%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP135-2.002 : 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 05:07:14 AM UTC 2025
% Result : Unsatisfiable 0.12s 0.39s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : GRP135-2.002 : TPTP v9.0.0. Released v1.2.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n015.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Jun 19 15:44:26 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.12/0.39 % SZS status Unsatisfiable
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 tff(product_type, type, (
% 0.12/0.39 product: ( $i * $i * $i ) > $o)).
% 0.12/0.39 tff(e_2_type, type, (
% 0.12/0.39 e_2: $i)).
% 0.12/0.39 tff(e_1_type, type, (
% 0.12/0.39 e_1: $i)).
% 0.12/0.39 tff(equalish_type, type, (
% 0.12/0.39 equalish: ( $i * $i ) > $o)).
% 0.12/0.39 tff(group_element_type, type, (
% 0.12/0.39 group_element: $i > $o)).
% 0.12/0.39 tff(1,assumption,(product(e_2, e_2, e_2)), introduced(assumption)).
% 0.12/0.39 tff(2,assumption,(product(e_1, e_2, e_1)), introduced(assumption)).
% 0.12/0.39 tff(3,plain,
% 0.12/0.39 (^[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))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(4,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[3])).
% 0.12/0.39 tff(5,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(6,plain,
% 0.12/0.39 (^[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)))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(7,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[6])).
% 0.12/0.39 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')).
% 0.12/0.39 tff(9,plain,
% 0.12/0.39 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[8, 7])).
% 0.12/0.39 tff(10,plain,
% 0.12/0.39 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.12/0.39 tff(11,plain,(
% 0.12/0.39 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.12/0.39 inference(skolemize,[status(sab)],[10])).
% 0.12/0.39 tff(12,plain,
% 0.12/0.39 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[11, 4])).
% 0.12/0.39 tff(13,plain,
% 0.12/0.39 ((~equalish(e_2, e_1)) <=> (~equalish(e_2, e_1))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(14,axiom,(~equalish(e_2, e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_2_is_not_e_1')).
% 0.12/0.39 tff(15,plain,
% 0.12/0.39 (~equalish(e_2, e_1)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.12/0.39 tff(16,plain,
% 0.12/0.39 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(17,plain,
% 0.12/0.39 ((equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1))) <=> (equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(18,plain,
% 0.12/0.39 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1))))),
% 0.12/0.39 inference(monotonicity,[status(thm)],[17])).
% 0.12/0.39 tff(19,plain,
% 0.12/0.39 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.12/0.39 inference(transitivity,[status(thm)],[18, 16])).
% 0.12/0.39 tff(20,plain,
% 0.12/0.39 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(21,plain,
% 0.12/0.39 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.12/0.40 tff(22,plain,
% 0.12/0.40 ((~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[21, 15, 12])).
% 0.12/0.40 tff(23,plain,
% 0.12/0.40 (~product(e_1, e_1, e_1)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[22, 2])).
% 0.12/0.40 tff(24,plain,
% 0.12/0.40 (^[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))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(25,plain,
% 0.12/0.40 (![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)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[24])).
% 0.12/0.40 tff(26,plain,
% 0.12/0.40 (![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)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(27,plain,
% 0.12/0.40 (^[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)))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(28,plain,
% 0.12/0.40 (![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)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[27])).
% 0.12/0.40 tff(29,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')).
% 0.12/0.40 tff(30,plain,
% 0.12/0.40 (![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.12/0.40 tff(31,plain,
% 0.12/0.40 (![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[30, 26])).
% 0.12/0.40 tff(32,plain,(
% 0.12/0.40 ![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.12/0.40 inference(skolemize,[status(sab)],[31])).
% 0.12/0.40 tff(33,plain,
% 0.12/0.40 (![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[32, 25])).
% 0.12/0.40 tff(34,plain,
% 0.12/0.40 (group_element(e_1) <=> group_element(e_1)),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(35,axiom,(group_element(e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_1')).
% 0.12/0.40 tff(36,plain,
% 0.12/0.40 (group_element(e_1)),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.12/0.40 tff(37,plain,
% 0.12/0.40 (((~![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))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(38,plain,
% 0.12/0.40 ((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))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(39,plain,
% 0.12/0.40 (((~![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)))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[38])).
% 0.12/0.40 tff(40,plain,
% 0.12/0.40 (((~![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))),
% 0.12/0.40 inference(transitivity,[status(thm)],[39, 37])).
