%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP131-2.002 : TPTP v9.0.0. Released v1.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n016.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 05:07:12 AM UTC 2025
% Result : Unsatisfiable 0.23s 0.41s
% Output : Proof 0.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.14 % Problem : GRP131-2.002 : TPTP v9.0.0. Released v1.2.0.
% 0.04/0.14 % Command : run_E %s %d THM
% 0.15/0.36 % Computer : n016.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Thu Jun 19 15:44:17 EDT 2025
% 0.15/0.36 % CPUTime :
% 0.23/0.41 % SZS status Unsatisfiable
% 0.23/0.41 % SZS output start Proof
% 0.23/0.41 tff(product_type, type, (
% 0.23/0.41 product: ( $i * $i * $i ) > $o)).
% 0.23/0.41 tff(e_2_type, type, (
% 0.23/0.41 e_2: $i)).
% 0.23/0.41 tff(e_1_type, type, (
% 0.23/0.41 e_1: $i)).
% 0.23/0.41 tff(equalish_type, type, (
% 0.23/0.41 equalish: ( $i * $i ) > $o)).
% 0.23/0.41 tff(group_element_type, type, (
% 0.23/0.41 group_element: $i > $o)).
% 0.23/0.41 tff(1,assumption,(product(e_1, e_2, e_2)), introduced(assumption)).
% 0.23/0.41 tff(2,assumption,(product(e_1, e_1, e_2)), introduced(assumption)).
% 0.23/0.41 tff(3,plain,
% 0.23/0.41 (^[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))))),
% 0.23/0.41 inference(bind,[status(th)],[])).
% 0.23/0.41 tff(4,plain,
% 0.23/0.41 (![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)))),
% 0.23/0.41 inference(quant_intro,[status(thm)],[3])).
% 0.23/0.41 tff(5,plain,
% 0.23/0.41 (![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)))),
% 0.23/0.41 inference(rewrite,[status(thm)],[])).
% 0.23/0.41 tff(6,plain,
% 0.23/0.41 (^[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)))))),
% 0.23/0.41 inference(bind,[status(th)],[])).
% 0.23/0.41 tff(7,plain,
% 0.23/0.41 (![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)))),
% 0.23/0.41 inference(quant_intro,[status(thm)],[6])).
% 0.23/0.41 tff(8,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')).
% 0.23/0.41 tff(9,plain,
% 0.23/0.41 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.23/0.41 inference(modus_ponens,[status(thm)],[8, 7])).
% 0.23/0.41 tff(10,plain,
% 0.23/0.41 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.23/0.41 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.23/0.41 tff(11,plain,(
% 0.23/0.41 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.23/0.41 inference(skolemize,[status(sab)],[10])).
% 0.23/0.41 tff(12,plain,
% 0.23/0.41 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.23/0.41 inference(modus_ponens,[status(thm)],[11, 4])).
% 0.23/0.41 tff(13,plain,
% 0.23/0.41 ((~equalish(e_1, e_2)) <=> (~equalish(e_1, e_2))),
% 0.23/0.41 inference(rewrite,[status(thm)],[])).
% 0.23/0.41 tff(14,axiom,(~equalish(e_1, e_2)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_1_is_not_e_2')).
% 0.23/0.41 tff(15,plain,
% 0.23/0.41 (~equalish(e_1, e_2)),
% 0.23/0.41 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.23/0.41 tff(16,plain,
% 0.23/0.41 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_2)))),
% 0.23/0.41 inference(rewrite,[status(thm)],[])).
% 0.23/0.41 tff(17,plain,
% 0.23/0.41 ((equalish(e_1, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_2))) <=> (equalish(e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_2)))),
% 0.23/0.41 inference(rewrite,[status(thm)],[])).
% 0.23/0.41 tff(18,plain,
% 0.23/0.41 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_1, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_2))))),
% 0.23/0.42 inference(monotonicity,[status(thm)],[17])).
