%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP127-1.004 : TPTP v9.0.0. Released v1.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.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 05:07:08 AM UTC 2025
% Result : Unsatisfiable 0.22s 0.43s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : GRP127-1.004 : TPTP v9.0.0. Released v1.2.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.13/0.35 % Computer : n015.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Thu Jun 19 15:43:10 EDT 2025
% 0.13/0.35 % CPUTime :
% 0.22/0.43 % SZS status Unsatisfiable
% 0.22/0.43 % SZS output start Proof
% 0.22/0.43 tff(product_type, type, (
% 0.22/0.43 product: ( $i * $i * $i ) > $o)).
% 0.22/0.43 tff(e_2_type, type, (
% 0.22/0.43 e_2: $i)).
% 0.22/0.43 tff(e_4_type, type, (
% 0.22/0.43 e_4: $i)).
% 0.22/0.43 tff(e_3_type, type, (
% 0.22/0.43 e_3: $i)).
% 0.22/0.43 tff(e_1_type, type, (
% 0.22/0.43 e_1: $i)).
% 0.22/0.43 tff(equalish_type, type, (
% 0.22/0.43 equalish: ( $i * $i ) > $o)).
% 0.22/0.43 tff(group_element_type, type, (
% 0.22/0.43 group_element: $i > $o)).
% 0.22/0.43 tff(1,plain,
% 0.22/0.43 (^[X: $i] : refl(product(X, X, X) <=> product(X, X, X))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(2,plain,
% 0.22/0.43 (![X: $i] : product(X, X, X) <=> ![X: $i] : product(X, X, X)),
% 0.22/0.43 inference(quant_intro,[status(thm)],[1])).
% 0.22/0.43 tff(3,plain,
% 0.22/0.43 (![X: $i] : product(X, X, X) <=> ![X: $i] : product(X, X, X)),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(4,axiom,(![X: $i] : product(X, X, X)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_idempotence')).
% 0.22/0.43 tff(5,plain,
% 0.22/0.43 (![X: $i] : product(X, X, X)),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.22/0.43 tff(6,plain,(
% 0.22/0.43 ![X: $i] : product(X, X, X)),
% 0.22/0.43 inference(skolemize,[status(sab)],[5])).
% 0.22/0.43 tff(7,plain,
% 0.22/0.43 (![X: $i] : product(X, X, X)),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.22/0.43 tff(8,plain,
% 0.22/0.43 ((~![X: $i] : product(X, X, X)) | product(e_4, e_4, e_4)),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(9,plain,
% 0.22/0.43 (product(e_4, e_4, e_4)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.22/0.43 tff(10,assumption,(product(e_4, e_3, e_4)), introduced(assumption)).
% 0.22/0.43 tff(11,plain,
% 0.22/0.43 (^[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))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(12,plain,
% 0.22/0.43 (![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)))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[11])).
% 0.22/0.43 tff(13,plain,
% 0.22/0.43 (![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)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(14,plain,
% 0.22/0.43 (^[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)))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(15,plain,
% 0.22/0.43 (![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)))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[14])).
% 0.22/0.43 tff(16,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_right_cancellation')).
% 0.22/0.43 tff(17,plain,
% 0.22/0.43 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.22/0.43 tff(18,plain,
% 0.22/0.43 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.22/0.43 tff(19,plain,(
% 0.22/0.43 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.22/0.43 inference(skolemize,[status(sab)],[18])).
% 0.22/0.43 tff(20,plain,
% 0.22/0.43 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[19, 12])).
% 0.22/0.43 tff(21,plain,
% 0.22/0.43 ((~equalish(e_3, e_4)) <=> (~equalish(e_3, e_4))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(22,axiom,(~equalish(e_3, e_4)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_3_is_not_e_4')).
% 0.22/0.43 tff(23,plain,
% 0.22/0.43 (~equalish(e_3, e_4)),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.22/0.43 tff(24,plain,
% 0.22/0.43 (((~![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_4, e_4)) | (~product(e_4, e_3, e_4)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(25,plain,
% 0.22/0.43 ((~![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)))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(26,plain,
% 0.22/0.43 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_4, e_3, e_4))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[25, 24])).
% 0.22/0.43 tff(27,plain,
% 0.22/0.43 ($false),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[26, 23, 20, 10, 9])).
% 0.22/0.43 tff(28,plain,(~product(e_4, e_3, e_4)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.43 tff(29,assumption,(product(e_1, e_3, e_2)), introduced(assumption)).
% 0.22/0.43 tff(30,assumption,(product(e_1, e_4, e_2)), introduced(assumption)).
% 0.22/0.43 tff(31,plain,
% 0.22/0.43 ((~equalish(e_4, e_3)) <=> (~equalish(e_4, e_3))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(32,axiom,(~equalish(e_4, e_3)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_4_is_not_e_3')).
% 0.22/0.43 tff(33,plain,
% 0.22/0.43 (~equalish(e_4, e_3)),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.22/0.43 tff(34,plain,
% 0.22/0.43 (((~![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_1, e_3, e_2)) | (~product(e_1, e_4, e_2)))) <=> ((~![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_1, e_3, e_2)) | (~product(e_1, e_4, e_2)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(35,plain,
% 0.22/0.43 ((~![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_1, e_3, e_2)) | (~product(e_1, e_4, e_2)))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(36,plain,
% 0.22/0.43 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_4, e_3) | (~product(e_1, e_3, e_2)) | (~product(e_1, e_4, e_2))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.22/0.43 tff(37,plain,
% 0.22/0.43 ($false),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[36, 33, 20, 29, 30])).
% 0.22/0.43 tff(38,plain,((~product(e_1, e_4, e_2)) | (~product(e_1, e_3, e_2))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.43 tff(39,plain,
% 0.22/0.43 (~product(e_1, e_4, e_2)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[38, 29])).
% 0.22/0.43 tff(40,assumption,(product(e_1, e_4, e_1)), introduced(assumption)).
% 0.22/0.43 tff(41,plain,
% 0.22/0.43 ((~![X: $i] : product(X, X, X)) | product(e_1, e_1, e_1)),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(42,plain,
% 0.22/0.43 (product(e_1, e_1, e_1)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[41, 7])).