% 0.12/0.40 tff(41,plain,
% 0.12/0.40 ((~![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)))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(42,plain,
% 0.12/0.40 ((~![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)),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.12/0.40 tff(43,plain,
% 0.12/0.40 (product(e_1, e_1, e_2)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[42, 36, 33, 23])).
% 0.12/0.40 tff(44,plain,
% 0.12/0.40 (^[W: $i, Z: $i, Y: $i, X: $i] : refl((equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X))) <=> (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(45,plain,
% 0.12/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[44])).
% 0.12/0.40 tff(46,plain,
% 0.12/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(47,plain,
% 0.12/0.40 (^[W: $i, Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~product(W, Y, X)) | (~product(Z, Y, X))) <=> ((~product(Z, Y, X)) | (~product(W, Y, X)))), ((((~product(W, Y, X)) | (~product(Z, Y, X))) | equalish(W, Z)) <=> (((~product(Z, Y, X)) | (~product(W, Y, X))) | equalish(W, Z)))), rewrite((((~product(Z, Y, X)) | (~product(W, Y, X))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))), ((((~product(W, Y, X)) | (~product(Z, Y, X))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(48,plain,
% 0.12/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product(W, Y, X)) | (~product(Z, Y, X))) | equalish(W, Z)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[47])).
% 0.21/0.40 tff(49,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')).
% 0.21/0.40 tff(50,plain,
% 0.21/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.21/0.40 tff(51,plain,
% 0.21/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[50, 46])).
% 0.21/0.40 tff(52,plain,(
% 0.21/0.40 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.21/0.40 inference(skolemize,[status(sab)],[51])).
% 0.21/0.40 tff(53,plain,
% 0.21/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[52, 45])).
% 0.21/0.40 tff(54,plain,
% 0.21/0.40 (((~![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)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(55,plain,
% 0.21/0.40 ((~![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)))),
% 0.21/0.40 inference(quant_inst,[status(thm)],[])).
% 0.21/0.40 tff(56,plain,
% 0.21/0.40 ((~![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))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[55, 54])).
% 0.21/0.40 tff(57,plain,
% 0.21/0.40 ((~product(e_1, e_1, e_2)) | (~product(e_2, e_1, e_2))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[56, 15, 53])).
% 0.21/0.40 tff(58,plain,
% 0.21/0.40 (~product(e_2, e_1, e_2)),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[57, 43])).
% 0.21/0.40 tff(59,plain,
% 0.21/0.40 (^[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))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(60,plain,
% 0.21/0.40 (![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)))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[59])).
% 0.21/0.40 tff(61,plain,
% 0.21/0.40 (![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)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(62,plain,
% 0.21/0.40 (^[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)))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(63,plain,
% 0.21/0.40 (![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)))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[62])).
% 0.21/0.40 tff(64,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')).
% 0.21/0.40 tff(65,plain,
% 0.21/0.40 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.21/0.40 tff(66,plain,
% 0.21/0.40 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[65, 61])).
% 0.21/0.40 tff(67,plain,(
% 0.21/0.40 ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))),
% 0.21/0.40 inference(skolemize,[status(sab)],[66])).
% 0.21/0.40 tff(68,plain,
% 0.21/0.40 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[67, 60])).
% 0.21/0.40 tff(69,plain,
% 0.21/0.40 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | ((~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | product(e_2, e_1, e_2))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | product(e_2, e_1, e_2))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(70,plain,
% 0.21/0.40 ((product(e_2, e_1, e_2) | (~product(e_1, e_1, e_2)) | (~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_2))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(71,plain,
% 0.21/0.40 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_2, e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | ((~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | product(e_2, e_1, e_2)))),
% 0.21/0.40 inference(monotonicity,[status(thm)],[70])).
% 0.21/0.40 tff(72,plain,
% 0.21/0.40 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_2, e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | product(e_2, e_1, e_2))),
% 0.21/0.40 inference(transitivity,[status(thm)],[71, 69])).
% 0.21/0.40 tff(73,plain,
% 0.21/0.40 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_2, e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)))),
% 0.21/0.40 inference(quant_inst,[status(thm)],[])).
% 0.21/0.40 tff(74,plain,
% 0.21/0.40 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | product(e_2, e_1, e_2)),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.21/0.40 tff(75,plain,
% 0.21/0.40 ($false),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[74, 68, 2, 43, 58])).
