%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP124-4.004 : TPTP v9.0.0. Bugfixed v1.2.1.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n014.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 05:07:04 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.41s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : GRP124-4.004 : TPTP v9.0.0. Bugfixed v1.2.1.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n014.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 15:41:54 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.41 % SZS status Unsatisfiable
% 0.19/0.41 % SZS output start Proof
% 0.19/0.41 tff(product_type, type, (
% 0.19/0.41 product: ( $i * $i * $i ) > $o)).
% 0.19/0.41 tff(e_2_type, type, (
% 0.19/0.41 e_2: $i)).
% 0.19/0.41 tff(e_4_type, type, (
% 0.19/0.41 e_4: $i)).
% 0.19/0.41 tff(e_3_type, type, (
% 0.19/0.41 e_3: $i)).
% 0.19/0.41 tff(e_1_type, type, (
% 0.19/0.41 e_1: $i)).
% 0.19/0.41 tff(equalish_type, type, (
% 0.19/0.41 equalish: ( $i * $i ) > $o)).
% 0.19/0.41 tff(group_element_type, type, (
% 0.19/0.41 group_element: $i > $o)).
% 0.19/0.41 tff(1,assumption,(~product(e_2, e_2, e_2)), introduced(assumption)).
% 0.19/0.41 tff(2,plain,
% 0.19/0.41 (^[X: $i] : refl(product(X, X, X) <=> product(X, X, X))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(3,plain,
% 0.19/0.41 (![X: $i] : product(X, X, X) <=> ![X: $i] : product(X, X, X)),
% 0.19/0.41 inference(quant_intro,[status(thm)],[2])).
% 0.19/0.41 tff(4,plain,
% 0.19/0.41 (![X: $i] : product(X, X, X) <=> ![X: $i] : product(X, X, X)),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(5,axiom,(![X: $i] : product(X, X, X)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','product_idempotence')).
% 0.19/0.41 tff(6,plain,
% 0.19/0.41 (![X: $i] : product(X, X, X)),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.19/0.41 tff(7,plain,(
% 0.19/0.41 ![X: $i] : product(X, X, X)),
% 0.19/0.41 inference(skolemize,[status(sab)],[6])).
% 0.19/0.41 tff(8,plain,
% 0.19/0.41 (![X: $i] : product(X, X, X)),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.19/0.41 tff(9,plain,
% 0.19/0.41 ((~![X: $i] : product(X, X, X)) | product(e_2, e_2, e_2)),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(10,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[9, 8, 1])).
% 0.19/0.41 tff(11,plain,(product(e_2, e_2, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(12,assumption,(product(e_1, e_2, e_2)), introduced(assumption)).
% 0.19/0.41 tff(13,plain,
% 0.19/0.41 (^[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))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(14,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[13])).
% 0.19/0.41 tff(15,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(16,plain,
% 0.19/0.41 (^[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)))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(17,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[16])).
% 0.19/0.41 tff(18,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(W, Y, X)) | (~product(Z, Y, X))) | equalish(W, Z))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','product_left_cancellation')).
% 0.19/0.41 tff(19,plain,
% 0.19/0.41 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.19/0.41 tff(20,plain,
% 0.19/0.41 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[19, 15])).
% 0.19/0.41 tff(21,plain,(
% 0.19/0.41 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.19/0.41 inference(skolemize,[status(sab)],[20])).
% 0.19/0.41 tff(22,plain,
% 0.19/0.41 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[21, 14])).
% 0.19/0.41 tff(23,plain,
% 0.19/0.41 ((~equalish(e_1, e_2)) <=> (~equalish(e_1, e_2))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(24,axiom,(~equalish(e_1, e_2)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','e_1_is_not_e_2')).
% 0.19/0.42 tff(25,plain,
% 0.19/0.42 (~equalish(e_1, e_2)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[24, 23])).
% 0.19/0.42 tff(26,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_1, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_1, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_2)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(27,plain,
% 0.19/0.42 ((equalish(e_1, e_2) | (~product(e_2, e_2, e_2)) | (~product(e_1, e_2, e_2))) <=> (equalish(e_1, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_2)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(28,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_1, e_2) | (~product(e_2, e_2, e_2)) | (~product(e_1, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_1, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_2))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[27])).
% 0.19/0.42 tff(29,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_1, e_2) | (~product(e_2, e_2, e_2)) | (~product(e_1, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_1, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_2)))),
% 0.19/0.42 inference(transitivity,[status(thm)],[28, 26])).
% 0.19/0.42 tff(30,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_1, e_2) | (~product(e_2, e_2, e_2)) | (~product(e_1, e_2, e_2)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(31,plain,
% 0.19/0.42 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_1, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_2))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[30, 29])).
% 0.19/0.42 tff(32,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[31, 25, 22, 12, 11])).
% 0.19/0.42 tff(33,plain,(~product(e_1, e_2, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.42 tff(34,assumption,(~product(e_2, e_3, e_4)), introduced(assumption)).
% 0.19/0.42 tff(35,assumption,(~product(e_1, e_1, e_1)), introduced(assumption)).
% 0.19/0.42 tff(36,plain,
% 0.19/0.42 ((~![X: $i] : product(X, X, X)) | product(e_1, e_1, e_1)),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(37,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[36, 8, 35])).
% 0.19/0.42 tff(38,plain,(product(e_1, e_1, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.42 tff(39,assumption,(product(e_1, e_2, e_3)), introduced(assumption)).
% 0.19/0.42 tff(40,plain,
% 0.19/0.42 (^[Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : refl(((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2)) <=> ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2)))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(41,plain,
% 0.19/0.42 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2)) <=> ![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[40])).
% 0.19/0.42 tff(42,plain,
% 0.19/0.42 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2)) <=> ![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(43,plain,
% 0.19/0.42 (^[Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, X1, Y1))) <=> ((~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)))), (((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, X1, Y1))) | (~product(Z2, X2, Y2))) <=> (((~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))) | (~product(Z2, X2, Y2))))), rewrite((((~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))) | (~product(Z2, X2, Y2))) <=> ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)))), (((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, X1, Y1))) | (~product(Z2, X2, Y2))) <=> ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))))), ((((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, X1, Y1))) | (~product(Z2, X2, Y2))) | equalish(Y1, Y2)) <=> (((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))) | equalish(Y1, Y2)))), rewrite((((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))) | equalish(Y1, Y2)) <=> ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))), ((((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, X1, Y1))) | (~product(Z2, X2, Y2))) | equalish(Y1, Y2)) <=> ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(44,plain,
% 0.19/0.42 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : (((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, X1, Y1))) | (~product(Z2, X2, Y2))) | equalish(Y1, Y2)) <=> ![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[43])).
% 0.19/0.42 tff(45,axiom,(![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : (((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, X1, Y1))) | (~product(Z2, X2, Y2))) | equalish(Y1, Y2))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','qg2_2')).
% 0.19/0.42 tff(46,plain,
% 0.19/0.42 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.19/0.42 tff(47,plain,
% 0.19/0.42 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[46, 42])).
% 0.19/0.42 tff(48,plain,(
% 0.19/0.42 ![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.19/0.42 inference(skolemize,[status(sab)],[47])).
% 0.19/0.42 tff(49,plain,
% 0.19/0.42 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[48, 41])).
% 0.19/0.42 tff(50,plain,
% 0.19/0.42 ((~equalish(e_3, e_1)) <=> (~equalish(e_3, e_1))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(51,axiom,(~equalish(e_3, e_1)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','e_3_is_not_e_1')).
% 0.19/0.42 tff(52,plain,
% 0.19/0.42 (~equalish(e_3, e_1)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.19/0.42 tff(53,plain,
% 0.19/0.42 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_3, e_1) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_3, e_1)))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_3, e_1) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_3, e_1)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(54,plain,
% 0.19/0.42 (((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_3, e_1)) | equalish(e_3, e_1)) <=> (equalish(e_3, e_1) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_3, e_1)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(55,plain,
% 0.19/0.42 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_3, e_1)) | equalish(e_3, e_1))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_3, e_1) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_3, e_1))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[54])).
% 0.19/0.42 tff(56,plain,
% 0.19/0.42 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_3, e_1)) | equalish(e_3, e_1))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_3, e_1) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_3, e_1)))),
% 0.19/0.42 inference(transitivity,[status(thm)],[55, 53])).
% 0.19/0.42 tff(57,plain,
% 0.19/0.42 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_3, e_1)) | equalish(e_3, e_1))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(58,plain,
% 0.19/0.42 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_3, e_1) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_3, e_1))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.19/0.42 tff(59,plain,
% 0.19/0.42 (~product(e_2, e_3, e_1)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[58, 52, 49, 39, 38])).
% 0.19/0.42 tff(60,plain,
% 0.19/0.42 ((~![X: $i] : product(X, X, X)) | product(e_3, e_3, e_3)),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(61,plain,
% 0.19/0.42 (product(e_3, e_3, e_3)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[60, 8])).