% 0.22/0.43 tff(43,plain,
% 0.22/0.43 ((~equalish(e_4, e_1)) <=> (~equalish(e_4, e_1))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(44,axiom,(~equalish(e_4, e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_4_is_not_e_1')).
% 0.22/0.43 tff(45,plain,
% 0.22/0.43 (~equalish(e_4, e_1)),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.22/0.43 tff(46,plain,
% 0.22/0.43 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_4, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_4, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_4, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_4, e_1)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(47,plain,
% 0.22/0.43 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_4, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_4, e_1)))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(48,plain,
% 0.22/0.43 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_4, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_4, e_1))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.22/0.43 tff(49,plain,
% 0.22/0.43 ($false),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[48, 45, 20, 42, 40])).
% 0.22/0.43 tff(50,plain,(~product(e_1, e_4, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.43 tff(51,assumption,(product(e_1, e_4, e_4)), introduced(assumption)).
% 0.22/0.43 tff(52,plain,
% 0.22/0.43 (^[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))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(53,plain,
% 0.22/0.43 (![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)))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[52])).
% 0.22/0.43 tff(54,plain,
% 0.22/0.43 (![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)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(55,plain,
% 0.22/0.43 (^[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)))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(56,plain,
% 0.22/0.43 (![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)))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[55])).
% 0.22/0.43 tff(57,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(W, Y, X)) | (~product(Z, Y, X))) | equalish(W, Z))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_left_cancellation')).
% 0.22/0.43 tff(58,plain,
% 0.22/0.43 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.22/0.43 tff(59,plain,
% 0.22/0.43 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[58, 54])).
% 0.22/0.43 tff(60,plain,(
% 0.22/0.43 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.22/0.43 inference(skolemize,[status(sab)],[59])).
% 0.22/0.43 tff(61,plain,
% 0.22/0.43 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[60, 53])).
% 0.22/0.43 tff(62,plain,
% 0.22/0.43 ((~equalish(e_1, e_4)) <=> (~equalish(e_1, e_4))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(63,axiom,(~equalish(e_1, e_4)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_1_is_not_e_4')).
% 0.22/0.43 tff(64,plain,
% 0.22/0.43 (~equalish(e_1, e_4)),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[63, 62])).
% 0.22/0.43 tff(65,plain,
% 0.22/0.43 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_1, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_1, 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_1, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_1, e_4, e_4)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(66,plain,
% 0.22/0.43 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_1, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_1, e_4, e_4)))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(67,plain,
% 0.22/0.43 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_1, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_1, e_4, e_4))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.22/0.44 tff(68,plain,
% 0.22/0.44 ($false),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[67, 64, 61, 9, 51])).
% 0.22/0.44 tff(69,plain,(~product(e_1, e_4, e_4)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.44 tff(70,plain,
% 0.22/0.44 (^[Y: $i, X: $i] : refl((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) | (~group_element(Y)) | (~group_element(X))))),
% 0.22/0.44 inference(bind,[status(th)],[])).
% 0.22/0.44 tff(71,plain,
% 0.22/0.44 (![Y: $i, X: $i] : (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))) <=> ![Y: $i, X: $i] : (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)))),
% 0.22/0.44 inference(quant_intro,[status(thm)],[70])).
% 0.22/0.44 tff(72,plain,
% 0.22/0.44 (![Y: $i, X: $i] : (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))) <=> ![Y: $i, X: $i] : (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)))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(73,plain,
% 0.22/0.44 (^[Y: $i, X: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) <=> (product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))), (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) <=> ((product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))) | product(X, Y, e_2)))), rewrite(((product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))) | product(X, Y, e_2)) <=> (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))), (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) <=> (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3)) <=> ((product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))) | product(X, Y, e_3)))), rewrite(((product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))) | product(X, Y, e_3)) <=> (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3)) <=> (product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))))), (((((((~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)) <=> ((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)))), rewrite(((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_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))), (((((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3)) | product(X, Y, e_4)) <=> (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)))))),
% 0.22/0.44 inference(bind,[status(th)],[])).
% 0.22/0.44 tff(74,plain,
% 0.22/0.44 (![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] : (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)))),
% 0.22/0.44 inference(quant_intro,[status(thm)],[73])).
% 0.22/0.44 tff(75,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/sandbox2/benchmark/theBenchmark.p','product_total_function1')).
% 0.22/0.44 tff(76,plain,
% 0.22/0.44 (![Y: $i, X: $i] : (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)))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.22/0.44 tff(77,plain,
% 0.22/0.44 (![Y: $i, X: $i] : (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)))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[76, 72])).
% 0.22/0.44 tff(78,plain,(
% 0.22/0.44 ![Y: $i, X: $i] : (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)))),
% 0.22/0.44 inference(skolemize,[status(sab)],[77])).
% 0.22/0.44 tff(79,plain,
% 0.22/0.44 (![Y: $i, X: $i] : (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)))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[78, 71])).
% 0.22/0.44 tff(80,plain,
% 0.22/0.44 (group_element(e_4) <=> group_element(e_4)),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(81,axiom,(group_element(e_4)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_4')).
% 0.22/0.44 tff(82,plain,
% 0.22/0.44 (group_element(e_4)),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[81, 80])).
% 0.22/0.44 tff(83,plain,
% 0.22/0.44 (group_element(e_1) <=> group_element(e_1)),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(84,axiom,(group_element(e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_1')).
% 0.22/0.44 tff(85,plain,
% 0.22/0.44 (group_element(e_1)),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.22/0.44 tff(86,plain,
% 0.22/0.44 (((~![Y: $i, X: $i] : (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)))) | ((~group_element(e_1)) | (~group_element(e_4)) | product(e_1, e_4, e_3) | product(e_1, e_4, e_1) | product(e_1, e_4, e_4) | product(e_1, e_4, e_2))) <=> ((~![Y: $i, X: $i] : (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)))) | (~group_element(e_1)) | (~group_element(e_4)) | product(e_1, e_4, e_3) | product(e_1, e_4, e_1) | product(e_1, e_4, e_4) | product(e_1, e_4, e_2))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(87,plain,
% 0.22/0.44 ((product(e_1, e_4, e_4) | product(e_1, e_4, e_3) | product(e_1, e_4, e_2) | product(e_1, e_4, e_1) | (~group_element(e_4)) | (~group_element(e_1))) <=> ((~group_element(e_1)) | (~group_element(e_4)) | product(e_1, e_4, e_3) | product(e_1, e_4, e_1) | product(e_1, e_4, e_4) | product(e_1, e_4, e_2))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(88,plain,
% 0.22/0.44 (((~![Y: $i, X: $i] : (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(e_1, e_4, e_4) | product(e_1, e_4, e_3) | product(e_1, e_4, e_2) | product(e_1, e_4, e_1) | (~group_element(e_4)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (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)))) | ((~group_element(e_1)) | (~group_element(e_4)) | product(e_1, e_4, e_3) | product(e_1, e_4, e_1) | product(e_1, e_4, e_4) | product(e_1, e_4, e_2)))),
% 0.22/0.44 inference(monotonicity,[status(thm)],[87])).
