108])).
% 0.19/0.40 tff(110,plain,
% 0.19/0.40 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_3, e_2)) | (~product(e_3, e_3, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_3, e_3, e_2)) | (~product(e_1, e_3, e_2)))),
% 0.19/0.40 inference(transitivity,[status(thm)],[109, 107])).
% 0.19/0.40 tff(111,plain,
% 0.19/0.40 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_3, e_2)) | (~product(e_3, e_3, e_2)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(112,plain,
% 0.19/0.40 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_3, e_3, e_2)) | (~product(e_1, e_3, e_2))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[111, 110])).
% 0.19/0.40 tff(113,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[112, 16, 106, 96, 95])).
% 0.19/0.40 tff(114,plain,((~product(e_3, e_3, e_2)) | (~product(e_1, e_3, e_2))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.40 tff(115,plain,
% 0.19/0.40 (~product(e_3, e_3, e_2)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[114, 94])).
% 0.19/0.40 tff(116,plain,
% 0.19/0.40 (((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_3, e_3) | product(e_3, e_3, e_2) | product(e_3, e_3, e_1) | (~group_element(e_3)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_3, e_3) | product(e_3, e_3, e_2) | product(e_3, e_3, e_1) | (~group_element(e_3)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(117,plain,
% 0.19/0.40 ((product(e_3, e_3, e_3) | product(e_3, e_3, e_2) | product(e_3, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_3))) <=> (product(e_3, e_3, e_3) | product(e_3, e_3, e_2) | product(e_3, e_3, e_1) | (~group_element(e_3)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(118,plain,
% 0.19/0.40 (((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_3, e_3) | product(e_3, e_3, e_2) | product(e_3, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_3)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_3, e_3) | product(e_3, e_3, e_2) | product(e_3, e_3, e_1) | (~group_element(e_3))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[117])).
% 0.19/0.41 tff(119,plain,
% 0.19/0.41 (((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_3, e_3) | product(e_3, e_3, e_2) | product(e_3, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_3)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_3, e_3) | product(e_3, e_3, e_2) | product(e_3, e_3, e_1) | (~group_element(e_3)))),
% 0.19/0.41 inference(transitivity,[status(thm)],[118, 116])).
% 0.19/0.41 tff(120,plain,
% 0.19/0.41 ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_3, e_3) | product(e_3, e_3, e_2) | product(e_3, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_3)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(121,plain,
% 0.19/0.41 ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_3, e_3) | product(e_3, e_3, e_2) | product(e_3, e_3, e_1) | (~group_element(e_3))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[120, 119])).
% 0.19/0.41 tff(122,plain,
% 0.19/0.41 (product(e_3, e_3, e_3) | product(e_3, e_3, e_2) | product(e_3, e_3, e_1)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[121, 83, 80])).
% 0.19/0.41 tff(123,plain,
% 0.19/0.41 (product(e_3, e_3, e_3) | product(e_3, e_3, e_2)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[122, 62])).
% 0.19/0.41 tff(124,plain,
% 0.19/0.41 (product(e_3, e_3, e_3)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[123, 115])).
% 0.19/0.41 tff(125,plain,
% 0.19/0.41 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_3, e_1, e_3)) | (~product(e_3, e_3, e_3)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_1) | (~product(e_3, e_1, e_3)) | (~product(e_3, e_3, e_3)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(126,plain,
% 0.19/0.41 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_3, e_1, e_3)) | (~product(e_3, e_3, e_3)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(127,plain,
% 0.19/0.41 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_1) | (~product(e_3, e_1, e_3)) | (~product(e_3, e_3, e_3))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[126, 125])).
% 0.19/0.41 tff(128,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[127, 16, 13, 124, 23])).
% 0.19/0.41 tff(129,plain,((~product(e_3, e_1, e_3)) | product(e_1, e_3, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(130,plain,
% 0.19/0.41 (~product(e_3, e_1, e_3)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[129, 22])).
% 0.19/0.41 tff(131,assumption,(product(e_3, e_1, e_1)), introduced(assumption)).