% 0.19/0.42 tff(62,assumption,(product(e_3, e_3, e_4)), introduced(assumption)).
% 0.19/0.42 tff(63,plain,
% 0.19/0.42 (^[W: $i, Z: $i, Y: $i, X: $i] : refl((equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W))) <=> (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(64,plain,
% 0.19/0.42 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[63])).
% 0.19/0.42 tff(65,plain,
% 0.19/0.42 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(66,plain,
% 0.19/0.42 (^[W: $i, Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~product(X, Y, W)) | (~product(X, Y, Z))) <=> ((~product(X, Y, Z)) | (~product(X, Y, W)))), ((((~product(X, Y, W)) | (~product(X, Y, Z))) | equalish(W, Z)) <=> (((~product(X, Y, Z)) | (~product(X, Y, W))) | equalish(W, Z)))), rewrite((((~product(X, Y, Z)) | (~product(X, Y, W))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))), ((((~product(X, Y, W)) | (~product(X, Y, Z))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(67,plain,
% 0.19/0.42 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, W)) | (~product(X, Y, Z))) | equalish(W, Z)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[66])).
% 0.19/0.42 tff(68,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, W)) | (~product(X, Y, Z))) | equalish(W, Z))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','product_total_function2')).
% 0.19/0.42 tff(69,plain,
% 0.19/0.42 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[68, 67])).
% 0.19/0.42 tff(70,plain,
% 0.19/0.42 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[69, 65])).
% 0.19/0.42 tff(71,plain,(
% 0.19/0.42 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.19/0.42 inference(skolemize,[status(sab)],[70])).
% 0.19/0.42 tff(72,plain,
% 0.19/0.42 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[71, 64])).
% 0.19/0.42 tff(73,plain,
% 0.19/0.42 ((~equalish(e_4, e_3)) <=> (~equalish(e_4, e_3))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(74,axiom,(~equalish(e_4, e_3)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','e_4_is_not_e_3')).
% 0.19/0.42 tff(75,plain,
% 0.19/0.42 (~equalish(e_4, e_3)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[74, 73])).
% 0.19/0.42 tff(76,plain,
% 0.19/0.42 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_4, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_3, e_4)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_4, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_3, e_4)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(77,plain,
% 0.19/0.42 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_4, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_3, e_4)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(78,plain,
% 0.19/0.42 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_4, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_3, e_4))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.19/0.42 tff(79,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[78, 75, 72, 62, 61])).
% 0.19/0.42 tff(80,plain,(~product(e_3, e_3, e_4)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.42 tff(81,assumption,(product(e_4, e_3, e_4)), introduced(assumption)).
% 0.19/0.42 tff(82,assumption,(~product(e_4, e_4, e_4)), introduced(assumption)).
% 0.19/0.42 tff(83,plain,
% 0.19/0.42 ((~![X: $i] : product(X, X, X)) | product(e_4, e_4, e_4)),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(84,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[83, 8, 82])).
% 0.19/0.42 tff(85,plain,(product(e_4, e_4, e_4)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.42 tff(86,plain,
% 0.19/0.42 (^[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))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(87,plain,
% 0.19/0.42 (![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)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[86])).
% 0.19/0.42 tff(88,plain,
% 0.19/0.42 (![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)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(89,plain,
% 0.19/0.42 (^[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)))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(90,plain,
% 0.19/0.43 (![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)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[89])).
% 0.19/0.43 tff(91,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','product_right_cancellation')).
% 0.19/0.43 tff(92,plain,
% 0.19/0.43 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.19/0.43 tff(93,plain,
% 0.19/0.43 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[92, 88])).
% 0.19/0.43 tff(94,plain,(
% 0.19/0.43 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.19/0.43 inference(skolemize,[status(sab)],[93])).
% 0.19/0.43 tff(95,plain,
% 0.19/0.43 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[94, 87])).
% 0.19/0.43 tff(96,plain,
% 0.19/0.43 ((~equalish(e_3, e_4)) <=> (~equalish(e_3, e_4))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(97,axiom,(~equalish(e_3, e_4)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','e_3_is_not_e_4')).
% 0.19/0.43 tff(98,plain,
% 0.19/0.43 (~equalish(e_3, e_4)),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[97, 96])).
% 0.19/0.43 tff(99,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_4) | (~product(e_4, e_3, e_4)) | (~product(e_4, e_4, e_4)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_4) | (~product(e_4, e_3, e_4)) | (~product(e_4, e_4, e_4)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(100,plain,
% 0.19/0.43 ((equalish(e_3, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_4, e_3, e_4))) <=> (equalish(e_3, e_4) | (~product(e_4, e_3, e_4)) | (~product(e_4, e_4, e_4)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(101,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_4) | (~product(e_4, e_4, e_4)) | (~product(e_4, e_3, e_4)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_4) | (~product(e_4, e_3, e_4)) | (~product(e_4, e_4, e_4))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[100])).
% 0.19/0.43 tff(102,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_4) | (~product(e_4, e_4, e_4)) | (~product(e_4, e_3, e_4)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_4) | (~product(e_4, e_3, e_4)) | (~product(e_4, e_4, e_4)))),
% 0.19/0.43 inference(transitivity,[status(thm)],[101, 99])).
% 0.19/0.43 tff(103,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_4) | (~product(e_4, e_4, e_4)) | (~product(e_4, e_3, e_4)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(104,plain,
% 0.19/0.43 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_4) | (~product(e_4, e_3, e_4)) | (~product(e_4, e_4, e_4))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[103, 102])).
% 0.19/0.43 tff(105,plain,
% 0.19/0.43 ($false),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[104, 98, 95, 85, 81])).
% 0.19/0.43 tff(106,plain,(~product(e_4, e_3, e_4)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.43 tff(107,plain,
% 0.19/0.43 (group_element(e_4) <=> group_element(e_4)),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(108,axiom,(group_element(e_4)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','element_4')).
% 0.19/0.43 tff(109,plain,
% 0.19/0.43 (group_element(e_4)),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[108, 107])).
% 0.19/0.43 tff(110,plain,
% 0.19/0.43 (group_element(e_3) <=> group_element(e_3)),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(111,axiom,(group_element(e_3)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','element_3')).
% 0.19/0.43 tff(112,plain,
% 0.19/0.43 (group_element(e_3)),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[111, 110])).
% 0.19/0.43 tff(113,plain,
% 0.19/0.43 (^[Y: $i, X: $i] : refl((product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) <=> (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(114,plain,
% 0.19/0.43 (![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) <=> ![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[113])).
% 0.19/0.43 tff(115,plain,
% 0.19/0.43 (![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) <=> ![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(116,plain,
% 0.19/0.43 (^[Y: $i, X: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) <=> (product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))), (((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) <=> ((product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) | product(e_2, X, Y)))), rewrite(((product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) | product(e_2, X, Y)) <=> (product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))), (((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) <=> (product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))))), ((((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) | product(e_3, X, Y)) <=> ((product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) | product(e_3, X, Y)))), rewrite(((product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) | product(e_3, X, Y)) <=> (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))), ((((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) | product(e_3, X, Y)) <=> (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))))), (((((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) | product(e_3, X, Y)) | product(e_4, X, Y)) <=> ((product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) | product(e_4, X, Y)))), rewrite(((product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) | product(e_4, X, Y)) <=> (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))), (((((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) | product(e_3, X, Y)) | product(e_4, X, Y)) <=> (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(117,plain,
% 0.19/0.43 (![Y: $i, X: $i] : ((((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) | product(e_3, X, Y)) | product(e_4, X, Y)) <=> ![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[116])).
% 0.19/0.43 tff(118,axiom,(![Y: $i, X: $i] : ((((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) | product(e_3, X, Y)) | product(e_4, X, Y))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','row_surjectivity')).
% 0.19/0.43 tff(119,plain,
% 0.19/0.43 (![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[118, 117])).
% 0.19/0.43 tff(120,plain,
% 0.19/0.43 (![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[119, 115])).
% 0.19/0.43 tff(121,plain,(
% 0.19/0.43 ![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.43 inference(skolemize,[status(sab)],[120])).
% 0.19/0.43 tff(122,plain,
% 0.19/0.43 (![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[121, 114])).
% 0.19/0.43 tff(123,plain,
% 0.19/0.43 (((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_3, e_4) | (~group_element(e_3)) | (~group_element(e_4)) | product(e_4, e_3, e_4) | product(e_2, e_3, e_4) | product(e_1, e_3, e_4))) <=> ((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_3, e_4) | (~group_element(e_3)) | (~group_element(e_4)) | product(e_4, e_3, e_4) | product(e_2, e_3, e_4) | product(e_1, e_3, e_4))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(124,plain,
% 0.19/0.43 ((product(e_4, e_3, e_4) | product(e_3, e_3, e_4) | product(e_2, e_3, e_4) | product(e_1, e_3, e_4) | (~group_element(e_4)) | (~group_element(e_3))) <=> (product(e_3, e_3, e_4) | (~group_element(e_3)) | (~group_element(e_4)) | product(e_4, e_3, e_4) | product(e_2, e_3, e_4) | product(e_1, e_3, e_4))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(125,plain,
% 0.19/0.43 (((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_4, e_3, e_4) | product(e_3, e_3, e_4) | product(e_2, e_3, e_4) | product(e_1, e_3, e_4) | (~group_element(e_4)) | (~group_element(e_3)))) <=> ((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_3, e_4) | (~group_element(e_3)) | (~group_element(e_4)) | product(e_4, e_3, e_4) | product(e_2, e_3, e_4) | product(e_1, e_3, e_4)))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[124])).