% 0.22/0.44 tff(89,plain,
% 0.22/0.44 (((~![Y: $i, X: $i] : (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(e_1, e_4, e_4) | product(e_1, e_4, e_3) | product(e_1, e_4, e_2) | product(e_1, e_4, e_1) | (~group_element(e_4)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (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)))) | (~group_element(e_1)) | (~group_element(e_4)) | product(e_1, e_4, e_3) | product(e_1, e_4, e_1) | product(e_1, e_4, e_4) | product(e_1, e_4, e_2))),
% 0.22/0.44 inference(transitivity,[status(thm)],[88, 86])).
% 0.22/0.44 tff(90,plain,
% 0.22/0.44 ((~![Y: $i, X: $i] : (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(e_1, e_4, e_4) | product(e_1, e_4, e_3) | product(e_1, e_4, e_2) | product(e_1, e_4, e_1) | (~group_element(e_4)) | (~group_element(e_1)))),
% 0.22/0.44 inference(quant_inst,[status(thm)],[])).
% 0.22/0.44 tff(91,plain,
% 0.22/0.44 ((~![Y: $i, X: $i] : (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)))) | (~group_element(e_1)) | (~group_element(e_4)) | product(e_1, e_4, e_3) | product(e_1, e_4, e_1) | product(e_1, e_4, e_4) | product(e_1, e_4, e_2)),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[90, 89])).
% 0.22/0.44 tff(92,plain,
% 0.22/0.44 (product(e_1, e_4, e_3) | product(e_1, e_4, e_1) | product(e_1, e_4, e_4) | product(e_1, e_4, e_2)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[91, 85, 82, 79])).
% 0.22/0.44 tff(93,plain,
% 0.22/0.44 (product(e_1, e_4, e_3) | product(e_1, e_4, e_2)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[92, 69, 50])).
% 0.22/0.44 tff(94,plain,
% 0.22/0.44 (product(e_1, e_4, e_3)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[93, 39])).
% 0.22/0.44 tff(95,assumption,(product(e_4, e_3, e_1)), introduced(assumption)).
% 0.22/0.44 tff(96,plain,
% 0.22/0.44 ((~![X: $i] : product(X, X, X)) | product(e_3, e_3, e_3)),
% 0.22/0.44 inference(quant_inst,[status(thm)],[])).
% 0.22/0.44 tff(97,plain,
% 0.22/0.44 (product(e_3, e_3, e_3)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[96, 7])).
% 0.22/0.44 tff(98,assumption,(product(e_1, e_3, e_3)), introduced(assumption)).
% 0.22/0.44 tff(99,plain,
% 0.22/0.44 ((~equalish(e_1, e_3)) <=> (~equalish(e_1, e_3))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(100,axiom,(~equalish(e_1, e_3)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_1_is_not_e_3')).
% 0.22/0.44 tff(101,plain,
% 0.22/0.44 (~equalish(e_1, e_3)),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[100, 99])).
% 0.22/0.44 tff(102,plain,
% 0.22/0.44 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_1, e_3) | (~product(e_1, e_3, e_3)) | (~product(e_3, e_3, e_3)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_1, e_3) | (~product(e_1, e_3, e_3)) | (~product(e_3, e_3, e_3)))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(103,plain,
% 0.22/0.44 ((equalish(e_1, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_1, e_3, e_3))) <=> (equalish(e_1, e_3) | (~product(e_1, e_3, e_3)) | (~product(e_3, e_3, e_3)))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(104,plain,
% 0.22/0.44 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_1, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_1, 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_1, e_3) | (~product(e_1, e_3, e_3)) | (~product(e_3, e_3, e_3))))),
% 0.22/0.44 inference(monotonicity,[status(thm)],[103])).
% 0.22/0.44 tff(105,plain,
% 0.22/0.44 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_1, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_1, 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_1, e_3) | (~product(e_1, e_3, e_3)) | (~product(e_3, e_3, e_3)))),
% 0.22/0.44 inference(transitivity,[status(thm)],[104, 102])).
% 0.22/0.44 tff(106,plain,
% 0.22/0.44 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_1, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_1, e_3, e_3)))),
% 0.22/0.44 inference(quant_inst,[status(thm)],[])).
% 0.22/0.44 tff(107,plain,
% 0.22/0.44 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_1, e_3) | (~product(e_1, e_3, e_3)) | (~product(e_3, e_3, e_3))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[106, 105])).
% 0.22/0.44 tff(108,plain,
% 0.22/0.44 ($false),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[107, 101, 61, 98, 97])).
% 0.22/0.44 tff(109,plain,(~product(e_1, e_3, e_3)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.44 tff(110,plain,
% 0.22/0.44 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : refl((product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1))) <=> (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1))))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(111,plain,
% 0.22/0.45 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[110])).
% 0.22/0.45 tff(112,plain,
% 0.22/0.45 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(113,plain,
% 0.22/0.45 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : trans(monotonicity(rewrite(((~product(Y, X, Z1)) | (~product(Z1, Y, Z2))) <=> ((~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))), ((((~product(Y, X, Z1)) | (~product(Z1, Y, Z2))) | product(Z2, Y, X)) <=> (((~product(Z1, Y, Z2)) | (~product(Y, X, Z1))) | product(Z2, Y, X)))), rewrite((((~product(Z1, Y, Z2)) | (~product(Y, X, Z1))) | product(Z2, Y, X)) <=> (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))), ((((~product(Y, X, Z1)) | (~product(Z1, Y, Z2))) | product(Z2, Y, X)) <=> (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(114,plain,
% 0.22/0.45 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (((~product(Y, X, Z1)) | (~product(Z1, Y, Z2))) | product(Z2, Y, X)) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[113])).