% 0.19/0.41 tff(132,plain,
% 0.19/0.41 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_3, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_3, e_1, e_1)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(133,plain,
% 0.19/0.41 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_3, e_1, e_1)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(134,plain,
% 0.19/0.41 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_3, e_1, e_1))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[133, 132])).
% 0.19/0.41 tff(135,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[134, 16, 106, 2, 131])).
% 0.19/0.41 tff(136,plain,((~product(e_3, e_1, e_1)) | (~product(e_1, e_1, e_1))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(137,plain,
% 0.19/0.41 (~product(e_3, e_1, e_1)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[136, 2])).
% 0.19/0.41 tff(138,plain,
% 0.19/0.41 (((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(139,plain,
% 0.19/0.41 ((product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_3))) <=> ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(140,plain,
% 0.19/0.41 (((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_3)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1)))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[139])).
% 0.19/0.41 tff(141,plain,
% 0.19/0.41 (((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_3)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1))),
% 0.19/0.41 inference(transitivity,[status(thm)],[140, 138])).
% 0.19/0.41 tff(142,plain,
% 0.19/0.41 ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_3)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(143,plain,
% 0.19/0.41 ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1)),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[142, 141])).
% 0.19/0.41 tff(144,plain,
% 0.19/0.41 (product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[143, 86, 83, 80])).
% 0.19/0.41 tff(145,plain,
% 0.19/0.41 (product(e_3, e_1, e_2)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[144, 137, 130])).
% 0.19/0.41 tff(146,plain,
% 0.19/0.41 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_3, e_3, e_2)) | (~product(e_3, e_1, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_1) | (~product(e_3, e_3, e_2)) | (~product(e_3, e_1, e_2)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(147,plain,
% 0.19/0.41 ((equalish(e_3, e_1) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_3, e_2))) <=> (equalish(e_3, e_1) | (~product(e_3, e_3, e_2)) | (~product(e_3, e_1, e_2)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(148,plain,
% 0.19/0.41 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_3, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_3, e_3, e_2)) | (~product(e_3, e_1, e_2))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[147])).
% 0.19/0.41 tff(149,plain,
% 0.19/0.41 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_3, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_1) | (~product(e_3, e_3, e_2)) | (~product(e_3, e_1, e_2)))),
% 0.19/0.41 inference(transitivity,[status(thm)],[148, 146])).
% 0.19/0.41 tff(150,plain,
% 0.19/0.41 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_3, e_2)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(151,plain,
% 0.19/0.41 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_1) | (~product(e_3, e_3, e_2)) | (~product(e_3, e_1, e_2))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[150, 149])).
% 0.19/0.41 tff(152,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[151, 16, 13, 96, 1])).
% 0.19/0.41 tff(153,plain,((~product(e_3, e_3, e_2)) | (~product(e_3, e_1, e_2))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(154,plain,
% 0.19/0.41 (~product(e_3, e_3, e_2)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[153, 145])).
% 0.19/0.41 tff(155,plain,
% 0.19/0.41 (product(e_3, e_3, e_3)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[123, 154])).
% 0.19/0.41 tff(156,assumption,(product(e_1, e_3, e_3)), introduced(assumption)).
% 0.19/0.41 tff(157,assumption,(product(e_3, e_3, e_3)), introduced(assumption)).
% 0.19/0.41 tff(158,plain,
% 0.19/0.41 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_3, e_3)) | (~product(e_3, e_3, e_3)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_1, e_3, e_3)) | (~product(e_3, e_3, e_3)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(159,plain,
% 0.19/0.41 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_3, e_3)) | (~product(e_3, e_3, e_3)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(160,plain,
% 0.19/0.41 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_1, e_3, e_3)) | (~product(e_3, e_3, e_3))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[159, 158])).
% 0.19/0.41 tff(161,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[160, 16, 106, 157, 156])).
% 0.19/0.41 tff(162,plain,((~product(e_3, e_3, e_3)) | (~product(e_1, e_3, e_3))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(163,plain,
% 0.19/0.41 (~product(e_1, e_3, e_3)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[162, 155])).
