%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP132-2.002 : TPTP v9.0.0. Released v1.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n012.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 0.12s 0.39s
% Output : Proof 0.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : GRP132-2.002 : TPTP v9.0.0. Released v1.2.0.
% 0.12/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n012.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 15:43:11 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_1, e_2, e_2)), introduced(assumption)).
% 0.12/0.39 tff(2,assumption,(product(e_1, e_1, e_2)), 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_1, e_2)) <=> (~equalish(e_1, e_2))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(14,axiom,(~equalish(e_1, e_2)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_1_is_not_e_2')).
% 0.12/0.39 tff(15,plain,
% 0.12/0.39 (~equalish(e_1, e_2)),
% 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_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)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(17,plain,
% 0.12/0.39 ((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)))),
% 0.12/0.39 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_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))))),
% 0.12/0.39 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_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)))),
% 0.12/0.39 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_1, e_2) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_2)))),
% 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: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_2))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.12/0.39 tff(22,plain,
% 0.12/0.39 ($false),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[21, 15, 12, 2, 1])).
% 0.12/0.39 tff(23,plain,((~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_2))), inference(lemma,lemma(discharge,[]))).
% 0.12/0.39 tff(24,plain,
% 0.12/0.39 (~product(e_1, e_1, e_2)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[23, 1])).
% 0.12/0.39 tff(25,plain,
% 0.12/0.39 (^[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))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(26,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[25])).
% 0.12/0.39 tff(27,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(28,plain,
% 0.12/0.39 (^[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)))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(29,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[28])).
% 0.12/0.39 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')).
% 0.12/0.39 tff(31,plain,
% 0.12/0.39 (![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[30, 29])).
% 0.12/0.39 tff(32,plain,
% 0.12/0.39 (![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[31, 27])).
% 0.12/0.39 tff(33,plain,(
% 0.12/0.39 ![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.12/0.39 inference(skolemize,[status(sab)],[32])).
% 0.12/0.39 tff(34,plain,
% 0.12/0.39 (![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[33, 26])).
% 0.12/0.39 tff(35,plain,
% 0.12/0.39 (group_element(e_1) <=> group_element(e_1)),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(36,axiom,(group_element(e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_1')).
% 0.12/0.39 tff(37,plain,
% 0.12/0.39 (group_element(e_1)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.12/0.39 tff(38,plain,
% 0.12/0.39 (((~![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))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(39,plain,
% 0.12/0.39 ((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(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(monotonicity,[status(thm)],[39])).
% 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)))) <=> ((~![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)],[40, 38])).
% 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) | 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(43,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)],[42, 41])).
% 0.12/0.40 tff(44,plain,
% 0.12/0.40 (product(e_1, e_1, e_2) | product(e_1, e_1, e_1)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[43, 37, 34])).
% 0.12/0.40 tff(45,plain,
% 0.12/0.40 (product(e_1, e_1, e_1)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[44, 24])).
% 0.12/0.40 tff(46,assumption,(product(e_2, e_2, e_1)), introduced(assumption)).
% 0.12/0.40 tff(47,assumption,(product(e_1, e_1, e_1)), introduced(assumption)).
% 0.12/0.40 tff(48,plain,
% 0.12/0.40 (^[Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : refl(((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2)) <=> ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2)))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(49,plain,
% 0.12/0.40 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2)) <=> ![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[48])).
% 0.12/0.40 tff(50,plain,
% 0.12/0.40 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2)) <=> ![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(51,plain,
% 0.12/0.40 (^[Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, X1, Y1))) <=> ((~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)))), (((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, X1, Y1))) | (~product(Z2, X2, Y2))) <=> (((~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))) | (~product(Z2, X2, Y2))))), rewrite((((~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))) | (~product(Z2, X2, Y2))) <=> ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)))), (((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, X1, Y1))) | (~product(Z2, X2, Y2))) <=> ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))))), ((((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, X1, Y1))) | (~product(Z2, X2, Y2))) | equalish(Y1, Y2)) <=> (((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))) | equalish(Y1, Y2)))), rewrite((((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1))) | equalish(Y1, Y2)) <=> ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))), ((((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, X1, Y1))) | (~product(Z2, X2, Y2))) | equalish(Y1, Y2)) <=> ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(52,plain,
% 0.12/0.40 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : (((((~product(X1, Y1, Z1)) | (~product(X2, Y2, Z1))) | (~product(Z2, X1, Y1))) | (~product(Z2, X2, Y2))) | equalish(Y1, Y2)) <=> ![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[51])).
% 0.12/0.40 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, X1, Y1))) | (~product(Z2, X2, Y2))) | equalish(Y1, Y2))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','qg2_2')).