% 0.22/0.45 tff(115,axiom,(![Y: $i, Z1: $i, X: $i, Z2: $i] : (((~product(Y, X, Z1)) | (~product(Z1, Y, Z2))) | product(Z2, Y, X))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','qg3')).
% 0.22/0.45 tff(116,plain,
% 0.22/0.45 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.22/0.45 tff(117,plain,
% 0.22/0.45 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[116, 112])).
% 0.22/0.45 tff(118,plain,(
% 0.22/0.45 ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))),
% 0.22/0.45 inference(skolemize,[status(sab)],[117])).
% 0.22/0.45 tff(119,plain,
% 0.22/0.45 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[118, 111])).
% 0.22/0.45 tff(120,plain,
% 0.22/0.45 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_1, e_3, e_3) | (~product(e_4, e_3, e_1)) | (~product(e_3, e_3, e_4)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | product(e_1, e_3, e_3) | (~product(e_4, e_3, e_1)) | (~product(e_3, e_3, e_4)))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(121,plain,
% 0.22/0.45 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_1, e_3, e_3) | (~product(e_4, e_3, e_1)) | (~product(e_3, e_3, e_4)))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(122,plain,
% 0.22/0.45 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | product(e_1, e_3, e_3) | (~product(e_4, e_3, e_1)) | (~product(e_3, e_3, e_4))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[121, 120])).
% 0.22/0.45 tff(123,plain,
% 0.22/0.45 (~product(e_3, e_3, e_4)),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[122, 119, 109, 95])).
% 0.22/0.45 tff(124,plain,
% 0.22/0.45 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_3, e_3, e_4) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_4, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | product(e_3, e_3, e_4) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_4, e_3)))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(125,plain,
% 0.22/0.45 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_3, e_3, e_4) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_4, e_3)))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(126,plain,
% 0.22/0.45 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | product(e_3, e_3, e_4) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_4, e_3))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[125, 124])).
% 0.22/0.45 tff(127,plain,
% 0.22/0.45 (~product(e_3, e_4, e_3)),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[126, 119, 97, 123])).
% 0.22/0.45 tff(128,assumption,(product(e_1, e_4, e_3)), introduced(assumption)).
% 0.22/0.45 tff(129,plain,
% 0.22/0.45 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | ((~product(e_4, e_3, e_1)) | product(e_3, e_4, e_3) | (~product(e_1, e_4, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_4, e_3, e_1)) | product(e_3, e_4, e_3) | (~product(e_1, e_4, e_3)))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(130,plain,
% 0.22/0.45 ((product(e_3, e_4, e_3) | (~product(e_1, e_4, e_3)) | (~product(e_4, e_3, e_1))) <=> ((~product(e_4, e_3, e_1)) | product(e_3, e_4, e_3) | (~product(e_1, e_4, e_3)))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(131,plain,
% 0.22/0.45 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_3, e_4, e_3) | (~product(e_1, e_4, e_3)) | (~product(e_4, e_3, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | ((~product(e_4, e_3, e_1)) | product(e_3, e_4, e_3) | (~product(e_1, e_4, e_3))))),
% 0.22/0.45 inference(monotonicity,[status(thm)],[130])).
% 0.22/0.45 tff(132,plain,
% 0.22/0.45 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_3, e_4, e_3) | (~product(e_1, e_4, e_3)) | (~product(e_4, e_3, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_4, e_3, e_1)) | product(e_3, e_4, e_3) | (~product(e_1, e_4, e_3)))),
% 0.22/0.45 inference(transitivity,[status(thm)],[131, 129])).
% 0.22/0.45 tff(133,plain,
% 0.22/0.45 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_3, e_4, e_3) | (~product(e_1, e_4, e_3)) | (~product(e_4, e_3, e_1)))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(134,plain,
% 0.22/0.45 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_4, e_3, e_1)) | product(e_3, e_4, e_3) | (~product(e_1, e_4, e_3))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[133, 132])).
% 0.22/0.45 tff(135,plain,
% 0.22/0.45 ($false),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[134, 119, 95, 128, 127])).
% 0.22/0.45 tff(136,plain,((~product(e_4, e_3, e_1)) | (~product(e_1, e_4, e_3))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.45 tff(137,plain,
% 0.22/0.45 (~product(e_4, e_3, e_1)),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[136, 94])).
% 0.22/0.45 tff(138,assumption,(product(e_4, e_3, e_3)), introduced(assumption)).
% 0.22/0.45 tff(139,plain,
% 0.22/0.45 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_4, e_3) | (~product(e_4, e_3, e_3)) | (~product(e_3, e_3, e_3)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_4, e_3) | (~product(e_4, e_3, e_3)) | (~product(e_3, e_3, e_3)))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(140,plain,
% 0.22/0.45 ((equalish(e_4, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_4, e_3, e_3))) <=> (equalish(e_4, e_3) | (~product(e_4, e_3, e_3)) | (~product(e_3, e_3, e_3)))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(141,plain,
% 0.22/0.45 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_4, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_4, 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_4, e_3) | (~product(e_4, e_3, e_3)) | (~product(e_3, e_3, e_3))))),
% 0.22/0.46 inference(monotonicity,[status(thm)],[140])).
% 0.22/0.46 tff(142,plain,
% 0.22/0.46 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_4, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_4, 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_4, e_3) | (~product(e_4, e_3, e_3)) | (~product(e_3, e_3, e_3)))),
% 0.22/0.46 inference(transitivity,[status(thm)],[141, 139])).
% 0.22/0.46 tff(143,plain,
% 0.22/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_4, e_3) | (~product(e_3, e_3, e_3)) | (~product(e_4, e_3, e_3)))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(144,plain,
% 0.22/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_4, e_3) | (~product(e_4, e_3, e_3)) | (~product(e_3, e_3, e_3))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[143, 142])).
% 0.22/0.46 tff(145,plain,
% 0.22/0.46 ($false),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[144, 33, 61, 138, 97])).