% 0.19/0.41 tff(164,plain,
% 0.19/0.41 (product(e_1, e_3, e_3) | product(e_1, e_3, e_2)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[92, 86, 83, 80, 22])).
% 0.19/0.41 tff(165,plain,
% 0.19/0.41 (product(e_1, e_3, e_2)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[164, 163])).
% 0.19/0.41 tff(166,assumption,(product(e_2, e_2, e_3)), introduced(assumption)).
% 0.19/0.41 tff(167,assumption,(~product(e_3, e_3, e_2)), introduced(assumption)).
% 0.19/0.41 tff(168,plain,
% 0.19/0.41 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (product(e_3, e_3, e_2) | (~product(e_2, e_2, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_3, e_3, e_2) | (~product(e_2, e_2, e_3)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(169,plain,
% 0.19/0.41 (((~product(e_2, e_2, e_3)) | product(e_3, e_3, e_2) | (~product(e_2, e_2, e_3))) <=> (product(e_3, e_3, e_2) | (~product(e_2, e_2, e_3)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(170,plain,
% 0.19/0.41 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_2, e_2, e_3)) | product(e_3, e_3, e_2) | (~product(e_2, e_2, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (product(e_3, e_3, e_2) | (~product(e_2, e_2, e_3))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[169])).
% 0.19/0.42 tff(171,plain,
% 0.19/0.42 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_2, e_2, e_3)) | product(e_3, e_3, e_2) | (~product(e_2, e_2, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_3, e_3, e_2) | (~product(e_2, e_2, e_3)))),
% 0.19/0.42 inference(transitivity,[status(thm)],[170, 168])).
% 0.19/0.42 tff(172,plain,
% 0.19/0.42 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_2, e_2, e_3)) | product(e_3, e_3, e_2) | (~product(e_2, e_2, e_3)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(173,plain,
% 0.19/0.42 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_3, e_3, e_2) | (~product(e_2, e_2, e_3))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[172, 171])).
% 0.19/0.42 tff(174,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[173, 54, 167, 166])).
% 0.19/0.42 tff(175,plain,((~product(e_2, e_2, e_3)) | product(e_3, e_3, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.42 tff(176,plain,
% 0.19/0.42 (~product(e_2, e_2, e_3)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[175, 154])).
% 0.19/0.42 tff(177,plain,
% 0.19/0.42 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (product(e_2, e_2, e_3) | (~product(e_1, e_3, e_2)) | (~product(e_3, e_1, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_2, e_2, e_3) | (~product(e_1, e_3, e_2)) | (~product(e_3, e_1, e_2)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(178,plain,
% 0.19/0.42 (((~product(e_1, e_3, e_2)) | product(e_2, e_2, e_3) | (~product(e_3, e_1, e_2))) <=> (product(e_2, e_2, e_3) | (~product(e_1, e_3, e_2)) | (~product(e_3, e_1, e_2)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(179,plain,
% 0.19/0.42 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_1, e_3, e_2)) | product(e_2, e_2, e_3) | (~product(e_3, e_1, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (product(e_2, e_2, e_3) | (~product(e_1, e_3, e_2)) | (~product(e_3, e_1, e_2))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[178])).
% 0.19/0.42 tff(180,plain,
% 0.19/0.42 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_1, e_3, e_2)) | product(e_2, e_2, e_3) | (~product(e_3, e_1, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_2, e_2, e_3) | (~product(e_1, e_3, e_2)) | (~product(e_3, e_1, e_2)))),
% 0.19/0.42 inference(transitivity,[status(thm)],[179, 177])).
% 0.19/0.42 tff(181,plain,
% 0.19/0.42 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_1, e_3, e_2)) | product(e_2, e_2, e_3) | (~product(e_3, e_1, e_2)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(182,plain,
% 0.19/0.42 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_2, e_2, e_3) | (~product(e_1, e_3, e_2)) | (~product(e_3, e_1, e_2))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[181, 180])).