% 0.19/0.43 tff(126,plain,
% 0.19/0.43 (((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_4, e_3, e_4) | product(e_3, e_3, e_4) | product(e_2, e_3, e_4) | product(e_1, e_3, e_4) | (~group_element(e_4)) | (~group_element(e_3)))) <=> ((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_3, e_4) | (~group_element(e_3)) | (~group_element(e_4)) | product(e_4, e_3, e_4) | product(e_2, e_3, e_4) | product(e_1, e_3, e_4))),
% 0.19/0.43 inference(transitivity,[status(thm)],[125, 123])).
% 0.19/0.43 tff(127,plain,
% 0.19/0.43 ((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_4, e_3, e_4) | product(e_3, e_3, e_4) | product(e_2, e_3, e_4) | product(e_1, e_3, e_4) | (~group_element(e_4)) | (~group_element(e_3)))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(128,plain,
% 0.19/0.44 ((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_3, e_4) | (~group_element(e_3)) | (~group_element(e_4)) | product(e_4, e_3, e_4) | product(e_2, e_3, e_4) | product(e_1, e_3, e_4)),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[127, 126])).
% 0.19/0.44 tff(129,plain,
% 0.19/0.44 (product(e_3, e_3, e_4) | product(e_2, e_3, e_4) | product(e_1, e_3, e_4)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[128, 122, 112, 109, 106])).
% 0.19/0.44 tff(130,plain,
% 0.19/0.44 (product(e_2, e_3, e_4) | product(e_1, e_3, e_4)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[129, 80])).
% 0.19/0.44 tff(131,plain,
% 0.19/0.44 (product(e_1, e_3, e_4)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[130, 34])).
% 0.19/0.44 tff(132,assumption,(product(e_3, e_4, e_4)), introduced(assumption)).
% 0.19/0.44 tff(133,plain,
% 0.19/0.44 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_4) | (~product(e_3, e_4, e_4)) | (~product(e_4, e_4, e_4)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_4) | (~product(e_3, e_4, e_4)) | (~product(e_4, e_4, e_4)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(134,plain,
% 0.19/0.44 ((equalish(e_3, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_3, e_4, e_4))) <=> (equalish(e_3, e_4) | (~product(e_3, e_4, e_4)) | (~product(e_4, e_4, e_4)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(135,plain,
% 0.19/0.44 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_3, e_4, e_4)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_4) | (~product(e_3, e_4, e_4)) | (~product(e_4, e_4, e_4))))),
% 0.19/0.44 inference(monotonicity,[status(thm)],[134])).
% 0.19/0.44 tff(136,plain,
% 0.19/0.44 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_3, e_4, e_4)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_4) | (~product(e_3, e_4, e_4)) | (~product(e_4, e_4, e_4)))),
% 0.19/0.44 inference(transitivity,[status(thm)],[135, 133])).
% 0.19/0.44 tff(137,plain,
% 0.19/0.44 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_3, e_4, e_4)))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(138,plain,
% 0.19/0.44 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_4) | (~product(e_3, e_4, e_4)) | (~product(e_4, e_4, e_4))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[137, 136])).
% 0.19/0.44 tff(139,plain,
% 0.19/0.44 ($false),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[138, 98, 22, 85, 132])).
% 0.19/0.44 tff(140,plain,(~product(e_3, e_4, e_4)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.44 tff(141,plain,
% 0.19/0.44 ((~equalish(e_4, e_1)) <=> (~equalish(e_4, e_1))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(142,axiom,(~equalish(e_4, e_1)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','e_4_is_not_e_1')).
% 0.19/0.44 tff(143,plain,
% 0.19/0.44 (~equalish(e_4, e_1)),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[142, 141])).
% 0.19/0.44 tff(144,plain,
% 0.19/0.44 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_4, e_1) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_1)) | (~product(e_1, e_1, e_1)))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_4, e_1) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(145,plain,
% 0.19/0.44 (((~product(e_1, e_1, e_1)) | (~product(e_1, e_3, e_4)) | (~product(e_1, e_1, e_1)) | (~product(e_3, e_4, e_1)) | equalish(e_4, e_1)) <=> (equalish(e_4, e_1) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(146,plain,
% 0.19/0.44 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_3, e_4)) | (~product(e_1, e_1, e_1)) | (~product(e_3, e_4, e_1)) | equalish(e_4, e_1))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_4, e_1) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_1)) | (~product(e_1, e_1, e_1))))),
% 0.19/0.44 inference(monotonicity,[status(thm)],[145])).
% 0.19/0.44 tff(147,plain,
% 0.19/0.44 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_3, e_4)) | (~product(e_1, e_1, e_1)) | (~product(e_3, e_4, e_1)) | equalish(e_4, e_1))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_4, e_1) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.19/0.44 inference(transitivity,[status(thm)],[146, 144])).
% 0.19/0.44 tff(148,plain,
% 0.19/0.44 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_3, e_4)) | (~product(e_1, e_1, e_1)) | (~product(e_3, e_4, e_1)) | equalish(e_4, e_1))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(149,plain,
% 0.19/0.44 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_4, e_1) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_1)) | (~product(e_1, e_1, e_1))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[148, 147])).
% 0.19/0.44 tff(150,plain,
% 0.19/0.44 (~product(e_3, e_4, e_1)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[149, 143, 49, 38, 131])).
% 0.19/0.44 tff(151,assumption,(product(e_3, e_4, e_3)), introduced(assumption)).
% 0.19/0.44 tff(152,plain,
% 0.19/0.44 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_4, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_4, e_3)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_4, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_4, e_3)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(153,plain,
% 0.19/0.44 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_4, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_4, e_3)))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(154,plain,
% 0.19/0.44 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_4, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_4, e_3))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[153, 152])).
% 0.19/0.44 tff(155,plain,
% 0.19/0.44 ($false),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[154, 75, 95, 61, 151])).
% 0.19/0.44 tff(156,plain,(~product(e_3, e_4, e_3)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.44 tff(157,plain,
% 0.19/0.44 (^[Y: $i, X: $i] : refl(((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | 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_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1)))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(158,plain,
% 0.19/0.44 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[157])).
% 0.19/0.44 tff(159,plain,
% 0.19/0.44 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(160,plain,
% 0.19/0.44 (^[Y: $i, X: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_1))), (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) <=> (((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_1)) | product(X, Y, e_2)))), rewrite((((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_1)) | product(X, Y, e_2)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_2) | product(X, Y, e_1))), (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_2) | product(X, Y, e_1)))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3)) <=> (((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_2) | product(X, Y, e_1)) | product(X, Y, e_3)))), rewrite((((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_2) | product(X, Y, e_1)) | product(X, Y, e_3)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1)))), (((((((~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_4)) <=> (((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1)) | product(X, Y, e_4)))), rewrite((((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1)) | product(X, Y, e_4)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))), (((((((~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_4)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(161,plain,
% 0.19/0.44 (![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)) | product(X, Y, e_4)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[160])).
% 0.19/0.44 tff(162,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)) | product(X, Y, e_4))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','product_total_function1')).
% 0.19/0.44 tff(163,plain,
% 0.19/0.44 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[162, 161])).
% 0.19/0.44 tff(164,plain,
% 0.19/0.44 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[163, 159])).
% 0.19/0.44 tff(165,plain,(
% 0.19/0.44 ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.19/0.45 inference(skolemize,[status(sab)],[164])).
% 0.19/0.45 tff(166,plain,
% 0.19/0.45 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[165, 158])).
% 0.19/0.45 tff(167,plain,
% 0.19/0.45 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (product(e_3, e_4, e_2) | (~group_element(e_3)) | (~group_element(e_4)) | product(e_3, e_4, e_4) | product(e_3, e_4, e_3) | product(e_3, e_4, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | product(e_3, e_4, e_2) | (~group_element(e_3)) | (~group_element(e_4)) | product(e_3, e_4, e_4) | product(e_3, e_4, e_3) | product(e_3, e_4, e_1))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(168,plain,
% 0.19/0.45 (((~group_element(e_4)) | (~group_element(e_3)) | product(e_3, e_4, e_4) | product(e_3, e_4, e_3) | product(e_3, e_4, e_2) | product(e_3, e_4, e_1)) <=> (product(e_3, e_4, e_2) | (~group_element(e_3)) | (~group_element(e_4)) | product(e_3, e_4, e_4) | product(e_3, e_4, e_3) | product(e_3, e_4, e_1))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(169,plain,
% 0.19/0.45 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_4)) | (~group_element(e_3)) | product(e_3, e_4, e_4) | product(e_3, e_4, e_3) | product(e_3, e_4, e_2) | product(e_3, e_4, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (product(e_3, e_4, e_2) | (~group_element(e_3)) | (~group_element(e_4)) | product(e_3, e_4, e_4) | product(e_3, e_4, e_3) | product(e_3, e_4, e_1)))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[168])).