% 0.23/0.42 tff(19,plain,
% 0.23/0.42 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_1, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_2)))),
% 0.23/0.42 inference(transitivity,[status(thm)],[18, 16])).
% 0.23/0.42 tff(20,plain,
% 0.23/0.42 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_1, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_2)))),
% 0.23/0.42 inference(quant_inst,[status(thm)],[])).
% 0.23/0.42 tff(21,plain,
% 0.23/0.42 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_2))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.23/0.42 tff(22,plain,
% 0.23/0.42 ($false),
% 0.23/0.42 inference(unit_resolution,[status(thm)],[21, 15, 12, 2, 1])).
% 0.23/0.42 tff(23,plain,((~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_2))), inference(lemma,lemma(discharge,[]))).
% 0.23/0.42 tff(24,plain,
% 0.23/0.42 (~product(e_1, e_1, e_2)),
% 0.23/0.42 inference(unit_resolution,[status(thm)],[23, 1])).
% 0.23/0.42 tff(25,plain,
% 0.23/0.42 (^[Y: $i, X: $i] : refl((product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))) <=> (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(26,plain,
% 0.23/0.42 (![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))) <=> ![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[25])).
% 0.23/0.42 tff(27,plain,
% 0.23/0.42 (![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X))) <=> ![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.23/0.42 inference(rewrite,[status(thm)],[])).
% 0.23/0.42 tff(28,plain,
% 0.23/0.42 (^[Y: $i, X: $i] : 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)))))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(29,plain,
% 0.23/0.42 (![Y: $i, X: $i] : ((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) <=> ![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[28])).
% 0.23/0.42 tff(30,axiom,(![Y: $i, X: $i] : ((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_total_function1')).
% 0.23/0.42 tff(31,plain,
% 0.23/0.42 (![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[30, 29])).
% 0.23/0.42 tff(32,plain,
% 0.23/0.42 (![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[31, 27])).
% 0.23/0.42 tff(33,plain,(
% 0.23/0.42 ![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.23/0.42 inference(skolemize,[status(sab)],[32])).
% 0.23/0.42 tff(34,plain,
% 0.23/0.42 (![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[33, 26])).
% 0.23/0.42 tff(35,plain,
% 0.23/0.42 (group_element(e_1) <=> group_element(e_1)),
% 0.23/0.42 inference(rewrite,[status(thm)],[])).
% 0.23/0.42 tff(36,axiom,(group_element(e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_1')).
% 0.23/0.42 tff(37,plain,
% 0.23/0.42 (group_element(e_1)),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.23/0.42 tff(38,plain,
% 0.23/0.42 (((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_1, e_2) | (~group_element(e_1)) | product(e_1, e_1, e_1))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_1, e_1, e_2) | (~group_element(e_1)) | product(e_1, e_1, e_1))),
% 0.23/0.42 inference(rewrite,[status(thm)],[])).
% 0.23/0.42 tff(39,plain,
% 0.23/0.42 ((product(e_1, e_1, e_2) | product(e_1, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_1))) <=> (product(e_1, e_1, e_2) | (~group_element(e_1)) | product(e_1, e_1, e_1))),
% 0.23/0.42 inference(rewrite,[status(thm)],[])).
% 0.23/0.42 tff(40,plain,
% 0.23/0.42 (((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_1, e_2) | product(e_1, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_1, e_2) | (~group_element(e_1)) | product(e_1, e_1, e_1)))),
% 0.23/0.42 inference(monotonicity,[status(thm)],[39])).
% 0.23/0.42 tff(41,plain,
% 0.23/0.42 (((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_1, e_2) | product(e_1, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_1, e_1, e_2) | (~group_element(e_1)) | product(e_1, e_1, e_1))),
% 0.23/0.42 inference(transitivity,[status(thm)],[40, 38])).
% 0.23/0.42 tff(42,plain,
% 0.23/0.42 ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_1, e_2) | product(e_1, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_1)))),
% 0.23/0.42 inference(quant_inst,[status(thm)],[])).