% 0.19/0.42 tff(183,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[182, 54, 145, 176, 165])).
% 0.19/0.42 tff(184,plain,(~product(e_1, e_1, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.42 tff(185,plain,
% 0.19/0.42 (((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_1, e_3) | product(e_1, e_1, e_2) | product(e_1, e_1, e_1) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_1, e_1, e_3) | product(e_1, e_1, e_2) | product(e_1, e_1, e_1) | (~group_element(e_1)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(186,plain,
% 0.19/0.42 ((product(e_1, e_1, e_3) | product(e_1, e_1, e_2) | product(e_1, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_1))) <=> (product(e_1, e_1, e_3) | product(e_1, e_1, e_2) | product(e_1, e_1, e_1) | (~group_element(e_1)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(187,plain,
% 0.19/0.42 (((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_1, e_3) | product(e_1, e_1, e_2) | product(e_1, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_1, e_3) | product(e_1, e_1, e_2) | product(e_1, e_1, e_1) | (~group_element(e_1))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[186])).
% 0.19/0.42 tff(188,plain,
% 0.19/0.42 (((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_1, e_3) | product(e_1, e_1, e_2) | product(e_1, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_1, e_1, e_3) | product(e_1, e_1, e_2) | product(e_1, e_1, e_1) | (~group_element(e_1)))),
% 0.19/0.42 inference(transitivity,[status(thm)],[187, 185])).
% 0.19/0.42 tff(189,plain,
% 0.19/0.42 ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_1, e_3) | product(e_1, e_1, e_2) | product(e_1, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_1)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(190,plain,
% 0.19/0.42 ((~![Y: $i, X: $i] : (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_1, e_1, e_3) | product(e_1, e_1, e_2) | product(e_1, e_1, e_1) | (~group_element(e_1))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[189, 188])).
% 0.19/0.42 tff(191,plain,
% 0.19/0.42 (product(e_1, e_1, e_2) | product(e_1, e_1, e_1)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[190, 86, 80, 44])).
% 0.19/0.42 tff(192,plain,
% 0.19/0.42 (product(e_1, e_1, e_2)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[191, 184])).
% 0.19/0.42 tff(193,plain,
% 0.19/0.42 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_3, e_1, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_3, e_1, e_2)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(194,plain,
% 0.19/0.42 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_3, e_1, e_2)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(195,plain,
% 0.19/0.42 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_3, e_1, e_2))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[194, 193])).
% 0.19/0.42 tff(196,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[195, 16, 106, 192, 1])).
% 0.19/0.42 tff(197,plain,(~product(e_3, e_1, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.42 tff(198,plain,
% 0.19/0.42 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_1, e_1, e_2)) | product(e_2, e_2, e_1))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (~product(e_1, e_1, e_2)) | product(e_2, e_2, e_1))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(199,plain,
% 0.19/0.42 (((~product(e_1, e_1, e_2)) | product(e_2, e_2, e_1) | (~product(e_1, e_1, e_2))) <=> ((~product(e_1, e_1, e_2)) | product(e_2, e_2, e_1))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(200,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_1, e_1, e_2)) | product(e_2, e_2, e_1) | (~product(e_1, e_1, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_1, e_1, e_2)) | product(e_2, e_2, e_1)))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[199])).
% 0.19/0.43 tff(201,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_1, e_1, e_2)) | product(e_2, e_2, e_1) | (~product(e_1, e_1, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (~product(e_1, e_1, e_2)) | product(e_2, e_2, e_1))),
% 0.19/0.43 inference(transitivity,[status(thm)],[200, 198])).
% 0.19/0.43 tff(202,plain,
% 0.19/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_1, e_1, e_2)) | product(e_2, e_2, e_1) | (~product(e_1, e_1, e_2)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(203,plain,
% 0.19/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (~product(e_1, e_1, e_2)) | product(e_2, e_2, e_1)),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[202, 201])).
% 0.19/0.43 tff(204,plain,
% 0.19/0.43 ((~product(e_1, e_1, e_2)) | product(e_2, e_2, e_1)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[203, 54])).