% 0.19/0.45 tff(170,plain,
% 0.19/0.45 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_4)) | (~group_element(e_3)) | product(e_3, e_4, e_4) | product(e_3, e_4, e_3) | product(e_3, e_4, e_2) | product(e_3, e_4, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | product(e_3, e_4, e_2) | (~group_element(e_3)) | (~group_element(e_4)) | product(e_3, e_4, e_4) | product(e_3, e_4, e_3) | product(e_3, e_4, e_1))),
% 0.19/0.45 inference(transitivity,[status(thm)],[169, 167])).
% 0.19/0.45 tff(171,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_4)) | (~group_element(e_3)) | product(e_3, e_4, e_4) | product(e_3, e_4, e_3) | product(e_3, e_4, e_2) | product(e_3, e_4, e_1))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(172,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | product(e_3, e_4, e_2) | (~group_element(e_3)) | (~group_element(e_4)) | product(e_3, e_4, e_4) | product(e_3, e_4, e_3) | product(e_3, e_4, e_1)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[171, 170])).
% 0.19/0.45 tff(173,plain,
% 0.19/0.45 (product(e_3, e_4, e_2) | product(e_3, e_4, e_4) | product(e_3, e_4, e_1)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[172, 112, 109, 166, 156])).
% 0.19/0.45 tff(174,plain,
% 0.19/0.45 (product(e_3, e_4, e_2)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[173, 150, 140])).
% 0.19/0.45 tff(175,plain,
% 0.19/0.45 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_3, e_4) | (~product(e_1, e_2, e_3)) | (~product(e_2, e_3, e_2)) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_2)))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_3, e_4) | (~product(e_1, e_2, e_3)) | (~product(e_2, e_3, e_2)) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_2)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(176,plain,
% 0.19/0.45 (((~product(e_1, e_3, e_4)) | (~product(e_1, e_2, e_3)) | (~product(e_3, e_4, e_2)) | (~product(e_2, e_3, e_2)) | equalish(e_3, e_4)) <=> (equalish(e_3, e_4) | (~product(e_1, e_2, e_3)) | (~product(e_2, e_3, e_2)) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_2)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(177,plain,
% 0.19/0.45 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_3, e_4)) | (~product(e_1, e_2, e_3)) | (~product(e_3, e_4, e_2)) | (~product(e_2, e_3, e_2)) | equalish(e_3, e_4))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_3, e_4) | (~product(e_1, e_2, e_3)) | (~product(e_2, e_3, e_2)) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_2))))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[176])).
% 0.19/0.45 tff(178,plain,
% 0.19/0.45 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_3, e_4)) | (~product(e_1, e_2, e_3)) | (~product(e_3, e_4, e_2)) | (~product(e_2, e_3, e_2)) | equalish(e_3, e_4))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_3, e_4) | (~product(e_1, e_2, e_3)) | (~product(e_2, e_3, e_2)) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_2)))),
% 0.19/0.45 inference(transitivity,[status(thm)],[177, 175])).
% 0.19/0.45 tff(179,plain,
% 0.19/0.45 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_3, e_4)) | (~product(e_1, e_2, e_3)) | (~product(e_3, e_4, e_2)) | (~product(e_2, e_3, e_2)) | equalish(e_3, e_4))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(180,plain,
% 0.19/0.45 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_3, e_4) | (~product(e_1, e_2, e_3)) | (~product(e_2, e_3, e_2)) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_2))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[179, 178])).
% 0.19/0.45 tff(181,plain,
% 0.19/0.45 (~product(e_2, e_3, e_2)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[180, 98, 49, 174, 39, 131])).
% 0.19/0.45 tff(182,assumption,(product(e_2, e_3, e_3)), introduced(assumption)).
% 0.19/0.45 tff(183,plain,
% 0.19/0.45 ((~equalish(e_2, e_3)) <=> (~equalish(e_2, e_3))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(184,axiom,(~equalish(e_2, e_3)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','e_2_is_not_e_3')).
% 0.19/0.45 tff(185,plain,
% 0.19/0.45 (~equalish(e_2, e_3)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[184, 183])).
% 0.19/0.45 tff(186,plain,
% 0.19/0.45 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_2, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_2, 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_2, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_2, e_3, e_3)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(187,plain,
% 0.19/0.45 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_2, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_2, e_3, e_3)))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(188,plain,
% 0.19/0.45 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_2, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_2, e_3, e_3))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[187, 186])).
% 0.19/0.46 tff(189,plain,
% 0.19/0.46 ($false),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[188, 185, 22, 61, 182])).
% 0.19/0.46 tff(190,plain,(~product(e_2, e_3, e_3)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.46 tff(191,plain,
% 0.19/0.46 (group_element(e_2) <=> group_element(e_2)),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(192,axiom,(group_element(e_2)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','element_2')).
% 0.19/0.46 tff(193,plain,
% 0.19/0.46 (group_element(e_2)),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[192, 191])).
% 0.19/0.46 tff(194,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_3)) | (~group_element(e_2)) | product(e_2, e_3, e_3) | product(e_2, e_3, e_4) | product(e_2, e_3, e_2) | product(e_2, e_3, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_3)) | (~group_element(e_2)) | product(e_2, e_3, e_3) | product(e_2, e_3, e_4) | product(e_2, e_3, e_2) | product(e_2, e_3, e_1))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(195,plain,
% 0.19/0.46 (((~group_element(e_3)) | (~group_element(e_2)) | product(e_2, e_3, e_4) | product(e_2, e_3, e_3) | product(e_2, e_3, e_2) | product(e_2, e_3, e_1)) <=> ((~group_element(e_3)) | (~group_element(e_2)) | product(e_2, e_3, e_3) | product(e_2, e_3, e_4) | product(e_2, e_3, e_2) | product(e_2, e_3, e_1))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(196,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_3)) | (~group_element(e_2)) | product(e_2, e_3, e_4) | product(e_2, e_3, e_3) | product(e_2, e_3, e_2) | product(e_2, e_3, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_3)) | (~group_element(e_2)) | product(e_2, e_3, e_3) | product(e_2, e_3, e_4) | product(e_2, e_3, e_2) | product(e_2, e_3, e_1)))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[195])).
% 0.19/0.46 tff(197,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_3)) | (~group_element(e_2)) | product(e_2, e_3, e_4) | product(e_2, e_3, e_3) | product(e_2, e_3, e_2) | product(e_2, e_3, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_3)) | (~group_element(e_2)) | product(e_2, e_3, e_3) | product(e_2, e_3, e_4) | product(e_2, e_3, e_2) | product(e_2, e_3, e_1))),
% 0.19/0.46 inference(transitivity,[status(thm)],[196, 194])).
% 0.19/0.46 tff(198,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_3)) | (~group_element(e_2)) | product(e_2, e_3, e_4) | product(e_2, e_3, e_3) | product(e_2, e_3, e_2) | product(e_2, e_3, e_1))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(199,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_3)) | (~group_element(e_2)) | product(e_2, e_3, e_3) | product(e_2, e_3, e_4) | product(e_2, e_3, e_2) | product(e_2, e_3, e_1)),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[198, 197])).
% 0.19/0.46 tff(200,plain,
% 0.19/0.46 (product(e_2, e_3, e_4) | product(e_2, e_3, e_2) | product(e_2, e_3, e_1)),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[199, 193, 112, 166, 190])).
% 0.19/0.46 tff(201,plain,
% 0.19/0.46 ($false),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[200, 181, 59, 34])).
% 0.19/0.46 tff(202,plain,((~product(e_1, e_2, e_3)) | product(e_2, e_3, e_4)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.46 tff(203,plain,
% 0.19/0.46 (~product(e_1, e_2, e_3)),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[202, 34])).
% 0.19/0.46 tff(204,assumption,(product(e_1, e_3, e_4)), introduced(assumption)).
% 0.19/0.46 tff(205,assumption,(product(e_1, e_2, e_4)), introduced(assumption)).
% 0.19/0.46 tff(206,plain,
% 0.19/0.46 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_3) | (~product(e_1, e_2, e_4)) | (~product(e_1, e_3, e_4)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_2, e_3) | (~product(e_1, e_2, e_4)) | (~product(e_1, e_3, e_4)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(207,plain,
% 0.19/0.46 ((equalish(e_2, e_3) | (~product(e_1, e_3, e_4)) | (~product(e_1, e_2, e_4))) <=> (equalish(e_2, e_3) | (~product(e_1, e_2, e_4)) | (~product(e_1, e_3, e_4)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(208,plain,
% 0.19/0.46 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_3) | (~product(e_1, e_3, e_4)) | (~product(e_1, e_2, e_4)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_3) | (~product(e_1, e_2, e_4)) | (~product(e_1, e_3, e_4))))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[207])).