% 0.22/0.46 tff(146,plain,(~product(e_4, e_3, e_3)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.46 tff(147,plain,
% 0.22/0.46 (group_element(e_3) <=> group_element(e_3)),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(148,axiom,(group_element(e_3)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_3')).
% 0.22/0.46 tff(149,plain,
% 0.22/0.46 (group_element(e_3)),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[148, 147])).
% 0.22/0.46 tff(150,plain,
% 0.22/0.46 (((~![Y: $i, X: $i] : (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)))) | ((~group_element(e_4)) | product(e_4, e_3, e_1) | product(e_4, e_3, e_2) | (~group_element(e_3)) | product(e_4, e_3, e_3) | product(e_4, e_3, e_4))) <=> ((~![Y: $i, X: $i] : (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)))) | (~group_element(e_4)) | product(e_4, e_3, e_1) | product(e_4, e_3, e_2) | (~group_element(e_3)) | product(e_4, e_3, e_3) | product(e_4, e_3, e_4))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(151,plain,
% 0.22/0.46 ((product(e_4, e_3, e_4) | product(e_4, e_3, e_3) | product(e_4, e_3, e_2) | product(e_4, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_4))) <=> ((~group_element(e_4)) | product(e_4, e_3, e_1) | product(e_4, e_3, e_2) | (~group_element(e_3)) | product(e_4, e_3, e_3) | product(e_4, e_3, e_4))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(152,plain,
% 0.22/0.46 (((~![Y: $i, X: $i] : (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(e_4, e_3, e_4) | product(e_4, e_3, e_3) | product(e_4, e_3, e_2) | product(e_4, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_4)))) <=> ((~![Y: $i, X: $i] : (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)))) | ((~group_element(e_4)) | product(e_4, e_3, e_1) | product(e_4, e_3, e_2) | (~group_element(e_3)) | product(e_4, e_3, e_3) | product(e_4, e_3, e_4)))),
% 0.22/0.46 inference(monotonicity,[status(thm)],[151])).
% 0.22/0.46 tff(153,plain,
% 0.22/0.46 (((~![Y: $i, X: $i] : (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(e_4, e_3, e_4) | product(e_4, e_3, e_3) | product(e_4, e_3, e_2) | product(e_4, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_4)))) <=> ((~![Y: $i, X: $i] : (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)))) | (~group_element(e_4)) | product(e_4, e_3, e_1) | product(e_4, e_3, e_2) | (~group_element(e_3)) | product(e_4, e_3, e_3) | product(e_4, e_3, e_4))),
% 0.22/0.46 inference(transitivity,[status(thm)],[152, 150])).
% 0.22/0.46 tff(154,plain,
% 0.22/0.46 ((~![Y: $i, X: $i] : (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(e_4, e_3, e_4) | product(e_4, e_3, e_3) | product(e_4, e_3, e_2) | product(e_4, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_4)))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(155,plain,
% 0.22/0.46 ((~![Y: $i, X: $i] : (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)))) | (~group_element(e_4)) | product(e_4, e_3, e_1) | product(e_4, e_3, e_2) | (~group_element(e_3)) | product(e_4, e_3, e_3) | product(e_4, e_3, e_4)),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[154, 153])).
% 0.22/0.46 tff(156,plain,
% 0.22/0.46 (product(e_4, e_3, e_1) | product(e_4, e_3, e_2) | product(e_4, e_3, e_3) | product(e_4, e_3, e_4)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[155, 149, 82, 79])).
% 0.22/0.46 tff(157,plain,
% 0.22/0.46 (product(e_4, e_3, e_1) | product(e_4, e_3, e_2) | product(e_4, e_3, e_3)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[156, 28])).
% 0.22/0.46 tff(158,plain,
% 0.22/0.46 (product(e_4, e_3, e_1) | product(e_4, e_3, e_2)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[157, 146])).
% 0.22/0.46 tff(159,plain,
% 0.22/0.46 (product(e_4, e_3, e_2)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[158, 137])).
% 0.22/0.46 tff(160,plain,
% 0.22/0.46 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_1, e_4) | (~product(e_4, e_3, e_2)) | (~product(e_1, e_3, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_1, e_4) | (~product(e_4, e_3, e_2)) | (~product(e_1, e_3, e_2)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(161,plain,
% 0.22/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_1, e_4) | (~product(e_4, e_3, e_2)) | (~product(e_1, e_3, e_2)))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(162,plain,
% 0.22/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_1, e_4) | (~product(e_4, e_3, e_2)) | (~product(e_1, e_3, e_2))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[161, 160])).
% 0.22/0.46 tff(163,plain,
% 0.22/0.46 ($false),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[162, 64, 61, 29, 159])).
% 0.22/0.46 tff(164,plain,(~product(e_1, e_3, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.46 tff(165,assumption,(product(e_1, e_3, e_1)), introduced(assumption)).
% 0.22/0.46 tff(166,plain,
% 0.22/0.46 ((~equalish(e_3, e_1)) <=> (~equalish(e_3, e_1))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(167,axiom,(~equalish(e_3, e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_3_is_not_e_1')).
% 0.22/0.46 tff(168,plain,
% 0.22/0.46 (~equalish(e_3, e_1)),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[167, 166])).
% 0.22/0.46 tff(169,plain,
% 0.22/0.46 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_3, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_3, e_1)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(170,plain,
% 0.22/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_3, e_1)))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(171,plain,
% 0.22/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_3, e_1))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[170, 169])).
% 0.22/0.46 tff(172,plain,
% 0.22/0.46 ($false),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[171, 168, 20, 165, 42])).
% 0.22/0.46 tff(173,plain,(~product(e_1, e_3, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.46 tff(174,plain,
% 0.22/0.46 (((~![Y: $i, X: $i] : (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(e_1, e_3, e_4) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | product(e_1, e_3, e_3) | (~group_element(e_1)) | (~group_element(e_3)))) <=> ((~![Y: $i, X: $i] : (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(e_1, e_3, e_4) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | product(e_1, e_3, e_3) | (~group_element(e_1)) | (~group_element(e_3)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(175,plain,
% 0.22/0.46 ((product(e_1, e_3, e_4) | product(e_1, e_3, e_3) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_1))) <=> (product(e_1, e_3, e_4) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | product(e_1, e_3, e_3) | (~group_element(e_1)) | (~group_element(e_3)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(176,plain,
% 0.22/0.46 (((~![Y: $i, X: $i] : (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(e_1, e_3, e_4) | product(e_1, e_3, e_3) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_3, e_4) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | product(e_1, e_3, e_3) | (~group_element(e_1)) | (~group_element(e_3))))),
% 0.22/0.46 inference(monotonicity,[status(thm)],[175])).
