%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP128-4.003 : TPTP v9.0.0. Bugfixed v1.2.1.
% 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 05:07:09 AM UTC 2025
% Result : Unsatisfiable 0.22s 0.42s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.14 % Problem : GRP128-4.003 : TPTP v9.0.0. Bugfixed v1.2.1.
% 0.08/0.14 % Command : run_E %s %d THM
% 0.14/0.36 % Computer : n015.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Thu Jun 19 15:43:40 EDT 2025
% 0.14/0.36 % CPUTime :
% 0.22/0.42 % SZS status Unsatisfiable
% 0.22/0.42 % SZS output start Proof
% 0.22/0.42 tff(product_type, type, (
% 0.22/0.42 product: ( $i * $i * $i ) > $o)).
% 0.22/0.42 tff(e_1_type, type, (
% 0.22/0.42 e_1: $i)).
% 0.22/0.42 tff(e_2_type, type, (
% 0.22/0.42 e_2: $i)).
% 0.22/0.42 tff(e_3_type, type, (
% 0.22/0.42 e_3: $i)).
% 0.22/0.42 tff(equalish_type, type, (
% 0.22/0.42 equalish: ( $i * $i ) > $o)).
% 0.22/0.42 tff(group_element_type, type, (
% 0.22/0.42 group_element: $i > $o)).
% 0.22/0.42 tff(1,assumption,(product(e_3, e_1, e_1)), introduced(assumption)).
% 0.22/0.42 tff(2,plain,
% 0.22/0.42 (^[W: $i, Z: $i, Y: $i, X: $i] : refl((equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X))) <=> (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(3,plain,
% 0.22/0.42 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[2])).
% 0.22/0.42 tff(4,plain,
% 0.22/0.42 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(5,plain,
% 0.22/0.42 (^[W: $i, Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~product(W, Y, X)) | (~product(Z, Y, X))) <=> ((~product(Z, Y, X)) | (~product(W, Y, X)))), ((((~product(W, Y, X)) | (~product(Z, Y, X))) | equalish(W, Z)) <=> (((~product(Z, Y, X)) | (~product(W, Y, X))) | equalish(W, Z)))), rewrite((((~product(Z, Y, X)) | (~product(W, Y, X))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))), ((((~product(W, Y, X)) | (~product(Z, Y, X))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(6,plain,
% 0.22/0.42 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product(W, Y, X)) | (~product(Z, Y, X))) | equalish(W, Z)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[5])).
% 0.22/0.42 tff(7,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(W, Y, X)) | (~product(Z, Y, X))) | equalish(W, Z))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_left_cancellation')).
% 0.22/0.42 tff(8,plain,
% 0.22/0.42 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[7, 6])).
% 0.22/0.42 tff(9,plain,
% 0.22/0.42 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[8, 4])).
% 0.22/0.42 tff(10,plain,(
% 0.22/0.42 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.22/0.42 inference(skolemize,[status(sab)],[9])).
% 0.22/0.42 tff(11,plain,
% 0.22/0.42 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[10, 3])).
% 0.22/0.42 tff(12,plain,
% 0.22/0.42 ((~equalish(e_3, e_1)) <=> (~equalish(e_3, e_1))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(13,axiom,(~equalish(e_3, e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_3_is_not_e_1')).
% 0.22/0.42 tff(14,plain,
% 0.22/0.42 (~equalish(e_3, e_1)),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.22/0.42 tff(15,plain,
% 0.22/0.42 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_3, e_1, e_1)) | (~product(e_1, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_3, e_1, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(16,plain,
% 0.22/0.42 ((equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_3, e_1, e_1))) <=> (equalish(e_3, e_1) | (~product(e_3, e_1, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(17,plain,
% 0.22/0.42 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_3, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_3, e_1, e_1)) | (~product(e_1, e_1, e_1))))),
% 0.22/0.42 inference(monotonicity,[status(thm)],[16])).
% 0.22/0.42 tff(18,plain,
% 0.22/0.42 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_3, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_3, e_1, e_1)) | (~product(e_1, e_1, e_1)))),
% 0.22/0.42 inference(transitivity,[status(thm)],[17, 15])).
% 0.22/0.42 tff(19,plain,
% 0.22/0.42 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_3, e_1, e_1)))),
% 0.22/0.42 inference(quant_inst,[status(thm)],[])).