% 0.23/0.42 tff(43,plain,
% 0.23/0.42 ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_1, e_1, e_2) | (~group_element(e_1)) | product(e_1, e_1, e_1)),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.23/0.42 tff(44,plain,
% 0.23/0.42 (product(e_1, e_1, e_2) | product(e_1, e_1, e_1)),
% 0.23/0.42 inference(unit_resolution,[status(thm)],[43, 37, 34])).
% 0.23/0.42 tff(45,plain,
% 0.23/0.42 (product(e_1, e_1, e_1)),
% 0.23/0.42 inference(unit_resolution,[status(thm)],[44, 24])).
% 0.23/0.42 tff(46,assumption,(product(e_1, e_1, e_1)), introduced(assumption)).
% 0.23/0.42 tff(47,assumption,(product(e_2, e_2, e_1)), introduced(assumption)).
% 0.23/0.42 tff(48,plain,
% 0.23/0.42 (^[Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : refl(((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2)) <=> ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2)))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(49,plain,
% 0.23/0.42 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~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, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[48])).
% 0.23/0.42 tff(50,plain,
% 0.23/0.42 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~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, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.23/0.42 inference(rewrite,[status(thm)],[])).
% 0.23/0.42 tff(51,plain,
% 0.23/0.42 (^[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, Y1, X1))) <=> ((~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)))), (((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, Y1, X1))) | (~product(Z2, Y2, X2))) <=> (((~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))) | (~product(Z2, Y2, X2))))), rewrite((((~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))) | (~product(Z2, Y2, X2))) <=> ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)))), (((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, Y1, X1))) | (~product(Z2, Y2, X2))) <=> ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))))), ((((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, Y1, X1))) | (~product(Z2, Y2, X2))) | equalish(Y1, Y2)) <=> (((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))) | equalish(Y1, Y2)))), rewrite((((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))) | equalish(Y1, Y2)) <=> ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))), ((((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, Y1, X1))) | (~product(Z2, Y2, X2))) | equalish(Y1, Y2)) <=> ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(52,plain,
% 0.23/0.42 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : (((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, Y1, X1))) | (~product(Z2, Y2, X2))) | equalish(Y1, Y2)) <=> ![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[51])).
% 0.23/0.42 tff(53,axiom,(![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : (((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, Y1, X1))) | (~product(Z2, Y2, X2))) | equalish(Y1, Y2))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','qg1_2')).
% 0.23/0.42 tff(54,plain,
% 0.23/0.42 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.23/0.42 tff(55,plain,
% 0.23/0.42 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[54, 50])).
% 0.23/0.42 tff(56,plain,(
% 0.23/0.42 ![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.23/0.42 inference(skolemize,[status(sab)],[55])).
% 0.23/0.42 tff(57,plain,
% 0.23/0.42 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[56, 49])).
% 0.23/0.42 tff(58,plain,
% 0.23/0.42 ((~equalish(e_2, e_1)) <=> (~equalish(e_2, e_1))),
% 0.23/0.42 inference(rewrite,[status(thm)],[])).
% 0.23/0.42 tff(59,axiom,(~equalish(e_2, e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_2_is_not_e_1')).
% 0.23/0.42 tff(60,plain,
% 0.23/0.42 (~equalish(e_2, e_1)),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.23/0.42 tff(61,plain,
% 0.23/0.42 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_2, e_1)))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_2, e_1)))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(62,plain,
% 0.23/0.43 (((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_2, e_1)) | equalish(e_2, e_1)) <=> (equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_2, e_1)))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(63,plain,
% 0.23/0.43 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_2, e_1)) | equalish(e_2, e_1))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_2, e_1))))),
% 0.23/0.43 inference(monotonicity,[status(thm)],[62])).