% 0.19/0.43 tff(205,plain,
% 0.19/0.43 (product(e_2, e_2, e_1)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[204, 192])).
% 0.19/0.43 tff(206,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (product(e_3, e_1, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_3, e_1, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_1)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(207,plain,
% 0.19/0.43 (((~product(e_2, e_2, e_1)) | product(e_3, e_1, e_2) | (~product(e_2, e_2, e_3))) <=> (product(e_3, e_1, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_1)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(208,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_2, e_2, e_1)) | product(e_3, e_1, e_2) | (~product(e_2, e_2, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (product(e_3, e_1, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_1))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[207])).
% 0.19/0.43 tff(209,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_2, e_2, e_1)) | product(e_3, e_1, e_2) | (~product(e_2, e_2, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_3, e_1, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_1)))),
% 0.19/0.43 inference(transitivity,[status(thm)],[208, 206])).
% 0.19/0.43 tff(210,plain,
% 0.19/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_2, e_2, e_1)) | product(e_3, e_1, e_2) | (~product(e_2, e_2, e_3)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(211,plain,
% 0.19/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_3, e_1, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_1))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[210, 209])).
% 0.19/0.43 tff(212,plain,
% 0.19/0.43 (~product(e_2, e_2, e_3)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[211, 54, 205, 197])).
% 0.19/0.43 tff(213,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (product(e_2, e_2, e_3) | (~product(e_3, e_3, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_2, e_2, e_3) | (~product(e_3, e_3, e_2)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(214,plain,
% 0.19/0.43 (((~product(e_3, e_3, e_2)) | product(e_2, e_2, e_3) | (~product(e_3, e_3, e_2))) <=> (product(e_2, e_2, e_3) | (~product(e_3, e_3, e_2)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(215,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_3, e_3, e_2)) | product(e_2, e_2, e_3) | (~product(e_3, e_3, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (product(e_2, e_2, e_3) | (~product(e_3, e_3, e_2))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[214])).
% 0.19/0.43 tff(216,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_3, e_3, e_2)) | product(e_2, e_2, e_3) | (~product(e_3, e_3, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_2, e_2, e_3) | (~product(e_3, e_3, e_2)))),
% 0.19/0.43 inference(transitivity,[status(thm)],[215, 213])).
% 0.19/0.43 tff(217,plain,
% 0.19/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_3, e_3, e_2)) | product(e_2, e_2, e_3) | (~product(e_3, e_3, e_2)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(218,plain,
% 0.19/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_2, e_2, e_3) | (~product(e_3, e_3, e_2))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[217, 216])).
% 0.19/0.43 tff(219,plain,
% 0.19/0.43 (product(e_2, e_2, e_3) | (~product(e_3, e_3, e_2))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[218, 54])).
% 0.19/0.43 tff(220,plain,
% 0.19/0.43 (~product(e_3, e_3, e_2)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[219, 212])).
% 0.19/0.43 tff(221,plain,
% 0.19/0.43 (product(e_3, e_3, e_3)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[123, 220])).
% 0.19/0.43 tff(222,plain,
% 0.19/0.43 (~product(e_1, e_3, e_3)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[162, 221])).
% 0.19/0.43 tff(223,plain,
% 0.19/0.43 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_3, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_3, e_2)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(224,plain,
% 0.19/0.43 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_3, e_2)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(225,plain,
% 0.19/0.43 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_3, e_2))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[224, 223])).
% 0.19/0.43 tff(226,plain,
% 0.19/0.43 ($false),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[225, 16, 13, 192, 95])).
% 0.19/0.43 tff(227,plain,(~product(e_1, e_3, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.43 tff(228,plain,
% 0.19/0.43 (product(e_1, e_3, e_3) | product(e_1, e_3, e_1)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[92, 86, 83, 80, 227])).
% 0.19/0.43 tff(229,plain,
% 0.19/0.43 (product(e_1, e_3, e_1)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[228, 222])).