% 0.19/0.46 tff(209,plain,
% 0.19/0.46 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_3) | (~product(e_1, e_3, e_4)) | (~product(e_1, e_2, e_4)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_2, e_3) | (~product(e_1, e_2, e_4)) | (~product(e_1, e_3, e_4)))),
% 0.19/0.46 inference(transitivity,[status(thm)],[208, 206])).
% 0.19/0.46 tff(210,plain,
% 0.19/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_3) | (~product(e_1, e_3, e_4)) | (~product(e_1, e_2, e_4)))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(211,plain,
% 0.19/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_2, e_3) | (~product(e_1, e_2, e_4)) | (~product(e_1, e_3, e_4))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[210, 209])).
% 0.19/0.46 tff(212,plain,
% 0.19/0.46 ($false),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[211, 185, 95, 205, 204])).
% 0.19/0.46 tff(213,plain,((~product(e_1, e_2, e_4)) | (~product(e_1, e_3, e_4))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.46 tff(214,plain,
% 0.19/0.46 (~product(e_1, e_2, e_4)),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[213, 131])).
% 0.19/0.46 tff(215,assumption,(product(e_1, e_2, e_1)), introduced(assumption)).
% 0.19/0.46 tff(216,plain,
% 0.19/0.46 ((~equalish(e_2, e_1)) <=> (~equalish(e_2, e_1))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(217,axiom,(~equalish(e_2, e_1)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','e_2_is_not_e_1')).
% 0.19/0.46 tff(218,plain,
% 0.19/0.46 (~equalish(e_2, e_1)),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[217, 216])).
% 0.19/0.46 tff(219,plain,
% 0.19/0.46 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(220,plain,
% 0.19/0.46 ((equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1))) <=> (equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(221,plain,
% 0.19/0.46 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1))))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[220])).
% 0.19/0.46 tff(222,plain,
% 0.19/0.46 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.19/0.46 inference(transitivity,[status(thm)],[221, 219])).
% 0.19/0.46 tff(223,plain,
% 0.19/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(224,plain,
% 0.19/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[223, 222])).
% 0.19/0.46 tff(225,plain,
% 0.19/0.46 ($false),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[224, 218, 95, 38, 215])).
% 0.19/0.46 tff(226,plain,(~product(e_1, e_2, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.46 tff(227,plain,
% 0.19/0.46 (group_element(e_1) <=> group_element(e_1)),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(228,axiom,(group_element(e_1)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','element_1')).
% 0.19/0.46 tff(229,plain,
% 0.19/0.46 (group_element(e_1)),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[228, 227])).
% 0.19/0.46 tff(230,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_1)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_4) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_1)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_4) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(231,plain,
% 0.19/0.46 (((~group_element(e_2)) | (~group_element(e_1)) | product(e_1, e_2, e_4) | product(e_1, e_2, e_3) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1)) <=> ((~group_element(e_1)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_4) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(232,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_1)) | product(e_1, e_2, e_4) | product(e_1, e_2, e_3) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_1)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_4) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1)))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[231])).
% 0.19/0.46 tff(233,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_1)) | product(e_1, e_2, e_4) | product(e_1, e_2, e_3) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_1)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_4) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1))),
% 0.19/0.46 inference(transitivity,[status(thm)],[232, 230])).
% 0.19/0.46 tff(234,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_1)) | product(e_1, e_2, e_4) | product(e_1, e_2, e_3) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(235,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_1)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_4) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1)),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[234, 233])).
% 0.19/0.46 tff(236,plain,
% 0.19/0.46 (product(e_1, e_2, e_3) | product(e_1, e_2, e_4) | product(e_1, e_2, e_2)),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[235, 229, 193, 166, 226])).
% 0.19/0.46 tff(237,plain,
% 0.19/0.46 ($false),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[236, 214, 203, 33])).
% 0.19/0.46 tff(238,plain,(product(e_2, e_3, e_4)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.46 tff(239,assumption,(product(e_2, e_3, e_4)), introduced(assumption)).
% 0.19/0.46 tff(240,assumption,(product(e_4, e_2, e_3)), introduced(assumption)).
% 0.19/0.46 tff(241,plain,
% 0.19/0.46 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_3, e_4) | (~product(e_4, e_2, e_3)) | (~product(e_2, e_3, e_4)) | (~product(e_4, e_4, e_4)))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_3, e_4) | (~product(e_4, e_2, e_3)) | (~product(e_2, e_3, e_4)) | (~product(e_4, e_4, e_4)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(242,plain,
% 0.19/0.46 (((~product(e_4, e_4, e_4)) | (~product(e_4, e_2, e_3)) | (~product(e_4, e_4, e_4)) | (~product(e_2, e_3, e_4)) | equalish(e_3, e_4)) <=> (equalish(e_3, e_4) | (~product(e_4, e_2, e_3)) | (~product(e_2, e_3, e_4)) | (~product(e_4, e_4, e_4)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(243,plain,
% 0.19/0.46 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_4, e_4, e_4)) | (~product(e_4, e_2, e_3)) | (~product(e_4, e_4, e_4)) | (~product(e_2, e_3, e_4)) | equalish(e_3, e_4))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_3, e_4) | (~product(e_4, e_2, e_3)) | (~product(e_2, e_3, e_4)) | (~product(e_4, e_4, e_4))))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[242])).
% 0.19/0.46 tff(244,plain,
% 0.19/0.46 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_4, e_4, e_4)) | (~product(e_4, e_2, e_3)) | (~product(e_4, e_4, e_4)) | (~product(e_2, e_3, e_4)) | equalish(e_3, e_4))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_3, e_4) | (~product(e_4, e_2, e_3)) | (~product(e_2, e_3, e_4)) | (~product(e_4, e_4, e_4)))),
% 0.19/0.46 inference(transitivity,[status(thm)],[243, 241])).
% 0.19/0.46 tff(245,plain,
% 0.19/0.46 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_4, e_4, e_4)) | (~product(e_4, e_2, e_3)) | (~product(e_4, e_4, e_4)) | (~product(e_2, e_3, e_4)) | equalish(e_3, e_4))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(246,plain,
% 0.19/0.46 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_3, e_4) | (~product(e_4, e_2, e_3)) | (~product(e_2, e_3, e_4)) | (~product(e_4, e_4, e_4))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[245, 244])).
% 0.19/0.46 tff(247,plain,
% 0.19/0.46 ($false),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[246, 98, 49, 85, 240, 239])).
% 0.19/0.46 tff(248,plain,((~product(e_4, e_2, e_3)) | (~product(e_2, e_3, e_4))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.47 tff(249,plain,
% 0.19/0.47 (~product(e_4, e_2, e_3)),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[248, 238])).
% 0.19/0.47 tff(250,assumption,(product(e_2, e_2, e_3)), introduced(assumption)).
% 0.19/0.47 tff(251,plain,
% 0.19/0.47 ((~equalish(e_3, e_2)) <=> (~equalish(e_3, e_2))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(252,axiom,(~equalish(e_3, e_2)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','e_3_is_not_e_2')).
% 0.19/0.47 tff(253,plain,
% 0.19/0.47 (~equalish(e_3, e_2)),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[252, 251])).
% 0.19/0.47 tff(254,plain,
% 0.19/0.47 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_3, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_3, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_2)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(255,plain,
% 0.19/0.47 ((equalish(e_3, e_2) | (~product(e_2, e_2, e_2)) | (~product(e_2, e_2, e_3))) <=> (equalish(e_3, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_2)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(256,plain,
% 0.19/0.47 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_3, e_2) | (~product(e_2, e_2, e_2)) | (~product(e_2, e_2, e_3)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_3, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_2))))),
% 0.19/0.47 inference(monotonicity,[status(thm)],[255])).
% 0.19/0.47 tff(257,plain,
% 0.19/0.47 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_3, e_2) | (~product(e_2, e_2, e_2)) | (~product(e_2, e_2, e_3)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_3, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_2)))),
% 0.19/0.47 inference(transitivity,[status(thm)],[256, 254])).
% 0.19/0.47 tff(258,plain,
% 0.19/0.47 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_3, e_2) | (~product(e_2, e_2, e_2)) | (~product(e_2, e_2, e_3)))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(259,plain,
% 0.19/0.47 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_3, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_2))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[258, 257])).
% 0.19/0.47 tff(260,plain,
% 0.19/0.47 ($false),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[259, 253, 72, 250, 11])).
% 0.19/0.47 tff(261,plain,(~product(e_2, e_2, e_3)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.47 tff(262,assumption,(product(e_3, e_2, e_3)), introduced(assumption)).