% 0.23/0.43 tff(64,plain,
% 0.23/0.43 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_2, e_1)) | equalish(e_2, e_1))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_2, e_1)))),
% 0.23/0.43 inference(transitivity,[status(thm)],[63, 61])).
% 0.23/0.43 tff(65,plain,
% 0.23/0.43 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_2, e_1)) | equalish(e_2, e_1))),
% 0.23/0.43 inference(quant_inst,[status(thm)],[])).
% 0.23/0.43 tff(66,plain,
% 0.23/0.43 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_2, e_1))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.23/0.43 tff(67,plain,
% 0.23/0.43 ($false),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[66, 60, 57, 1, 47, 46])).
% 0.23/0.43 tff(68,plain,((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1))), inference(lemma,lemma(discharge,[]))).
% 0.23/0.43 tff(69,plain,
% 0.23/0.43 (~product(e_2, e_2, e_1)),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[68, 45, 1])).
% 0.23/0.43 tff(70,plain,
% 0.23/0.43 (group_element(e_2) <=> group_element(e_2)),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(71,axiom,(group_element(e_2)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_2')).
% 0.23/0.43 tff(72,plain,
% 0.23/0.43 (group_element(e_2)),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.23/0.43 tff(73,plain,
% 0.23/0.43 (((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_2, e_2, e_2) | (~group_element(e_2)) | product(e_2, e_2, e_1))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_2, e_2, e_2) | (~group_element(e_2)) | product(e_2, e_2, e_1))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(74,plain,
% 0.23/0.43 ((product(e_2, e_2, e_2) | product(e_2, e_2, e_1) | (~group_element(e_2)) | (~group_element(e_2))) <=> (product(e_2, e_2, e_2) | (~group_element(e_2)) | product(e_2, e_2, e_1))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(75,plain,
% 0.23/0.43 (((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_2, e_2, e_2) | product(e_2, e_2, e_1) | (~group_element(e_2)) | (~group_element(e_2)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_2, e_2, e_2) | (~group_element(e_2)) | product(e_2, e_2, e_1)))),
% 0.23/0.43 inference(monotonicity,[status(thm)],[74])).
% 0.23/0.43 tff(76,plain,
% 0.23/0.43 (((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_2, e_2, e_2) | product(e_2, e_2, e_1) | (~group_element(e_2)) | (~group_element(e_2)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_2, e_2, e_2) | (~group_element(e_2)) | product(e_2, e_2, e_1))),
% 0.23/0.43 inference(transitivity,[status(thm)],[75, 73])).
% 0.23/0.43 tff(77,plain,
% 0.23/0.43 ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_2, e_2, e_2) | product(e_2, e_2, e_1) | (~group_element(e_2)) | (~group_element(e_2)))),
% 0.23/0.43 inference(quant_inst,[status(thm)],[])).
% 0.23/0.43 tff(78,plain,
% 0.23/0.43 ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_2, e_2, e_2) | (~group_element(e_2)) | product(e_2, e_2, e_1)),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.23/0.43 tff(79,plain,
% 0.23/0.43 (product(e_2, e_2, e_2) | product(e_2, e_2, e_1)),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[78, 72, 34])).
% 0.23/0.43 tff(80,plain,
% 0.23/0.43 (product(e_2, e_2, e_2)),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[79, 69])).
% 0.23/0.43 tff(81,plain,
% 0.23/0.43 (^[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))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(82,plain,
% 0.23/0.43 (![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)))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[81])).
% 0.23/0.43 tff(83,plain,
% 0.23/0.43 (![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)))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(84,plain,
% 0.23/0.43 (^[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)))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(85,plain,
% 0.23/0.43 (![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)))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[84])).
% 0.23/0.43 tff(86,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')).
% 0.23/0.43 tff(87,plain,
% 0.23/0.43 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[86, 85])).
% 0.23/0.43 tff(88,plain,
% 0.23/0.43 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[87, 83])).
% 0.23/0.43 tff(89,plain,(
% 0.23/0.43 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.23/0.43 inference(skolemize,[status(sab)],[88])).