% 0.22/0.42 tff(20,plain,
% 0.22/0.42 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_3, e_1, e_1)) | (~product(e_1, e_1, e_1))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[19, 18])).
% 0.22/0.42 tff(21,plain,
% 0.22/0.42 ((~product(e_3, e_1, e_1)) | (~product(e_1, e_1, e_1))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[20, 14, 11])).
% 0.22/0.42 tff(22,plain,
% 0.22/0.42 (~product(e_1, e_1, e_1)),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[21, 1])).
% 0.22/0.42 tff(23,plain,
% 0.22/0.42 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : refl((product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1))) <=> (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(24,plain,
% 0.22/0.42 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[23])).
% 0.22/0.42 tff(25,plain,
% 0.22/0.42 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(26,plain,
% 0.22/0.42 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : trans(monotonicity(rewrite((product(Z1, Y, Z2) | (~product(X, Z1, Z2))) <=> (product(Z1, Y, Z2) | (~product(X, Z1, Z2)))), (((product(Z1, Y, Z2) | (~product(X, Z1, Z2))) | (~product(X, Y, Z1))) <=> ((product(Z1, Y, Z2) | (~product(X, Z1, Z2))) | (~product(X, Y, Z1))))), rewrite(((product(Z1, Y, Z2) | (~product(X, Z1, Z2))) | (~product(X, Y, Z1))) <=> (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))), (((product(Z1, Y, Z2) | (~product(X, Z1, Z2))) | (~product(X, Y, Z1))) <=> (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(27,plain,
% 0.22/0.42 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((product(Z1, Y, Z2) | (~product(X, Z1, Z2))) | (~product(X, Y, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[26])).
% 0.22/0.42 tff(28,axiom,(![Y: $i, Z1: $i, X: $i, Z2: $i] : ((product(Z1, Y, Z2) | (~product(X, Z1, Z2))) | (~product(X, Y, Z1)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','qg3_2')).
% 0.22/0.42 tff(29,plain,
% 0.22/0.42 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[28, 27])).
% 0.22/0.42 tff(30,plain,
% 0.22/0.42 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[29, 25])).
% 0.22/0.42 tff(31,plain,(
% 0.22/0.42 ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))),
% 0.22/0.42 inference(skolemize,[status(sab)],[30])).
% 0.22/0.42 tff(32,plain,
% 0.22/0.42 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[31, 24])).
% 0.22/0.42 tff(33,plain,
% 0.22/0.42 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | ((~product(e_3, e_1, e_1)) | product(e_1, e_1, e_1))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (~product(e_3, e_1, e_1)) | product(e_1, e_1, e_1))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(34,plain,
% 0.22/0.43 ((product(e_1, e_1, e_1) | (~product(e_3, e_1, e_1)) | (~product(e_3, e_1, e_1))) <=> ((~product(e_3, e_1, e_1)) | product(e_1, e_1, e_1))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(35,plain,
% 0.22/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_1, e_1, e_1) | (~product(e_3, e_1, e_1)) | (~product(e_3, e_1, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | ((~product(e_3, e_1, e_1)) | product(e_1, e_1, e_1)))),
% 0.22/0.43 inference(monotonicity,[status(thm)],[34])).
% 0.22/0.43 tff(36,plain,
% 0.22/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_1, e_1, e_1) | (~product(e_3, e_1, e_1)) | (~product(e_3, e_1, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (~product(e_3, e_1, e_1)) | product(e_1, e_1, e_1))),
% 0.22/0.43 inference(transitivity,[status(thm)],[35, 33])).
% 0.22/0.43 tff(37,plain,
% 0.22/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_1, e_1, e_1) | (~product(e_3, e_1, e_1)) | (~product(e_3, e_1, e_1)))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(38,plain,
% 0.22/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (~product(e_3, e_1, e_1)) | product(e_1, e_1, e_1)),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.22/0.43 tff(39,plain,
% 0.22/0.43 ($false),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[38, 32, 1, 22])).
% 0.22/0.43 tff(40,plain,(~product(e_3, e_1, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.43 tff(41,assumption,(~product(e_3, e_1, e_1)), introduced(assumption)).
% 0.22/0.43 tff(42,assumption,(~product(e_3, e_2, e_2)), introduced(assumption)).
% 0.22/0.43 tff(43,assumption,(~product(e_3, e_1, e_2)), introduced(assumption)).
% 0.22/0.43 tff(44,plain,
% 0.22/0.43 (group_element(e_3) <=> group_element(e_3)),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(45,axiom,(group_element(e_3)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_3')).