% 0.22/0.46 tff(177,plain,
% 0.22/0.46 (((~![Y: $i, X: $i] : (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(e_1, e_3, e_4) | product(e_1, e_3, e_3) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_4) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_1, e_3, e_4) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | product(e_1, e_3, e_3) | (~group_element(e_1)) | (~group_element(e_3)))),
% 0.22/0.46 inference(transitivity,[status(thm)],[176, 174])).
% 0.22/0.46 tff(178,plain,
% 0.22/0.46 ((~![Y: $i, X: $i] : (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(e_1, e_3, e_4) | product(e_1, e_3, e_3) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | (~group_element(e_3)) | (~group_element(e_1)))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(179,plain,
% 0.22/0.46 ((~![Y: $i, X: $i] : (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(e_1, e_3, e_4) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | product(e_1, e_3, e_3) | (~group_element(e_1)) | (~group_element(e_3))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[178, 177])).
% 0.22/0.46 tff(180,plain,
% 0.22/0.46 (product(e_1, e_3, e_4) | product(e_1, e_3, e_2) | product(e_1, e_3, e_1) | product(e_1, e_3, e_3)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[179, 85, 149, 79])).
% 0.22/0.46 tff(181,plain,
% 0.22/0.46 (product(e_1, e_3, e_4) | product(e_1, e_3, e_2)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[180, 173, 109])).
% 0.22/0.46 tff(182,plain,
% 0.22/0.46 (product(e_1, e_3, e_4)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[181, 164])).
% 0.22/0.46 tff(183,plain,
% 0.22/0.46 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_4, e_3, e_4) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | product(e_4, e_3, e_4) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_1)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(184,plain,
% 0.22/0.46 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_4, e_3, e_4) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_1)))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(185,plain,
% 0.22/0.46 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | product(e_4, e_3, e_4) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_4, e_1))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[184, 183])).
% 0.22/0.47 tff(186,plain,
% 0.22/0.47 (~product(e_3, e_4, e_1)),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[185, 119, 182, 28])).
% 0.22/0.47 tff(187,plain,
% 0.22/0.47 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_4, e_3, e_3) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_3, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | product(e_4, e_3, e_3) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_3, e_1)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(188,plain,
% 0.22/0.47 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_4, e_3, e_3) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_3, e_1)))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(189,plain,
% 0.22/0.47 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | product(e_4, e_3, e_3) | (~product(e_1, e_3, e_4)) | (~product(e_3, e_3, e_1))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[188, 187])).
% 0.22/0.47 tff(190,plain,
% 0.22/0.47 (~product(e_3, e_3, e_1)),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[189, 119, 182, 146])).
% 0.22/0.47 tff(191,plain,
% 0.22/0.47 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | ((~product(e_3, e_1, e_3)) | product(e_3, e_3, e_1) | (~product(e_3, e_3, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_3, e_1, e_3)) | product(e_3, e_3, e_1) | (~product(e_3, e_3, e_3)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(192,plain,
% 0.22/0.47 ((product(e_3, e_3, e_1) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_1, e_3))) <=> ((~product(e_3, e_1, e_3)) | product(e_3, e_3, e_1) | (~product(e_3, e_3, e_3)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(193,plain,
% 0.22/0.47 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_3, e_3, e_1) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_1, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | ((~product(e_3, e_1, e_3)) | product(e_3, e_3, e_1) | (~product(e_3, e_3, e_3))))),
% 0.22/0.47 inference(monotonicity,[status(thm)],[192])).
% 0.22/0.47 tff(194,plain,
% 0.22/0.47 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_3, e_3, e_1) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_1, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_3, e_1, e_3)) | product(e_3, e_3, e_1) | (~product(e_3, e_3, e_3)))),
% 0.22/0.47 inference(transitivity,[status(thm)],[193, 191])).
% 0.22/0.47 tff(195,plain,
% 0.22/0.47 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_3, e_3, e_1) | (~product(e_3, e_3, e_3)) | (~product(e_3, e_1, e_3)))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(196,plain,
% 0.22/0.47 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_3, e_1, e_3)) | product(e_3, e_3, e_1) | (~product(e_3, e_3, e_3))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[195, 194])).
% 0.22/0.47 tff(197,plain,
% 0.22/0.47 (~product(e_3, e_1, e_3)),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[196, 119, 97, 190])).
% 0.22/0.47 tff(198,assumption,(product(e_1, e_3, e_4)), introduced(assumption)).
% 0.22/0.47 tff(199,assumption,(product(e_4, e_1, e_3)), introduced(assumption)).
% 0.22/0.47 tff(200,assumption,(~product(e_3, e_1, e_3)), introduced(assumption)).
% 0.22/0.47 tff(201,plain,
% 0.22/0.47 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | ((~product(e_4, e_1, e_3)) | product(e_3, e_1, e_3) | (~product(e_1, e_3, e_4)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_4, e_1, e_3)) | product(e_3, e_1, e_3) | (~product(e_1, e_3, e_4)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(202,plain,
% 0.22/0.47 ((product(e_3, e_1, e_3) | (~product(e_4, e_1, e_3)) | (~product(e_1, e_3, e_4))) <=> ((~product(e_4, e_1, e_3)) | product(e_3, e_1, e_3) | (~product(e_1, e_3, e_4)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(203,plain,
% 0.22/0.47 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_3, e_1, e_3) | (~product(e_4, e_1, e_3)) | (~product(e_1, e_3, e_4)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | ((~product(e_4, e_1, e_3)) | product(e_3, e_1, e_3) | (~product(e_1, e_3, e_4))))),
% 0.22/0.47 inference(monotonicity,[status(thm)],[202])).
% 0.22/0.47 tff(204,plain,
% 0.22/0.47 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_3, e_1, e_3) | (~product(e_4, e_1, e_3)) | (~product(e_1, e_3, e_4)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_4, e_1, e_3)) | product(e_3, e_1, e_3) | (~product(e_1, e_3, e_4)))),
% 0.22/0.47 inference(transitivity,[status(thm)],[203, 201])).