% 0.23/0.43 tff(90,plain,
% 0.23/0.43 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[89, 82])).
% 0.23/0.43 tff(91,plain,
% 0.23/0.43 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_2, e_1) | (~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_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_2)))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(92,plain,
% 0.23/0.44 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_2)))),
% 0.23/0.44 inference(quant_inst,[status(thm)],[])).
% 0.23/0.44 tff(93,plain,
% 0.23/0.44 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_2))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[92, 91])).
% 0.23/0.44 tff(94,plain,
% 0.23/0.44 ($false),
% 0.23/0.44 inference(unit_resolution,[status(thm)],[93, 60, 90, 1, 80])).
% 0.23/0.44 tff(95,plain,(~product(e_1, e_2, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.23/0.44 tff(96,plain,
% 0.23/0.44 (((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_2, e_1) | product(e_1, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_1, e_2, e_1) | product(e_1, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_1)))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(97,plain,
% 0.23/0.44 ((product(e_1, e_2, e_2) | product(e_1, e_2, e_1) | (~group_element(e_2)) | (~group_element(e_1))) <=> (product(e_1, e_2, e_1) | product(e_1, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_1)))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(98,plain,
% 0.23/0.44 (((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_2, e_2) | product(e_1, e_2, e_1) | (~group_element(e_2)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_2, e_1) | product(e_1, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_1))))),
% 0.23/0.44 inference(monotonicity,[status(thm)],[97])).
% 0.23/0.44 tff(99,plain,
% 0.23/0.44 (((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_2, e_2) | product(e_1, e_2, e_1) | (~group_element(e_2)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_1, e_2, e_1) | product(e_1, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_1)))),
% 0.23/0.44 inference(transitivity,[status(thm)],[98, 96])).
% 0.23/0.44 tff(100,plain,
% 0.23/0.44 ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_1, e_2, e_2) | product(e_1, e_2, e_1) | (~group_element(e_2)) | (~group_element(e_1)))),
% 0.23/0.44 inference(quant_inst,[status(thm)],[])).
% 0.23/0.44 tff(101,plain,
% 0.23/0.44 ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_1, e_2, e_1) | product(e_1, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_1))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[100, 99])).
% 0.23/0.44 tff(102,plain,
% 0.23/0.44 (product(e_1, e_2, e_1)),
% 0.23/0.44 inference(unit_resolution,[status(thm)],[101, 37, 72, 34, 95])).
% 0.23/0.44 tff(103,plain,
% 0.23/0.44 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(104,plain,
% 0.23/0.44 (((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)) | equalish(e_2, e_1)) <=> (equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(105,plain,
% 0.23/0.45 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)) | equalish(e_2, e_1))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1))))),
% 0.23/0.45 inference(monotonicity,[status(thm)],[104])).
% 0.23/0.45 tff(106,plain,
% 0.23/0.45 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)) | equalish(e_2, e_1))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.23/0.45 inference(transitivity,[status(thm)],[105, 103])).
% 0.23/0.45 tff(107,plain,
% 0.23/0.45 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)) | equalish(e_2, e_1))),
% 0.23/0.45 inference(quant_inst,[status(thm)],[])).
% 0.23/0.45 tff(108,plain,
% 0.23/0.45 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.23/0.45 tff(109,plain,
% 0.23/0.45 ($false),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[108, 60, 57, 102, 46])).
% 0.23/0.45 tff(110,plain,(~product(e_1, e_1, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.23/0.45 tff(111,plain,
% 0.23/0.45 (product(e_1, e_1, e_2)),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[44, 110])).
% 0.23/0.45 tff(112,plain,
% 0.23/0.45 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~product(e_2, e_1, e_1)))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~product(e_2, e_1, e_1)))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(113,plain,
% 0.23/0.45 (((~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~product(e_2, e_1, e_1)) | equalish(e_1, e_2)) <=> (equalish(e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~product(e_2, e_1, e_1)))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(114,plain,
% 0.23/0.45 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~product(e_2, e_1, e_1)) | equalish(e_1, e_2))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~product(e_2, e_1, e_1))))),
% 0.23/0.45 inference(monotonicity,[status(thm)],[113])).