% 0.22/0.43 tff(46,plain,
% 0.22/0.43 (group_element(e_3)),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.22/0.43 tff(47,plain,
% 0.22/0.43 (group_element(e_2) <=> group_element(e_2)),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(48,axiom,(group_element(e_2)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_2')).
% 0.22/0.43 tff(49,plain,
% 0.22/0.43 (group_element(e_2)),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[48, 47])).
% 0.22/0.43 tff(50,plain,
% 0.22/0.43 (^[Y: $i, X: $i] : refl(((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(51,plain,
% 0.22/0.43 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[50])).
% 0.22/0.43 tff(52,plain,
% 0.22/0.43 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(53,plain,
% 0.22/0.43 (^[Y: $i, X: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_1, Y))), (((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) <=> (((~group_element(Y)) | (~group_element(X)) | product(X, e_1, Y)) | product(X, e_2, Y)))), rewrite((((~group_element(Y)) | (~group_element(X)) | product(X, e_1, Y)) | product(X, e_2, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_2, Y) | product(X, e_1, Y))), (((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_2, Y) | product(X, e_1, Y)))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) | product(X, e_3, Y)) <=> (((~group_element(Y)) | (~group_element(X)) | product(X, e_2, Y) | product(X, e_1, Y)) | product(X, e_3, Y)))), rewrite((((~group_element(Y)) | (~group_element(X)) | product(X, e_2, Y) | product(X, e_1, Y)) | product(X, e_3, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) | product(X, e_3, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(54,plain,
% 0.22/0.43 (![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) | product(X, e_3, Y)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[53])).
% 0.22/0.43 tff(55,axiom,(![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) | product(X, e_3, Y))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','column_surjectivity')).
% 0.22/0.43 tff(56,plain,
% 0.22/0.43 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[55, 54])).
% 0.22/0.43 tff(57,plain,
% 0.22/0.43 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[56, 52])).
% 0.22/0.43 tff(58,plain,(
% 0.22/0.43 ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.22/0.43 inference(skolemize,[status(sab)],[57])).
% 0.22/0.43 tff(59,plain,
% 0.22/0.43 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[58, 51])).
% 0.22/0.43 tff(60,plain,
% 0.22/0.43 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | (product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_3, e_2) | product(e_3, e_1, e_2))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_3, e_2) | product(e_3, e_1, e_2))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(61,plain,
% 0.22/0.43 (((~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_3, e_2) | product(e_3, e_2, e_2) | product(e_3, e_1, e_2)) <=> (product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_3, e_2) | product(e_3, e_1, e_2))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(62,plain,
% 0.22/0.43 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_3, e_2) | product(e_3, e_2, e_2) | product(e_3, e_1, e_2))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | (product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_3, e_2) | product(e_3, e_1, e_2)))),
% 0.22/0.43 inference(monotonicity,[status(thm)],[61])).
% 0.22/0.43 tff(63,plain,
% 0.22/0.43 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_3, e_2) | product(e_3, e_2, e_2) | product(e_3, e_1, e_2))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_3, e_2) | product(e_3, e_1, e_2))),
% 0.22/0.43 inference(transitivity,[status(thm)],[62, 60])).
% 0.22/0.43 tff(64,plain,
% 0.22/0.43 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_3, e_2) | product(e_3, e_2, e_2) | product(e_3, e_1, e_2))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(65,plain,
% 0.22/0.43 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_3, e_2) | product(e_3, e_1, e_2)),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.22/0.43 tff(66,plain,
% 0.22/0.43 (product(e_3, e_2, e_2) | product(e_3, e_3, e_2) | product(e_3, e_1, e_2)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[65, 59, 49, 46])).
% 0.22/0.43 tff(67,plain,
% 0.22/0.43 (product(e_3, e_3, e_2)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[66, 43, 42])).
% 0.22/0.43 tff(68,plain,
% 0.22/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_1, e_2) | (~product(e_3, e_3, e_2)) | (~product(e_3, e_1, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | product(e_3, e_1, e_2) | (~product(e_3, e_3, e_2)) | (~product(e_3, e_1, e_3)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(69,plain,
% 0.22/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_1, e_2) | (~product(e_3, e_3, e_2)) | (~product(e_3, e_1, e_3)))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(70,plain,
% 0.22/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | product(e_3, e_1, e_2) | (~product(e_3, e_3, e_2)) | (~product(e_3, e_1, e_3))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.22/0.43 tff(71,plain,
% 0.22/0.43 (~product(e_3, e_1, e_3)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[70, 32, 43, 67])).