% 0.12/0.40 tff(54,plain,
% 0.12/0.40 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.12/0.40 tff(55,plain,
% 0.12/0.40 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[54, 50])).
% 0.12/0.40 tff(56,plain,(
% 0.12/0.40 ![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.12/0.40 inference(skolemize,[status(sab)],[55])).
% 0.12/0.40 tff(57,plain,
% 0.12/0.40 (![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[56, 49])).
% 0.12/0.40 tff(58,plain,
% 0.12/0.40 ((~equalish(e_2, e_1)) <=> (~equalish(e_2, e_1))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(59,axiom,(~equalish(e_2, e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_2_is_not_e_1')).
% 0.12/0.40 tff(60,plain,
% 0.12/0.40 (~equalish(e_2, e_1)),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.12/0.40 tff(61,plain,
% 0.12/0.40 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_2, 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, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(62,plain,
% 0.12/0.40 (((~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_2, e_2, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(63,plain,
% 0.12/0.40 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_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, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1)) | (~product(e_1, e_1, e_1))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[62])).
% 0.12/0.40 tff(64,plain,
% 0.12/0.40 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_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, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.12/0.40 inference(transitivity,[status(thm)],[63, 61])).
% 0.12/0.40 tff(65,plain,
% 0.12/0.40 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_2)) | (~product(e_1, e_1, e_1)) | (~product(e_2, e_2, e_1)) | equalish(e_2, e_1))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(66,plain,
% 0.12/0.40 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_1)) | (~product(e_1, e_1, e_1))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.12/0.40 tff(67,plain,
% 0.12/0.40 ($false),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[66, 60, 57, 1, 47, 46])).
% 0.12/0.40 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,[]))).
% 0.12/0.40 tff(69,plain,
% 0.12/0.40 (~product(e_2, e_2, e_1)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[68, 45, 1])).
% 0.12/0.40 tff(70,plain,
% 0.12/0.40 (group_element(e_2) <=> group_element(e_2)),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(71,axiom,(group_element(e_2)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_2')).
% 0.12/0.40 tff(72,plain,
% 0.12/0.40 (group_element(e_2)),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.12/0.40 tff(73,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_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))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(74,plain,
% 0.12/0.40 ((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))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(75,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_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)))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[74])).
% 0.12/0.40 tff(76,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_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))),
% 0.12/0.40 inference(transitivity,[status(thm)],[75, 73])).
% 0.12/0.40 tff(77,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_2, e_2, e_2) | product(e_2, e_2, e_1) | (~group_element(e_2)) | (~group_element(e_2)))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(78,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_2, e_2, e_2) | (~group_element(e_2)) | product(e_2, e_2, e_1)),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.12/0.40 tff(79,plain,
% 0.12/0.40 (product(e_2, e_2, e_2)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[78, 72, 34, 69])).
% 0.12/0.40 tff(80,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(81,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)],[80])).
% 0.12/0.40 tff(82,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(83,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(84,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)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[83])).
% 0.12/0.40 tff(85,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')).
% 0.12/0.40 tff(86,plain,
% 0.12/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.12/0.40 tff(87,plain,
% 0.12/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[86, 82])).
% 0.12/0.40 tff(88,plain,(
% 0.12/0.40 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.12/0.40 inference(skolemize,[status(sab)],[87])).
% 0.12/0.40 tff(89,plain,
% 0.12/0.40 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[88, 81])).
% 0.12/0.40 tff(90,plain,
% 0.12/0.40 (((~![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_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)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(91,plain,
% 0.12/0.41 ((~![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)))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(92,plain,
% 0.12/0.41 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_2, e_1) | (~product(e_1, e_2, e_2)) | (~product(e_2, e_2, e_2))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.12/0.41 tff(93,plain,
% 0.12/0.41 ($false),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[92, 60, 89, 1, 79])).
% 0.12/0.41 tff(94,plain,(~product(e_1, e_2, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.12/0.41 tff(95,plain,
% 0.12/0.41 (((~![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)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(96,plain,
% 0.12/0.41 ((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)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(97,plain,
% 0.12/0.41 (((~![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))))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[96])).
% 0.12/0.41 tff(98,plain,
% 0.12/0.41 (((~![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)))),
% 0.12/0.41 inference(transitivity,[status(thm)],[97, 95])).
% 0.12/0.41 tff(99,plain,
% 0.12/0.41 ((~![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)))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(100,plain,
% 0.12/0.41 ((~![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))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.12/0.41 tff(101,plain,
% 0.12/0.41 (product(e_1, e_2, e_1)),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[100, 37, 72, 34, 94])).