% 0.23/0.45 tff(115,plain,
% 0.23/0.45 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~product(e_2, e_1, e_1)) | equalish(e_1, e_2))) <=> ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~product(e_2, e_1, e_1)))),
% 0.23/0.45 inference(transitivity,[status(thm)],[114, 112])).
% 0.23/0.45 tff(116,plain,
% 0.23/0.45 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~product(e_2, e_1, e_1)) | equalish(e_1, e_2))),
% 0.23/0.45 inference(quant_inst,[status(thm)],[])).
% 0.23/0.45 tff(117,plain,
% 0.23/0.45 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, Y2, X2)) | (~product(Z2, Y1, X1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~product(e_2, e_1, e_1))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[116, 115])).
% 0.23/0.45 tff(118,plain,
% 0.23/0.45 (~product(e_2, e_1, e_1)),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[117, 15, 57, 111, 102])).
% 0.23/0.45 tff(119,plain,
% 0.23/0.45 (((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_2, e_1, e_1) | product(e_2, e_1, e_2) | (~group_element(e_2)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_2, e_1, e_1) | product(e_2, e_1, e_2) | (~group_element(e_2)) | (~group_element(e_1)))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(120,plain,
% 0.23/0.45 ((product(e_2, e_1, e_2) | product(e_2, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_2))) <=> (product(e_2, e_1, e_1) | product(e_2, e_1, e_2) | (~group_element(e_2)) | (~group_element(e_1)))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(121,plain,
% 0.23/0.45 (((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_2, e_1, e_2) | product(e_2, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_2)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_2, e_1, e_1) | product(e_2, e_1, e_2) | (~group_element(e_2)) | (~group_element(e_1))))),
% 0.23/0.45 inference(monotonicity,[status(thm)],[120])).
% 0.23/0.45 tff(122,plain,
% 0.23/0.45 (((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_2, e_1, e_2) | product(e_2, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_2)))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_2, e_1, e_1) | product(e_2, e_1, e_2) | (~group_element(e_2)) | (~group_element(e_1)))),
% 0.23/0.45 inference(transitivity,[status(thm)],[121, 119])).
% 0.23/0.45 tff(123,plain,
% 0.23/0.45 ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_2, e_1, e_2) | product(e_2, e_1, e_1) | (~group_element(e_1)) | (~group_element(e_2)))),
% 0.23/0.45 inference(quant_inst,[status(thm)],[])).
% 0.23/0.45 tff(124,plain,
% 0.23/0.45 ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | product(e_2, e_1, e_1) | product(e_2, e_1, e_2) | (~group_element(e_2)) | (~group_element(e_1))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[123, 122])).
% 0.23/0.45 tff(125,plain,
% 0.23/0.45 (product(e_2, e_1, e_2)),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[124, 37, 72, 34, 118])).
% 0.23/0.45 tff(126,plain,
% 0.23/0.45 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_2, e_1, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_2, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_2, e_1, e_2)))),
% 0.23/0.46 inference(rewrite,[status(thm)],[])).
% 0.23/0.46 tff(127,plain,
% 0.23/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_2, e_1, e_2)))),
% 0.23/0.46 inference(quant_inst,[status(thm)],[])).
% 0.23/0.46 tff(128,plain,
% 0.23/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_2, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_2, e_1, e_2))),
% 0.23/0.46 inference(modus_ponens,[status(thm)],[127, 126])).
% 0.23/0.46 tff(129,plain,
% 0.23/0.46 ($false),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[128, 60, 90, 111, 125])).
% 0.23/0.46 % SZS output end Proof
% 0.23/0.46 % E exiting
%------------------------------------------------------------------------------