% 0.22/0.43 tff(72,plain,
% 0.22/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_2, e_2) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_3, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | product(e_3, e_2, e_2) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_3, e_2)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(73,plain,
% 0.22/0.43 ((product(e_3, e_2, e_2) | (~product(e_3, e_3, e_2)) | (~product(e_3, e_2, e_3))) <=> (product(e_3, e_2, e_2) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_3, e_2)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(74,plain,
% 0.22/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_2, e_2) | (~product(e_3, e_3, e_2)) | (~product(e_3, e_2, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_2, e_2) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_3, e_2))))),
% 0.22/0.43 inference(monotonicity,[status(thm)],[73])).
% 0.22/0.43 tff(75,plain,
% 0.22/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_2, e_2) | (~product(e_3, e_3, e_2)) | (~product(e_3, e_2, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | product(e_3, e_2, e_2) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_3, e_2)))),
% 0.22/0.43 inference(transitivity,[status(thm)],[74, 72])).
% 0.22/0.43 tff(76,plain,
% 0.22/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_2, e_2) | (~product(e_3, e_3, e_2)) | (~product(e_3, e_2, e_3)))),
% 0.22/0.44 inference(quant_inst,[status(thm)],[])).
% 0.22/0.44 tff(77,plain,
% 0.22/0.44 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | product(e_3, e_2, e_2) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_3, e_2))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.22/0.44 tff(78,plain,
% 0.22/0.44 (product(e_3, e_2, e_2) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_3, e_2))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[77, 32])).
% 0.22/0.44 tff(79,plain,
% 0.22/0.44 (~product(e_3, e_2, e_3)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[78, 67, 42])).
% 0.22/0.44 tff(80,plain,
% 0.22/0.44 (^[Y: $i, X: $i] : refl(((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1)))),
% 0.22/0.44 inference(bind,[status(th)],[])).
% 0.22/0.44 tff(81,plain,
% 0.22/0.44 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.22/0.44 inference(quant_intro,[status(thm)],[80])).
% 0.22/0.44 tff(82,plain,
% 0.22/0.44 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(83,plain,
% 0.22/0.44 (^[Y: $i, X: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_1))), (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) <=> (((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_1)) | product(X, Y, e_2)))), rewrite((((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_1)) | product(X, Y, e_2)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_2) | product(X, Y, e_1))), (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_2) | product(X, Y, e_1)))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3)) <=> (((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_2) | product(X, Y, e_1)) | product(X, Y, e_3)))), rewrite((((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_2) | product(X, Y, e_1)) | product(X, Y, e_3)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))))),
% 0.22/0.44 inference(bind,[status(th)],[])).
% 0.22/0.44 tff(84,plain,
% 0.22/0.44 (![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.22/0.44 inference(quant_intro,[status(thm)],[83])).
% 0.22/0.44 tff(85,axiom,(![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_total_function1')).
% 0.22/0.44 tff(86,plain,
% 0.22/0.44 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.22/0.44 tff(87,plain,
% 0.22/0.44 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[86, 82])).
% 0.22/0.44 tff(88,plain,(
% 0.22/0.44 ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.22/0.44 inference(skolemize,[status(sab)],[87])).
% 0.22/0.44 tff(89,plain,
% 0.22/0.44 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[88, 81])).
% 0.22/0.44 tff(90,plain,
% 0.22/0.44 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_1))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(91,plain,
% 0.22/0.44 (((~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | product(e_3, e_2, e_1)) <=> (product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_1))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(92,plain,
% 0.22/0.44 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | product(e_3, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_1)))),
% 0.22/0.44 inference(monotonicity,[status(thm)],[91])).
% 0.22/0.44 tff(93,plain,
% 0.22/0.44 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | product(e_3, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_1))),
% 0.22/0.44 inference(transitivity,[status(thm)],[92, 90])).
% 0.22/0.44 tff(94,plain,
% 0.22/0.44 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | product(e_3, e_2, e_1))),
% 0.22/0.44 inference(quant_inst,[status(thm)],[])).
% 0.22/0.44 tff(95,plain,
% 0.22/0.44 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_1)),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[94, 93])).