% 0.12/0.41 tff(102,plain,
% 0.12/0.41 (((~![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_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_1, e_2) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(103,plain,
% 0.12/0.41 ((~![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_1)) | (~product(e_1, e_1, e_1)))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(104,plain,
% 0.12/0.41 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_1, e_2) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_1, e_1))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[103, 102])).
% 0.12/0.41 tff(105,plain,
% 0.12/0.41 ($false),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[104, 15, 12, 101, 47])).
% 0.12/0.41 tff(106,plain,(~product(e_1, e_1, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.12/0.41 tff(107,plain,
% 0.12/0.41 (product(e_1, e_1, e_2)),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[44, 106])).
% 0.12/0.41 tff(108,plain,
% 0.12/0.41 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | (equalish(e_1, e_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, X2, Y2)) | (~product(Z2, X1, Y1)) | (~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)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(109,plain,
% 0.12/0.41 (((~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~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)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(110,plain,
% 0.12/0.41 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~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, X2, Y2)) | (~product(Z2, X1, Y1)) | (~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))))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[109])).
% 0.12/0.41 tff(111,plain,
% 0.12/0.41 (((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~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, X2, Y2)) | (~product(Z2, X1, Y1)) | (~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)))),
% 0.12/0.41 inference(transitivity,[status(thm)],[110, 108])).
% 0.12/0.41 tff(112,plain,
% 0.12/0.41 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | ((~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_2, e_1)) | (~product(e_2, e_1, e_1)) | equalish(e_1, e_2))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(113,plain,
% 0.12/0.41 ((~![Y2: $i, Y1: $i, Z1: $i, X2: $i, Z2: $i, X1: $i] : ((~product(Z2, X2, Y2)) | (~product(Z2, X1, Y1)) | (~product(X2, Y2, Z1)) | (~product(X1, Y1, Z1)) | equalish(Y1, Y2))) | equalish(e_1, e_2) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_2, e_1)) | (~product(e_2, e_1, e_1))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[112, 111])).
% 0.12/0.41 tff(114,plain,
% 0.12/0.41 (~product(e_2, e_1, e_1)),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[113, 15, 57, 107, 101])).
% 0.12/0.41 tff(115,plain,
% 0.12/0.41 (((~![Y: $i, X: $i] : (product(X, Y, e_2) | product(X, Y, e_1) | (~group_element(Y)) | (~group_element(X)))) | (product(e_2, e_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)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(116,plain,
% 0.12/0.41 ((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)))),
% 0.12/0.42 inference(rewrite,[status(thm)],[])).
% 0.12/0.42 tff(117,plain,
% 0.12/0.42 (((~![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))))),
% 0.12/0.42 inference(monotonicity,[status(thm)],[116])).
% 0.12/0.42 tff(118,plain,
% 0.12/0.42 (((~![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)))),
% 0.12/0.42 inference(transitivity,[status(thm)],[117, 115])).
% 0.12/0.42 tff(119,plain,
% 0.12/0.42 ((~![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)))),
% 0.12/0.42 inference(quant_inst,[status(thm)],[])).
% 0.12/0.42 tff(120,plain,
% 0.12/0.42 ((~![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))),
% 0.12/0.42 inference(modus_ponens,[status(thm)],[119, 118])).
% 0.12/0.42 tff(121,plain,
% 0.12/0.42 (product(e_2, e_1, e_1) | product(e_2, e_1, e_2)),
% 0.12/0.42 inference(unit_resolution,[status(thm)],[120, 37, 72, 34])).
% 0.12/0.42 tff(122,plain,
% 0.12/0.42 (product(e_2, e_1, e_2)),
% 0.12/0.42 inference(unit_resolution,[status(thm)],[121, 114])).
% 0.12/0.42 tff(123,plain,
% 0.12/0.42 (((~![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)))),
% 0.12/0.42 inference(rewrite,[status(thm)],[])).
% 0.12/0.42 tff(124,plain,
% 0.12/0.42 ((~![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)))),
% 0.12/0.42 inference(quant_inst,[status(thm)],[])).
% 0.12/0.42 tff(125,plain,
% 0.12/0.42 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_2, e_1) | (~product(e_1, e_1, e_2)) | (~product(e_2, e_1, e_2))),
% 0.12/0.42 inference(modus_ponens,[status(thm)],[124, 123])).
% 0.12/0.42 tff(126,plain,
% 0.12/0.42 ($false),
% 0.12/0.42 inference(unit_resolution,[status(thm)],[125, 60, 89, 107, 122])).
% 0.12/0.42 % SZS output end Proof
% 0.12/0.42 % E exiting
%------------------------------------------------------------------------------