% 0.21/0.40 tff(76,plain,(~product(e_1, e_2, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.21/0.40 tff(77,plain,
% 0.21/0.40 (group_element(e_2) <=> group_element(e_2)),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(78,axiom,(group_element(e_2)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_2')).
% 0.21/0.40 tff(79,plain,
% 0.21/0.40 (group_element(e_2)),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[78, 77])).
% 0.21/0.40 tff(80,plain,
% 0.21/0.40 (((~![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_2) | product(e_1, e_2, e_1) | (~group_element(e_2)) | (~group_element(e_1)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(81,plain,
% 0.21/0.40 ((~![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)))),
% 0.21/0.40 inference(quant_inst,[status(thm)],[])).
% 0.21/0.40 tff(82,plain,
% 0.21/0.40 ((~![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))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[81, 80])).
% 0.21/0.41 tff(83,plain,
% 0.21/0.41 (product(e_1, e_2, e_2) | product(e_1, e_2, e_1)),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[82, 36, 79, 33])).
% 0.21/0.41 tff(84,plain,
% 0.21/0.41 (product(e_1, e_2, e_2)),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[83, 76])).
% 0.21/0.41 tff(85,plain,
% 0.21/0.41 (((~![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)))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(86,plain,
% 0.21/0.41 ((~![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)))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(87,plain,
% 0.21/0.41 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_2))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[86, 85])).
% 0.21/0.41 tff(88,plain,
% 0.21/0.41 ($false),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[87, 15, 53, 84, 1])).
% 0.21/0.41 tff(89,plain,(~product(e_2, e_2, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.21/0.41 tff(90,plain,
% 0.21/0.41 (((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | ((~group_element(e_2)) | product(e_2, e_2, e_1) | product(e_2, e_2, e_2))) <=> ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (~group_element(e_2)) | product(e_2, e_2, e_1) | product(e_2, e_2, e_2))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(91,plain,
% 0.21/0.41 ((product(e_2, e_2, e_2) | product(e_2, e_2, e_1) | (~group_element(e_2)) | (~group_element(e_2))) <=> ((~group_element(e_2)) | product(e_2, e_2, e_1) | product(e_2, e_2, e_2))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(92,plain,
% 0.21/0.41 (((~![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)))) | ((~group_element(e_2)) | product(e_2, e_2, e_1) | product(e_2, e_2, e_2)))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[91])).
% 0.21/0.41 tff(93,plain,
% 0.21/0.41 (((~![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)))) | (~group_element(e_2)) | product(e_2, e_2, e_1) | product(e_2, e_2, e_2))),
% 0.21/0.41 inference(transitivity,[status(thm)],[92, 90])).
% 0.21/0.41 tff(94,plain,
% 0.21/0.41 ((~![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)))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(95,plain,
% 0.21/0.41 ((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (~group_element(e_2)) | product(e_2, e_2, e_1) | product(e_2, e_2, e_2)),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[94, 93])).
% 0.21/0.41 tff(96,plain,
% 0.21/0.41 (product(e_2, e_2, e_1) | product(e_2, e_2, e_2)),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[95, 79, 33])).
% 0.21/0.41 tff(97,plain,
% 0.21/0.41 (product(e_2, e_2, e_1)),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[96, 89])).
% 0.21/0.41 tff(98,plain,
% 0.21/0.41 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | ((~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1)) | product(e_2, e_2, e_2))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1)) | product(e_2, e_2, e_2))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(99,plain,
% 0.21/0.41 ((product(e_2, e_2, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1))) <=> ((~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1)) | product(e_2, e_2, e_2))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(100,plain,
% 0.21/0.41 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_2, e_2, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | ((~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1)) | product(e_2, e_2, e_2)))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[99])).
% 0.21/0.41 tff(101,plain,
% 0.21/0.41 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_2, e_2, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1)) | product(e_2, e_2, e_2))),
% 0.21/0.41 inference(transitivity,[status(thm)],[100, 98])).
% 0.21/0.41 tff(102,plain,
% 0.21/0.41 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (product(e_2, e_2, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1)))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(103,plain,
% 0.21/0.41 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z2, Y, X) | (~product(Z1, Y, Z2)) | (~product(Y, X, Z1)))) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1)) | product(e_2, e_2, e_2)),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.21/0.41 tff(104,plain,
% 0.21/0.41 ($false),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[103, 68, 84, 97, 89])).
% 0.21/0.41 % SZS output end Proof
% 0.21/0.41 % E exiting
%------------------------------------------------------------------------------