% 0.19/0.47 tff(263,plain,
% 0.19/0.47 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_2, e_3)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_2, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_2, e_3)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(264,plain,
% 0.19/0.47 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_2, e_3)))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(265,plain,
% 0.19/0.47 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_2, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_2, e_3))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[264, 263])).
% 0.19/0.47 tff(266,plain,
% 0.19/0.47 ($false),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[265, 185, 95, 61, 262])).
% 0.19/0.47 tff(267,plain,(~product(e_3, e_2, e_3)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.47 tff(268,plain,
% 0.19/0.47 (((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | ((~group_element(e_3)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_4, e_2, e_3) | product(e_3, e_2, e_3) | product(e_2, e_2, e_3))) <=> ((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (~group_element(e_3)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_4, e_2, e_3) | product(e_3, e_2, e_3) | product(e_2, e_2, e_3))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(269,plain,
% 0.19/0.47 ((product(e_4, e_2, e_3) | product(e_3, e_2, e_3) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3) | (~group_element(e_3)) | (~group_element(e_2))) <=> ((~group_element(e_3)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_4, e_2, e_3) | product(e_3, e_2, e_3) | product(e_2, e_2, e_3))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(270,plain,
% 0.19/0.47 (((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_4, e_2, e_3) | product(e_3, e_2, e_3) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3) | (~group_element(e_3)) | (~group_element(e_2)))) <=> ((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | ((~group_element(e_3)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_4, e_2, e_3) | product(e_3, e_2, e_3) | product(e_2, e_2, e_3)))),
% 0.19/0.47 inference(monotonicity,[status(thm)],[269])).
% 0.19/0.47 tff(271,plain,
% 0.19/0.47 (((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_4, e_2, e_3) | product(e_3, e_2, e_3) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3) | (~group_element(e_3)) | (~group_element(e_2)))) <=> ((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (~group_element(e_3)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_4, e_2, e_3) | product(e_3, e_2, e_3) | product(e_2, e_2, e_3))),
% 0.19/0.47 inference(transitivity,[status(thm)],[270, 268])).
% 0.19/0.47 tff(272,plain,
% 0.19/0.47 ((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_4, e_2, e_3) | product(e_3, e_2, e_3) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3) | (~group_element(e_3)) | (~group_element(e_2)))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(273,plain,
% 0.19/0.47 ((~![Y: $i, X: $i] : (product(e_4, X, Y) | product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (~group_element(e_3)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_4, e_2, e_3) | product(e_3, e_2, e_3) | product(e_2, e_2, e_3)),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[272, 271])).
% 0.19/0.47 tff(274,plain,
% 0.19/0.47 (product(e_1, e_2, e_3) | product(e_4, e_2, e_3) | product(e_2, e_2, e_3)),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[273, 122, 193, 112, 267])).
% 0.19/0.47 tff(275,plain,
% 0.19/0.47 (product(e_1, e_2, e_3)),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[274, 261, 249])).
% 0.19/0.47 tff(276,assumption,(~product(e_1, e_3, e_4)), introduced(assumption)).
% 0.19/0.47 tff(277,plain,
% 0.19/0.47 (product(e_2, e_3, e_4)),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[130, 276])).
% 0.19/0.47 tff(278,assumption,(product(e_1, e_4, e_4)), introduced(assumption)).
% 0.19/0.47 tff(279,plain,
% 0.19/0.47 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_2, e_3)) | (~product(e_1, e_4, e_4)) | (~product(e_2, e_3, e_4)) | (~product(e_4, e_4, e_4)) | equalish(e_4, e_3))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_4, e_4)) | (~product(e_2, e_3, e_4)) | (~product(e_4, e_4, e_4)) | equalish(e_4, e_3))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(280,plain,
% 0.19/0.47 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_2, e_3)) | (~product(e_1, e_4, e_4)) | (~product(e_2, e_3, e_4)) | (~product(e_4, e_4, e_4)) | equalish(e_4, e_3))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(281,plain,
% 0.19/0.47 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_4, e_4)) | (~product(e_2, e_3, e_4)) | (~product(e_4, e_4, e_4)) | equalish(e_4, e_3)),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[280, 279])).
% 0.19/0.47 tff(282,plain,
% 0.19/0.47 ($false),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[281, 75, 49, 85, 39, 239, 278])).
% 0.19/0.47 tff(283,plain,((~product(e_1, e_4, e_4)) | (~product(e_1, e_2, e_3)) | (~product(e_2, e_3, e_4))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.47 tff(284,plain,
% 0.19/0.47 (~product(e_1, e_4, e_4)),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[283, 39, 277])).
% 0.19/0.47 tff(285,plain,
% 0.19/0.47 ((~equalish(e_1, e_3)) <=> (~equalish(e_1, e_3))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(286,axiom,(~equalish(e_1, e_3)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','e_1_is_not_e_3')).
% 0.19/0.47 tff(287,plain,
% 0.19/0.47 (~equalish(e_1, e_3)),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[286, 285])).
% 0.19/0.47 tff(288,plain,
% 0.19/0.47 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_1, e_3) | (~product(e_1, e_2, e_3)) | (~product(e_2, e_3, e_4)) | (~product(e_1, e_1, e_4)) | (~product(e_1, e_1, e_1)))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_1, e_3) | (~product(e_1, e_2, e_3)) | (~product(e_2, e_3, e_4)) | (~product(e_1, e_1, e_4)) | (~product(e_1, e_1, e_1)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(289,plain,
% 0.19/0.47 (((~product(e_1, e_2, e_3)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_3, e_4)) | (~product(e_1, e_1, e_4)) | equalish(e_1, e_3)) <=> (equalish(e_1, e_3) | (~product(e_1, e_2, e_3)) | (~product(e_2, e_3, e_4)) | (~product(e_1, e_1, e_4)) | (~product(e_1, e_1, e_1)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(290,plain,
% 0.19/0.47 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_2, e_3)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_3, e_4)) | (~product(e_1, e_1, e_4)) | equalish(e_1, e_3))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_1, e_3) | (~product(e_1, e_2, e_3)) | (~product(e_2, e_3, e_4)) | (~product(e_1, e_1, e_4)) | (~product(e_1, e_1, e_1))))),
% 0.19/0.47 inference(monotonicity,[status(thm)],[289])).
% 0.19/0.47 tff(291,plain,
% 0.19/0.47 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_2, e_3)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_3, e_4)) | (~product(e_1, e_1, e_4)) | equalish(e_1, e_3))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_1, e_3) | (~product(e_1, e_2, e_3)) | (~product(e_2, e_3, e_4)) | (~product(e_1, e_1, e_4)) | (~product(e_1, e_1, e_1)))),
% 0.19/0.47 inference(transitivity,[status(thm)],[290, 288])).
% 0.19/0.47 tff(292,plain,
% 0.19/0.47 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_2, e_3)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_3, e_4)) | (~product(e_1, e_1, e_4)) | equalish(e_1, e_3))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(293,plain,
% 0.19/0.47 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_1, e_3) | (~product(e_1, e_2, e_3)) | (~product(e_2, e_3, e_4)) | (~product(e_1, e_1, e_4)) | (~product(e_1, e_1, e_1))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[292, 291])).
% 0.19/0.47 tff(294,plain,
% 0.19/0.47 (~product(e_1, e_1, e_4)),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[293, 287, 49, 39, 38, 277])).
% 0.19/0.47 tff(295,plain,
% 0.19/0.47 (^[Y: $i, X: $i] : refl(((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)))),
% 0.19/0.47 inference(bind,[status(th)],[])).
% 0.19/0.47 tff(296,plain,
% 0.19/0.47 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.19/0.47 inference(quant_intro,[status(thm)],[295])).
% 0.19/0.47 tff(297,plain,
% 0.19/0.47 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(298,plain,
% 0.19/0.47 (^[Y: $i, X: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_1, Y))), (((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) <=> (((~group_element(Y)) | (~group_element(X)) | product(X, e_1, Y)) | product(X, e_2, Y)))), rewrite((((~group_element(Y)) | (~group_element(X)) | product(X, e_1, Y)) | product(X, e_2, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_2, Y) | product(X, e_1, Y))), (((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_2, Y) | product(X, e_1, Y)))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) | product(X, e_3, Y)) <=> (((~group_element(Y)) | (~group_element(X)) | product(X, e_2, Y) | product(X, e_1, Y)) | product(X, e_3, Y)))), rewrite((((~group_element(Y)) | (~group_element(X)) | product(X, e_2, Y) | product(X, e_1, Y)) | product(X, e_3, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) | product(X, e_3, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)))), (((((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) | product(X, e_3, Y)) | product(X, e_4, Y)) <=> (((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)) | product(X, e_4, Y)))), rewrite((((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)) | product(X, e_4, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))), (((((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) | product(X, e_3, Y)) | product(X, e_4, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))))),
% 0.19/0.47 inference(bind,[status(th)],[])).