% 0.19/0.43 tff(230,plain,
% 0.19/0.43 ((~equalish(e_1, e_3)) <=> (~equalish(e_1, e_3))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(231,axiom,(~equalish(e_1, e_3)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_1_is_not_e_3')).
% 0.19/0.43 tff(232,plain,
% 0.19/0.43 (~equalish(e_1, e_3)),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[231, 230])).
% 0.19/0.43 tff(233,plain,
% 0.19/0.43 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_1, e_3) | (~product(e_3, e_1, e_3)) | (~product(e_3, e_3, e_3)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_1, e_3) | (~product(e_3, e_1, e_3)) | (~product(e_3, e_3, e_3)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(234,plain,
% 0.19/0.43 ((equalish(e_1, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_1, e_3))) <=> (equalish(e_1, e_3) | (~product(e_3, e_1, e_3)) | (~product(e_3, e_3, e_3)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(235,plain,
% 0.19/0.43 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_1, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_1, e_3)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_1, e_3) | (~product(e_3, e_1, e_3)) | (~product(e_3, e_3, e_3))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[234])).
% 0.19/0.43 tff(236,plain,
% 0.19/0.43 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_1, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_1, e_3)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_1, e_3) | (~product(e_3, e_1, e_3)) | (~product(e_3, e_3, e_3)))),
% 0.19/0.43 inference(transitivity,[status(thm)],[235, 233])).
% 0.19/0.43 tff(237,plain,
% 0.19/0.43 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_1, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_1, e_3)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(238,plain,
% 0.19/0.43 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_1, e_3) | (~product(e_3, e_1, e_3)) | (~product(e_3, e_3, e_3))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[237, 236])).
% 0.19/0.43 tff(239,plain,
% 0.19/0.43 ($false),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[238, 232, 13, 157, 23])).
% 0.19/0.43 tff(240,plain,((~product(e_3, e_1, e_3)) | (~product(e_3, e_3, e_3))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.43 tff(241,plain,
% 0.19/0.43 (~product(e_3, e_1, e_3)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[240, 221])).
% 0.19/0.43 tff(242,plain,
% 0.19/0.43 (product(e_3, e_1, e_1)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[144, 241, 197])).
% 0.19/0.43 tff(243,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (product(e_1, e_1, e_3) | (~product(e_3, e_1, e_1)) | (~product(e_1, e_3, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_1, e_1, e_3) | (~product(e_3, e_1, e_1)) | (~product(e_1, e_3, e_1)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(244,plain,
% 0.19/0.43 (((~product(e_1, e_3, e_1)) | product(e_1, e_1, e_3) | (~product(e_3, e_1, e_1))) <=> (product(e_1, e_1, e_3) | (~product(e_3, e_1, e_1)) | (~product(e_1, e_3, e_1)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(245,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_1, e_3, e_1)) | product(e_1, e_1, e_3) | (~product(e_3, e_1, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | (product(e_1, e_1, e_3) | (~product(e_3, e_1, e_1)) | (~product(e_1, e_3, e_1))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[244])).
% 0.19/0.43 tff(246,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_1, e_3, e_1)) | product(e_1, e_1, e_3) | (~product(e_3, e_1, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_1, e_1, e_3) | (~product(e_3, e_1, e_1)) | (~product(e_1, e_3, e_1)))),
% 0.19/0.43 inference(transitivity,[status(thm)],[245, 243])).
% 0.19/0.43 tff(247,plain,
% 0.19/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | ((~product(e_1, e_3, e_1)) | product(e_1, e_1, e_3) | (~product(e_3, e_1, e_1)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(248,plain,
% 0.19/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Y, X, Z2)) | product(Z1, Z2, X) | (~product(X, Y, Z1)))) | product(e_1, e_1, e_3) | (~product(e_3, e_1, e_1)) | (~product(e_1, e_3, e_1))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[247, 246])).
% 0.19/0.44 tff(249,plain,
% 0.19/0.44 ($false),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[248, 54, 44, 242, 229])).
% 0.19/0.44 % SZS output end Proof
% 0.19/0.44 % E exiting
%------------------------------------------------------------------------------