% 0.22/0.47 tff(205,plain,
% 0.22/0.47 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_3, e_1, e_3) | (~product(e_4, e_1, e_3)) | (~product(e_1, e_3, e_4)))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(206,plain,
% 0.22/0.47 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_4, e_1, e_3)) | product(e_3, e_1, e_3) | (~product(e_1, e_3, e_4))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[205, 204])).
% 0.22/0.47 tff(207,plain,
% 0.22/0.47 ($false),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[206, 119, 200, 199, 198])).
% 0.22/0.47 tff(208,plain,((~product(e_4, e_1, e_3)) | product(e_3, e_1, e_3) | (~product(e_1, e_3, e_4))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.47 tff(209,plain,
% 0.22/0.47 (~product(e_4, e_1, e_3)),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[208, 197, 182])).
% 0.22/0.47 tff(210,assumption,(product(e_4, e_1, e_4)), introduced(assumption)).
% 0.22/0.47 tff(211,plain,
% 0.22/0.47 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_1, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_4, e_1, e_4)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_1, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_4, e_1, e_4)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(212,plain,
% 0.22/0.47 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_1, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_4, e_1, e_4)))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(213,plain,
% 0.22/0.47 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_1, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_4, e_1, e_4))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[212, 211])).
% 0.22/0.47 tff(214,plain,
% 0.22/0.47 ($false),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[213, 64, 20, 9, 210])).
% 0.22/0.47 tff(215,plain,(~product(e_4, e_1, e_4)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.47 tff(216,assumption,(product(e_4, e_1, e_1)), introduced(assumption)).
% 0.22/0.47 tff(217,plain,
% 0.22/0.47 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_4, e_1) | (~product(e_4, e_1, e_1)) | (~product(e_1, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_4, e_1) | (~product(e_4, e_1, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(218,plain,
% 0.22/0.47 ((equalish(e_4, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_4, e_1, e_1))) <=> (equalish(e_4, e_1) | (~product(e_4, e_1, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(219,plain,
% 0.22/0.47 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_4, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_4, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_4, e_1) | (~product(e_4, e_1, e_1)) | (~product(e_1, e_1, e_1))))),
% 0.22/0.48 inference(monotonicity,[status(thm)],[218])).
% 0.22/0.48 tff(220,plain,
% 0.22/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_4, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_4, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_4, e_1) | (~product(e_4, e_1, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.22/0.48 inference(transitivity,[status(thm)],[219, 217])).
% 0.22/0.48 tff(221,plain,
% 0.22/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_4, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_4, e_1, e_1)))),
% 0.22/0.48 inference(quant_inst,[status(thm)],[])).
% 0.22/0.48 tff(222,plain,
% 0.22/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_4, e_1) | (~product(e_4, e_1, e_1)) | (~product(e_1, e_1, e_1))),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[221, 220])).
% 0.22/0.48 tff(223,plain,
% 0.22/0.48 ($false),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[222, 45, 61, 42, 216])).
% 0.22/0.48 tff(224,plain,(~product(e_4, e_1, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.48 tff(225,plain,
% 0.22/0.48 (((~![Y: $i, X: $i] : (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(e_4, e_1, e_4) | product(e_4, e_1, e_2) | product(e_4, e_1, e_3) | product(e_4, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_4)))) <=> ((~![Y: $i, X: $i] : (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(e_4, e_1, e_4) | product(e_4, e_1, e_2) | product(e_4, e_1, e_3) | product(e_4, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_4)))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(226,plain,
% 0.22/0.48 ((product(e_4, e_1, e_4) | product(e_4, e_1, e_3) | product(e_4, e_1, e_2) | product(e_4, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_4))) <=> (product(e_4, e_1, e_4) | product(e_4, e_1, e_2) | product(e_4, e_1, e_3) | product(e_4, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_4)))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(227,plain,
% 0.22/0.48 (((~![Y: $i, X: $i] : (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(e_4, e_1, e_4) | product(e_4, e_1, e_3) | product(e_4, e_1, e_2) | product(e_4, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_4)))) <=> ((~![Y: $i, X: $i] : (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(e_4, e_1, e_4) | product(e_4, e_1, e_2) | product(e_4, e_1, e_3) | product(e_4, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_4))))),
% 0.22/0.48 inference(monotonicity,[status(thm)],[226])).
% 0.22/0.48 tff(228,plain,
% 0.22/0.48 (((~![Y: $i, X: $i] : (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(e_4, e_1, e_4) | product(e_4, e_1, e_3) | product(e_4, e_1, e_2) | product(e_4, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_4)))) <=> ((~![Y: $i, X: $i] : (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(e_4, e_1, e_4) | product(e_4, e_1, e_2) | product(e_4, e_1, e_3) | product(e_4, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_4)))),
% 0.22/0.48 inference(transitivity,[status(thm)],[227, 225])).
% 0.22/0.48 tff(229,plain,
% 0.22/0.48 ((~![Y: $i, X: $i] : (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(e_4, e_1, e_4) | product(e_4, e_1, e_3) | product(e_4, e_1, e_2) | product(e_4, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_4)))),
% 0.22/0.48 inference(quant_inst,[status(thm)],[])).
% 0.22/0.48 tff(230,plain,
% 0.22/0.48 ((~![Y: $i, X: $i] : (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(e_4, e_1, e_4) | product(e_4, e_1, e_2) | product(e_4, e_1, e_3) | product(e_4, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_4))),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[229, 228])).
% 0.22/0.48 tff(231,plain,
% 0.22/0.48 (product(e_4, e_1, e_4) | product(e_4, e_1, e_2) | product(e_4, e_1, e_3)),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[230, 85, 82, 79, 224])).
% 0.22/0.48 tff(232,plain,
% 0.22/0.48 (product(e_4, e_1, e_2) | product(e_4, e_1, e_3)),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[231, 215])).
% 0.22/0.48 tff(233,plain,
% 0.22/0.48 (product(e_4, e_1, e_2)),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[232, 209])).
% 0.22/0.48 tff(234,assumption,(product(e_4, e_1, e_2)), introduced(assumption)).