% 0.22/0.44 tff(96,plain,
% 0.22/0.44 (product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | product(e_3, e_2, e_1)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[95, 49, 46, 89])).
% 0.22/0.44 tff(97,plain,
% 0.22/0.44 (product(e_3, e_2, e_1)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[96, 79, 42])).
% 0.22/0.44 tff(98,plain,
% 0.22/0.44 (^[W: $i, Z: $i, Y: $i, X: $i] : refl((equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y))) <=> (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y))))),
% 0.22/0.44 inference(bind,[status(th)],[])).
% 0.22/0.44 tff(99,plain,
% 0.22/0.44 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.22/0.44 inference(quant_intro,[status(thm)],[98])).
% 0.22/0.44 tff(100,plain,
% 0.22/0.44 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(101,plain,
% 0.22/0.44 (^[W: $i, Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~product(X, W, Y)) | (~product(X, Z, Y))) <=> ((~product(X, Z, Y)) | (~product(X, W, Y)))), ((((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z)) <=> (((~product(X, Z, Y)) | (~product(X, W, Y))) | equalish(W, Z)))), rewrite((((~product(X, Z, Y)) | (~product(X, W, Y))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))), ((((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))))),
% 0.22/0.44 inference(bind,[status(th)],[])).
% 0.22/0.44 tff(102,plain,
% 0.22/0.44 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.22/0.44 inference(quant_intro,[status(thm)],[101])).
% 0.22/0.44 tff(103,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_right_cancellation')).
% 0.22/0.44 tff(104,plain,
% 0.22/0.44 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[103, 102])).
% 0.22/0.44 tff(105,plain,
% 0.22/0.44 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[104, 100])).
% 0.22/0.44 tff(106,plain,(
% 0.22/0.44 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.22/0.44 inference(skolemize,[status(sab)],[105])).
% 0.22/0.44 tff(107,plain,
% 0.22/0.44 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[106, 99])).
% 0.22/0.44 tff(108,plain,
% 0.22/0.44 ((~equalish(e_1, e_2)) <=> (~equalish(e_1, e_2))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(109,axiom,(~equalish(e_1, e_2)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_1_is_not_e_2')).
% 0.22/0.44 tff(110,plain,
% 0.22/0.44 (~equalish(e_1, e_2)),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[109, 108])).
% 0.22/0.44 tff(111,plain,
% 0.22/0.44 (((~![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_3, e_2, e_1)) | (~product(e_3, 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_3, e_2, e_1)) | (~product(e_3, e_1, e_1)))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(112,plain,
% 0.22/0.44 ((~![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_3, e_2, e_1)) | (~product(e_3, e_1, e_1)))),
% 0.22/0.44 inference(quant_inst,[status(thm)],[])).
% 0.22/0.44 tff(113,plain,
% 0.22/0.44 ((~![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_3, e_2, e_1)) | (~product(e_3, e_1, e_1))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[112, 111])).
% 0.22/0.44 tff(114,plain,
% 0.22/0.44 ((~product(e_3, e_2, e_1)) | (~product(e_3, e_1, e_1))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[113, 110, 107])).
% 0.22/0.44 tff(115,plain,
% 0.22/0.44 (~product(e_3, e_1, e_1)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[114, 97])).
% 0.22/0.44 tff(116,plain,
% 0.22/0.44 (group_element(e_1) <=> group_element(e_1)),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(117,axiom,(group_element(e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_1')).
% 0.22/0.44 tff(118,plain,
% 0.22/0.44 (group_element(e_1)),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[117, 116])).
% 0.22/0.44 tff(119,plain,
% 0.22/0.44 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_2) | product(e_3, e_1, e_3) | product(e_3, e_1, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_2) | product(e_3, e_1, e_3) | product(e_3, e_1, e_1))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(120,plain,
% 0.22/0.44 (((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1)) <=> ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_2) | product(e_3, e_1, e_3) | product(e_3, e_1, e_1))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(121,plain,
% 0.22/0.45 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_2) | product(e_3, e_1, e_3) | product(e_3, e_1, e_1)))),
% 0.22/0.45 inference(monotonicity,[status(thm)],[120])).
% 0.22/0.45 tff(122,plain,
% 0.22/0.45 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_2) | product(e_3, e_1, e_3) | product(e_3, e_1, e_1))),
% 0.22/0.45 inference(transitivity,[status(thm)],[121, 119])).