% 0.19/0.48 tff(299,plain,
% 0.19/0.48 (![Y: $i, X: $i] : ((((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) | product(X, e_3, Y)) | product(X, e_4, Y)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.19/0.48 inference(quant_intro,[status(thm)],[298])).
% 0.19/0.48 tff(300,axiom,(![Y: $i, X: $i] : ((((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) | product(X, e_3, Y)) | product(X, e_4, Y))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','column_surjectivity')).
% 0.19/0.48 tff(301,plain,
% 0.19/0.48 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[300, 299])).
% 0.19/0.48 tff(302,plain,
% 0.19/0.48 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[301, 297])).
% 0.19/0.48 tff(303,plain,(
% 0.19/0.48 ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.19/0.48 inference(skolemize,[status(sab)],[302])).
% 0.19/0.48 tff(304,plain,
% 0.19/0.48 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[303, 296])).
% 0.19/0.48 tff(305,plain,
% 0.19/0.48 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_1)) | product(e_1, e_2, e_4) | (~group_element(e_4)) | product(e_1, e_3, e_4) | product(e_1, e_1, e_4) | product(e_1, e_4, e_4))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | (~group_element(e_1)) | product(e_1, e_2, e_4) | (~group_element(e_4)) | product(e_1, e_3, e_4) | product(e_1, e_1, e_4) | product(e_1, e_4, e_4))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(306,plain,
% 0.19/0.48 (((~group_element(e_4)) | (~group_element(e_1)) | product(e_1, e_4, e_4) | product(e_1, e_3, e_4) | product(e_1, e_2, e_4) | product(e_1, e_1, e_4)) <=> ((~group_element(e_1)) | product(e_1, e_2, e_4) | (~group_element(e_4)) | product(e_1, e_3, e_4) | product(e_1, e_1, e_4) | product(e_1, e_4, e_4))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(307,plain,
% 0.19/0.48 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_4)) | (~group_element(e_1)) | product(e_1, e_4, e_4) | product(e_1, e_3, e_4) | product(e_1, e_2, e_4) | product(e_1, e_1, e_4))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_1)) | product(e_1, e_2, e_4) | (~group_element(e_4)) | product(e_1, e_3, e_4) | product(e_1, e_1, e_4) | product(e_1, e_4, e_4)))),
% 0.19/0.48 inference(monotonicity,[status(thm)],[306])).
% 0.19/0.48 tff(308,plain,
% 0.19/0.48 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_4)) | (~group_element(e_1)) | product(e_1, e_4, e_4) | product(e_1, e_3, e_4) | product(e_1, e_2, e_4) | product(e_1, e_1, e_4))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | (~group_element(e_1)) | product(e_1, e_2, e_4) | (~group_element(e_4)) | product(e_1, e_3, e_4) | product(e_1, e_1, e_4) | product(e_1, e_4, e_4))),
% 0.19/0.48 inference(transitivity,[status(thm)],[307, 305])).
% 0.19/0.48 tff(309,plain,
% 0.19/0.48 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_4)) | (~group_element(e_1)) | product(e_1, e_4, e_4) | product(e_1, e_3, e_4) | product(e_1, e_2, e_4) | product(e_1, e_1, e_4))),
% 0.19/0.48 inference(quant_inst,[status(thm)],[])).
% 0.19/0.48 tff(310,plain,
% 0.19/0.48 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | (~group_element(e_1)) | product(e_1, e_2, e_4) | (~group_element(e_4)) | product(e_1, e_3, e_4) | product(e_1, e_1, e_4) | product(e_1, e_4, e_4)),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[309, 308])).
% 0.19/0.48 tff(311,plain,
% 0.19/0.48 (product(e_1, e_2, e_4) | product(e_1, e_4, e_4)),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[310, 304, 229, 109, 276, 294])).
% 0.19/0.48 tff(312,plain,
% 0.19/0.48 (product(e_1, e_2, e_4)),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[311, 284])).
% 0.19/0.48 tff(313,plain,
% 0.19/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_4, e_3) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_2, e_4)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_4, e_3) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_2, e_4)))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(314,plain,
% 0.19/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_4, e_3) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_2, e_4)))),
% 0.19/0.48 inference(quant_inst,[status(thm)],[])).
% 0.19/0.48 tff(315,plain,
% 0.19/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_4, e_3) | (~product(e_1, e_2, e_3)) | (~product(e_1, e_2, e_4))),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[314, 313])).
% 0.19/0.48 tff(316,plain,
% 0.19/0.48 ($false),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[315, 75, 72, 312, 39])).
% 0.19/0.48 tff(317,plain,((~product(e_1, e_2, e_3)) | product(e_1, e_3, e_4)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.48 tff(318,plain,
% 0.19/0.48 (product(e_1, e_3, e_4)),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[317, 275])).
% 0.19/0.48 tff(319,assumption,(product(e_1, e_3, e_2)), introduced(assumption)).
% 0.19/0.48 tff(320,plain,
% 0.19/0.48 ((~equalish(e_4, e_2)) <=> (~equalish(e_4, e_2))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(321,axiom,(~equalish(e_4, e_2)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','e_4_is_not_e_2')).
% 0.19/0.48 tff(322,plain,
% 0.19/0.48 (~equalish(e_4, e_2)),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[321, 320])).
% 0.19/0.48 tff(323,plain,
% 0.19/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_4, e_2) | (~product(e_1, e_3, e_2)) | (~product(e_1, e_3, e_4)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_4, e_2) | (~product(e_1, e_3, e_2)) | (~product(e_1, e_3, e_4)))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(324,plain,
% 0.19/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_4, e_2) | (~product(e_1, e_3, e_2)) | (~product(e_1, e_3, e_4)))),
% 0.19/0.48 inference(quant_inst,[status(thm)],[])).
% 0.19/0.48 tff(325,plain,
% 0.19/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_4, e_2) | (~product(e_1, e_3, e_2)) | (~product(e_1, e_3, e_4))),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[324, 323])).
% 0.19/0.48 tff(326,plain,
% 0.19/0.48 ($false),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[325, 322, 72, 319, 204])).
% 0.19/0.48 tff(327,plain,((~product(e_1, e_3, e_4)) | (~product(e_1, e_3, e_2))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.48 tff(328,plain,
% 0.19/0.48 (~product(e_1, e_3, e_2)),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[327, 318])).
% 0.19/0.48 tff(329,assumption,(product(e_1, e_1, e_2)), introduced(assumption)).
% 0.19/0.48 tff(330,plain,
% 0.19/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_2, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_1, e_1)))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(331,plain,
% 0.19/0.48 ((equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_1, e_2))) <=> (equalish(e_2, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_1, e_1)))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(332,plain,
% 0.19/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_1, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_1, e_1))))),
% 0.19/0.48 inference(monotonicity,[status(thm)],[331])).
% 0.19/0.48 tff(333,plain,
% 0.19/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_1, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_2, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_1, e_1)))),
% 0.19/0.48 inference(transitivity,[status(thm)],[332, 330])).
% 0.19/0.48 tff(334,plain,
% 0.19/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_1, e_2)))),
% 0.19/0.48 inference(quant_inst,[status(thm)],[])).
% 0.19/0.48 tff(335,plain,
% 0.19/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_2, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_1, e_1))),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[334, 333])).
% 0.19/0.48 tff(336,plain,
% 0.19/0.48 ($false),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[335, 218, 72, 329, 38])).
% 0.19/0.48 tff(337,plain,(~product(e_1, e_1, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.48 tff(338,plain,
% 0.19/0.48 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_1)) | product(e_1, e_1, e_2) | (~group_element(e_2)) | product(e_1, e_2, e_2) | product(e_1, e_4, e_2) | product(e_1, e_3, e_2))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | (~group_element(e_1)) | product(e_1, e_1, e_2) | (~group_element(e_2)) | product(e_1, e_2, e_2) | product(e_1, e_4, e_2) | product(e_1, e_3, e_2))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(339,plain,
% 0.19/0.48 (((~group_element(e_2)) | (~group_element(e_1)) | product(e_1, e_4, e_2) | product(e_1, e_3, e_2) | product(e_1, e_2, e_2) | product(e_1, e_1, e_2)) <=> ((~group_element(e_1)) | product(e_1, e_1, e_2) | (~group_element(e_2)) | product(e_1, e_2, e_2) | product(e_1, e_4, e_2) | product(e_1, e_3, e_2))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(340,plain,
% 0.19/0.48 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_2)) | (~group_element(e_1)) | product(e_1, e_4, e_2) | product(e_1, e_3, e_2) | product(e_1, e_2, e_2) | product(e_1, e_1, e_2))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_1)) | product(e_1, e_1, e_2) | (~group_element(e_2)) | product(e_1, e_2, e_2) | product(e_1, e_4, e_2) | product(e_1, e_3, e_2)))),
% 0.19/0.48 inference(monotonicity,[status(thm)],[339])).