% 0.22/0.48 tff(235,assumption,(product(e_4, e_3, e_2)), introduced(assumption)).
% 0.22/0.48 tff(236,plain,
% 0.22/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_4, e_1, e_2)) | (~product(e_4, e_3, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_1) | (~product(e_4, e_1, e_2)) | (~product(e_4, e_3, e_2)))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(237,plain,
% 0.22/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_3, e_1) | (~product(e_4, e_1, e_2)) | (~product(e_4, e_3, e_2)))),
% 0.22/0.48 inference(quant_inst,[status(thm)],[])).
% 0.22/0.48 tff(238,plain,
% 0.22/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_3, e_1) | (~product(e_4, e_1, e_2)) | (~product(e_4, e_3, e_2))),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[237, 236])).
% 0.22/0.48 tff(239,plain,
% 0.22/0.48 ($false),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[238, 168, 20, 235, 234])).
% 0.22/0.48 tff(240,plain,((~product(e_4, e_3, e_2)) | (~product(e_4, e_1, e_2))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.48 tff(241,plain,
% 0.22/0.48 (~product(e_4, e_3, e_2)),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[240, 233])).
% 0.22/0.48 tff(242,plain,
% 0.22/0.48 (product(e_4, e_3, e_1)),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[158, 241])).
% 0.22/0.48 tff(243,plain,
% 0.22/0.48 (~product(e_3, e_3, e_4)),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[122, 119, 109, 242])).
% 0.22/0.48 tff(244,plain,
% 0.22/0.48 (~product(e_3, e_4, e_3)),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[126, 119, 97, 243])).
% 0.22/0.48 tff(245,assumption,(product(e_3, e_4, e_4)), introduced(assumption)).
% 0.22/0.48 tff(246,plain,
% 0.22/0.48 (((~![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_4, e_4, e_4)) | (~product(e_3, e_4, e_4)))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(247,plain,
% 0.22/0.48 ((~![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)))),
% 0.22/0.48 inference(quant_inst,[status(thm)],[])).
% 0.22/0.48 tff(248,plain,
% 0.22/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_4) | (~product(e_4, e_4, e_4)) | (~product(e_3, e_4, e_4))),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[247, 246])).
% 0.22/0.48 tff(249,plain,
% 0.22/0.48 ($false),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[248, 23, 61, 9, 245])).
% 0.22/0.48 tff(250,plain,(~product(e_3, e_4, e_4)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.48 tff(251,plain,
% 0.22/0.48 (((~![Y: $i, X: $i] : (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)))) | ((~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] : (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)))) | (~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))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(252,plain,
% 0.22/0.48 ((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) | (~group_element(e_4)) | (~group_element(e_3))) <=> ((~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))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(253,plain,
% 0.22/0.48 (((~![Y: $i, X: $i] : (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(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) | (~group_element(e_4)) | (~group_element(e_3)))) <=> ((~![Y: $i, X: $i] : (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)))) | ((~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)))),
% 0.22/0.48 inference(monotonicity,[status(thm)],[252])).
% 0.22/0.48 tff(254,plain,
% 0.22/0.48 (((~![Y: $i, X: $i] : (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(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) | (~group_element(e_4)) | (~group_element(e_3)))) <=> ((~![Y: $i, X: $i] : (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)))) | (~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))),
% 0.22/0.48 inference(transitivity,[status(thm)],[253, 251])).
% 0.22/0.48 tff(255,plain,
% 0.22/0.48 ((~![Y: $i, X: $i] : (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(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) | (~group_element(e_4)) | (~group_element(e_3)))),
% 0.22/0.48 inference(quant_inst,[status(thm)],[])).
% 0.22/0.48 tff(256,plain,
% 0.22/0.48 ((~![Y: $i, X: $i] : (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)))) | (~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)),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[255, 254])).
% 0.22/0.48 tff(257,plain,
% 0.22/0.48 (product(e_3, e_4, e_3) | product(e_3, e_4, e_2) | product(e_3, e_4, e_1)),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[256, 149, 82, 79, 250])).
% 0.22/0.48 tff(258,plain,
% 0.22/0.48 (product(e_3, e_4, e_2)),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[257, 244, 186])).
% 0.22/0.48 tff(259,plain,
% 0.22/0.48 (~product(e_1, e_4, e_3)),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[136, 242])).
% 0.22/0.48 tff(260,plain,
% 0.22/0.48 (product(e_1, e_4, e_2)),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[93, 259])).
% 0.22/0.48 tff(261,plain,
% 0.22/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_3, e_4, e_2)) | (~product(e_1, e_4, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_3, e_4, e_2)) | (~product(e_1, e_4, e_2)))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(262,plain,
% 0.22/0.48 ((equalish(e_3, e_1) | (~product(e_1, e_4, e_2)) | (~product(e_3, e_4, e_2))) <=> (equalish(e_3, e_1) | (~product(e_3, e_4, e_2)) | (~product(e_1, e_4, e_2)))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(263,plain,
% 0.22/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_4, e_2)) | (~product(e_3, e_4, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_3, e_4, e_2)) | (~product(e_1, e_4, e_2))))),
% 0.22/0.49 inference(monotonicity,[status(thm)],[262])).
% 0.22/0.49 tff(264,plain,
% 0.22/0.49 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_4, e_2)) | (~product(e_3, e_4, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_3, e_4, e_2)) | (~product(e_1, e_4, e_2)))),
% 0.22/0.49 inference(transitivity,[status(thm)],[263, 261])).
% 0.22/0.49 tff(265,plain,
% 0.22/0.49 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_4, e_2)) | (~product(e_3, e_4, e_2)))),
% 0.22/0.49 inference(quant_inst,[status(thm)],[])).
% 0.22/0.49 tff(266,plain,
% 0.22/0.49 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_3, e_4, e_2)) | (~product(e_1, e_4, e_2))),
% 0.22/0.49 inference(modus_ponens,[status(thm)],[265, 264])).
% 0.22/0.49 tff(267,plain,
% 0.22/0.49 ($false),
% 0.22/0.49 inference(unit_resolution,[status(thm)],[266, 168, 61, 260, 258])).
% 0.22/0.49 % SZS output end Proof
% 0.22/0.49 % E exiting
%------------------------------------------------------------------------------