% 0.22/0.45 tff(123,plain,
% 0.22/0.45 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_3) | product(e_3, e_1, e_2) | product(e_3, e_1, e_1))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(124,plain,
% 0.22/0.45 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_1, e_2) | product(e_3, e_1, e_3) | product(e_3, e_1, e_1)),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[123, 122])).
% 0.22/0.45 tff(125,plain,
% 0.22/0.45 ($false),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[124, 118, 46, 89, 43, 115, 71])).
% 0.22/0.45 tff(126,plain,(product(e_3, e_1, e_2) | product(e_3, e_2, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.45 tff(127,plain,
% 0.22/0.45 (product(e_3, e_1, e_2)),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[126, 42])).
% 0.22/0.45 tff(128,plain,
% 0.22/0.45 (^[W: $i, Z: $i, Y: $i, X: $i] : refl((equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W))) <=> (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W))))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(129,plain,
% 0.22/0.45 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[128])).
% 0.22/0.45 tff(130,plain,
% 0.22/0.45 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(131,plain,
% 0.22/0.45 (^[W: $i, Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~product(X, Y, W)) | (~product(X, Y, Z))) <=> ((~product(X, Y, Z)) | (~product(X, Y, W)))), ((((~product(X, Y, W)) | (~product(X, Y, Z))) | equalish(W, Z)) <=> (((~product(X, Y, Z)) | (~product(X, Y, W))) | equalish(W, Z)))), rewrite((((~product(X, Y, Z)) | (~product(X, Y, W))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))), ((((~product(X, Y, W)) | (~product(X, Y, Z))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(132,plain,
% 0.22/0.45 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, W)) | (~product(X, Y, Z))) | equalish(W, Z)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[131])).
% 0.22/0.45 tff(133,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, W)) | (~product(X, Y, Z))) | equalish(W, Z))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_total_function2')).
% 0.22/0.45 tff(134,plain,
% 0.22/0.45 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[133, 132])).
% 0.22/0.45 tff(135,plain,
% 0.22/0.45 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[134, 130])).
% 0.22/0.45 tff(136,plain,(
% 0.22/0.45 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.22/0.45 inference(skolemize,[status(sab)],[135])).
% 0.22/0.45 tff(137,plain,
% 0.22/0.45 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[136, 129])).
% 0.22/0.45 tff(138,plain,
% 0.22/0.45 ((~equalish(e_2, e_1)) <=> (~equalish(e_2, e_1))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(139,axiom,(~equalish(e_2, e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_2_is_not_e_1')).
% 0.22/0.45 tff(140,plain,
% 0.22/0.45 (~equalish(e_2, e_1)),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[139, 138])).
% 0.22/0.45 tff(141,plain,
% 0.22/0.45 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_2, e_1) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_2, e_1) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_1, e_1)))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(142,plain,
% 0.22/0.45 ((equalish(e_2, e_1) | (~product(e_3, e_1, e_1)) | (~product(e_3, e_1, e_2))) <=> (equalish(e_2, e_1) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_1, e_1)))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(143,plain,
% 0.22/0.45 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_2, e_1) | (~product(e_3, e_1, e_1)) | (~product(e_3, e_1, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_2, e_1) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_1, e_1))))),
% 0.22/0.45 inference(monotonicity,[status(thm)],[142])).
% 0.22/0.45 tff(144,plain,
% 0.22/0.45 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_2, e_1) | (~product(e_3, e_1, e_1)) | (~product(e_3, e_1, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_2, e_1) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_1, e_1)))),
% 0.22/0.45 inference(transitivity,[status(thm)],[143, 141])).
% 0.22/0.45 tff(145,plain,
% 0.22/0.45 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_2, e_1) | (~product(e_3, e_1, e_1)) | (~product(e_3, e_1, e_2)))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(146,plain,
% 0.22/0.45 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_2, e_1) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_1, e_1))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[145, 144])).
% 0.22/0.45 tff(147,plain,
% 0.22/0.45 ((~product(e_3, e_1, e_2)) | (~product(e_3, e_1, e_1))),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[146, 140, 137])).
% 0.22/0.45 tff(148,plain,
% 0.22/0.45 (~product(e_3, e_1, e_1)),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[147, 127])).
% 0.22/0.45 tff(149,assumption,(product(e_3, e_2, e_3)), introduced(assumption)).
% 0.22/0.45 tff(150,assumption,(~product(e_3, e_2, e_1)), introduced(assumption)).