% 0.19/0.48 tff(341,plain,
% 0.19/0.48 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_2)) | (~group_element(e_1)) | product(e_1, e_4, e_2) | product(e_1, e_3, e_2) | product(e_1, e_2, e_2) | product(e_1, e_1, e_2))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | (~group_element(e_1)) | product(e_1, e_1, e_2) | (~group_element(e_2)) | product(e_1, e_2, e_2) | product(e_1, e_4, e_2) | product(e_1, e_3, e_2))),
% 0.19/0.48 inference(transitivity,[status(thm)],[340, 338])).
% 0.19/0.48 tff(342,plain,
% 0.19/0.48 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_2)) | (~group_element(e_1)) | product(e_1, e_4, e_2) | product(e_1, e_3, e_2) | product(e_1, e_2, e_2) | product(e_1, e_1, e_2))),
% 0.19/0.48 inference(quant_inst,[status(thm)],[])).
% 0.19/0.48 tff(343,plain,
% 0.19/0.48 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_4, Y) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | (~group_element(e_1)) | product(e_1, e_1, e_2) | (~group_element(e_2)) | product(e_1, e_2, e_2) | product(e_1, e_4, e_2) | product(e_1, e_3, e_2)),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[342, 341])).
% 0.19/0.48 tff(344,plain,
% 0.19/0.48 (product(e_1, e_2, e_2) | product(e_1, e_4, e_2) | product(e_1, e_3, e_2)),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[343, 304, 229, 193, 337])).
% 0.19/0.48 tff(345,plain,
% 0.19/0.48 (product(e_1, e_4, e_2)),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[344, 328, 33])).
% 0.19/0.48 tff(346,plain,
% 0.19/0.48 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_4, e_2)) | (~product(e_4, e_2, e_1)))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_4, e_2)) | (~product(e_4, e_2, e_1)))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(347,plain,
% 0.19/0.48 (((~product(e_1, e_1, e_1)) | (~product(e_1, e_4, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_4, e_2, e_1)) | equalish(e_2, e_1)) <=> (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_4, e_2)) | (~product(e_4, e_2, e_1)))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(348,plain,
% 0.19/0.48 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_4, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_4, e_2, e_1)) | equalish(e_2, e_1))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_4, e_2)) | (~product(e_4, e_2, e_1))))),
% 0.19/0.48 inference(monotonicity,[status(thm)],[347])).
% 0.19/0.48 tff(349,plain,
% 0.19/0.48 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_4, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_4, e_2, e_1)) | equalish(e_2, e_1))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_4, e_2)) | (~product(e_4, e_2, e_1)))),
% 0.19/0.48 inference(transitivity,[status(thm)],[348, 346])).
% 0.19/0.48 tff(350,plain,
% 0.19/0.48 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_4, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_4, e_2, e_1)) | equalish(e_2, e_1))),
% 0.19/0.48 inference(quant_inst,[status(thm)],[])).
% 0.19/0.48 tff(351,plain,
% 0.19/0.48 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_4, e_2)) | (~product(e_4, e_2, e_1))),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[350, 349])).
% 0.19/0.49 tff(352,plain,
% 0.19/0.49 (~product(e_4, e_2, e_1)),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[351, 218, 49, 38, 345])).
% 0.19/0.49 tff(353,assumption,(product(e_4, e_2, e_4)), introduced(assumption)).
% 0.19/0.49 tff(354,plain,
% 0.19/0.49 ((~equalish(e_2, e_4)) <=> (~equalish(e_2, e_4))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(355,axiom,(~equalish(e_2, e_4)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','e_2_is_not_e_4')).
% 0.19/0.49 tff(356,plain,
% 0.19/0.49 (~equalish(e_2, e_4)),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[355, 354])).
% 0.19/0.49 tff(357,plain,
% 0.19/0.49 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_4, e_2, e_4)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_2, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_4, e_2, e_4)))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(358,plain,
% 0.19/0.49 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_4, e_2, e_4)))),
% 0.19/0.49 inference(quant_inst,[status(thm)],[])).
% 0.19/0.49 tff(359,plain,
% 0.19/0.49 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_2, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_4, e_2, e_4))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[358, 357])).
% 0.19/0.49 tff(360,plain,
% 0.19/0.49 ($false),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[359, 356, 95, 85, 353])).
% 0.19/0.49 tff(361,plain,(~product(e_4, e_2, e_4)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.49 tff(362,plain,
% 0.19/0.49 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | product(e_4, e_2, e_3) | (~group_element(e_4)) | product(e_4, e_2, e_2) | product(e_4, e_2, e_4) | product(e_4, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_2)) | product(e_4, e_2, e_3) | (~group_element(e_4)) | product(e_4, e_2, e_2) | product(e_4, e_2, e_4) | product(e_4, e_2, e_1))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(363,plain,
% 0.19/0.49 (((~group_element(e_2)) | (~group_element(e_4)) | product(e_4, e_2, e_4) | product(e_4, e_2, e_3) | product(e_4, e_2, e_2) | product(e_4, e_2, e_1)) <=> ((~group_element(e_2)) | product(e_4, e_2, e_3) | (~group_element(e_4)) | product(e_4, e_2, e_2) | product(e_4, e_2, e_4) | product(e_4, e_2, e_1))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(364,plain,
% 0.19/0.49 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_4)) | product(e_4, e_2, e_4) | product(e_4, e_2, e_3) | product(e_4, e_2, e_2) | product(e_4, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | product(e_4, e_2, e_3) | (~group_element(e_4)) | product(e_4, e_2, e_2) | product(e_4, e_2, e_4) | product(e_4, e_2, e_1)))),
% 0.19/0.49 inference(monotonicity,[status(thm)],[363])).
% 0.19/0.49 tff(365,plain,
% 0.19/0.49 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_4)) | product(e_4, e_2, e_4) | product(e_4, e_2, e_3) | product(e_4, e_2, e_2) | product(e_4, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_2)) | product(e_4, e_2, e_3) | (~group_element(e_4)) | product(e_4, e_2, e_2) | product(e_4, e_2, e_4) | product(e_4, e_2, e_1))),
% 0.19/0.49 inference(transitivity,[status(thm)],[364, 362])).
% 0.19/0.49 tff(366,plain,
% 0.19/0.49 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_4)) | product(e_4, e_2, e_4) | product(e_4, e_2, e_3) | product(e_4, e_2, e_2) | product(e_4, e_2, e_1))),
% 0.19/0.49 inference(quant_inst,[status(thm)],[])).
% 0.19/0.49 tff(367,plain,
% 0.19/0.49 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_2)) | product(e_4, e_2, e_3) | (~group_element(e_4)) | product(e_4, e_2, e_2) | product(e_4, e_2, e_4) | product(e_4, e_2, e_1)),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[366, 365])).
% 0.19/0.49 tff(368,plain,
% 0.19/0.49 (product(e_4, e_2, e_3) | product(e_4, e_2, e_2) | product(e_4, e_2, e_1)),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[367, 193, 109, 166, 361])).
% 0.19/0.49 tff(369,plain,
% 0.19/0.49 (product(e_4, e_2, e_2)),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[368, 352, 249])).
% 0.19/0.49 tff(370,plain,
% 0.19/0.49 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_4, e_2) | (~product(e_4, e_2, e_2)) | (~product(e_2, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_4, e_2) | (~product(e_4, e_2, e_2)) | (~product(e_2, e_2, e_2)))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(371,plain,
% 0.19/0.49 ((equalish(e_4, e_2) | (~product(e_2, e_2, e_2)) | (~product(e_4, e_2, e_2))) <=> (equalish(e_4, e_2) | (~product(e_4, e_2, e_2)) | (~product(e_2, e_2, e_2)))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(372,plain,
% 0.19/0.49 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_4, e_2) | (~product(e_2, e_2, e_2)) | (~product(e_4, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_4, e_2) | (~product(e_4, e_2, e_2)) | (~product(e_2, e_2, e_2))))),
% 0.19/0.49 inference(monotonicity,[status(thm)],[371])).
% 0.19/0.49 tff(373,plain,
% 0.19/0.49 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_4, e_2) | (~product(e_2, e_2, e_2)) | (~product(e_4, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_4, e_2) | (~product(e_4, e_2, e_2)) | (~product(e_2, e_2, e_2)))),
% 0.19/0.49 inference(transitivity,[status(thm)],[372, 370])).
% 0.19/0.49 tff(374,plain,
% 0.19/0.49 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_4, e_2) | (~product(e_2, e_2, e_2)) | (~product(e_4, e_2, e_2)))),
% 0.19/0.49 inference(quant_inst,[status(thm)],[])).
% 0.19/0.49 tff(375,plain,
% 0.19/0.49 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_4, e_2) | (~product(e_4, e_2, e_2)) | (~product(e_2, e_2, e_2))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[374, 373])).
% 0.19/0.49 tff(376,plain,
% 0.19/0.49 ($false),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[375, 322, 22, 369, 11])).
% 0.19/0.49 % SZS output end Proof
% 0.19/0.49 % E exiting
%------------------------------------------------------------------------------