% 0.22/0.45 tff(151,plain,
% 0.22/0.45 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_2, e_1) | product(e_3, e_3, e_1) | product(e_3, e_1, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | (~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_2, e_1) | product(e_3, e_3, e_1) | product(e_3, e_1, e_1))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(152,plain,
% 0.22/0.45 (((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_3, e_1) | product(e_3, e_2, e_1) | product(e_3, e_1, e_1)) <=> ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_2, e_1) | product(e_3, e_3, e_1) | product(e_3, e_1, e_1))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(153,plain,
% 0.22/0.45 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_3, e_1) | product(e_3, e_2, e_1) | product(e_3, e_1, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_2, e_1) | product(e_3, e_3, e_1) | product(e_3, e_1, e_1)))),
% 0.22/0.45 inference(monotonicity,[status(thm)],[152])).
% 0.22/0.45 tff(154,plain,
% 0.22/0.45 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_3, e_1) | product(e_3, e_2, e_1) | product(e_3, e_1, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | (~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_2, e_1) | product(e_3, e_3, e_1) | product(e_3, e_1, e_1))),
% 0.22/0.46 inference(transitivity,[status(thm)],[153, 151])).
% 0.22/0.46 tff(155,plain,
% 0.22/0.46 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_3, e_1) | product(e_3, e_2, e_1) | product(e_3, e_1, e_1))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(156,plain,
% 0.22/0.46 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | (~group_element(e_1)) | (~group_element(e_3)) | product(e_3, e_2, e_1) | product(e_3, e_3, e_1) | product(e_3, e_1, e_1)),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[155, 154])).
% 0.22/0.46 tff(157,plain,
% 0.22/0.46 (product(e_3, e_2, e_1) | product(e_3, e_3, e_1)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[156, 59, 118, 46, 41])).
% 0.22/0.46 tff(158,plain,
% 0.22/0.46 (product(e_3, e_3, e_1)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[157, 150])).
% 0.22/0.46 tff(159,plain,
% 0.22/0.46 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_2, e_1) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_3, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | product(e_3, e_2, e_1) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_3, e_1)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(160,plain,
% 0.22/0.46 ((product(e_3, e_2, e_1) | (~product(e_3, e_3, e_1)) | (~product(e_3, e_2, e_3))) <=> (product(e_3, e_2, e_1) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_3, e_1)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(161,plain,
% 0.22/0.46 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_2, e_1) | (~product(e_3, e_3, e_1)) | (~product(e_3, e_2, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_2, e_1) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_3, e_1))))),
% 0.22/0.46 inference(monotonicity,[status(thm)],[160])).
% 0.22/0.46 tff(162,plain,
% 0.22/0.46 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_2, e_1) | (~product(e_3, e_3, e_1)) | (~product(e_3, e_2, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | product(e_3, e_2, e_1) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_3, e_1)))),
% 0.22/0.46 inference(transitivity,[status(thm)],[161, 159])).
% 0.22/0.46 tff(163,plain,
% 0.22/0.46 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_2, e_1) | (~product(e_3, e_3, e_1)) | (~product(e_3, e_2, e_3)))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(164,plain,
% 0.22/0.46 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | product(e_3, e_2, e_1) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_3, e_1))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[163, 162])).
% 0.22/0.46 tff(165,plain,
% 0.22/0.46 ($false),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[164, 32, 149, 150, 158])).
% 0.22/0.46 tff(166,plain,(product(e_3, e_2, e_1) | (~product(e_3, e_2, e_3)) | product(e_3, e_1, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.46 tff(167,plain,
% 0.22/0.46 (product(e_3, e_2, e_1)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[166, 149, 41])).
% 0.22/0.46 tff(168,plain,
% 0.22/0.46 ((~equalish(e_1, e_3)) <=> (~equalish(e_1, e_3))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(169,axiom,(~equalish(e_1, e_3)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_1_is_not_e_3')).
% 0.22/0.46 tff(170,plain,
% 0.22/0.46 (~equalish(e_1, e_3)),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[169, 168])).
% 0.22/0.46 tff(171,plain,
% 0.22/0.46 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_1, e_3) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_2, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_1, e_3) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_2, e_1)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(172,plain,
% 0.22/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_1, e_3) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_2, e_1)))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(173,plain,
% 0.22/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_1, e_3) | (~product(e_3, e_2, e_3)) | (~product(e_3, e_2, e_1))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[172, 171])).
% 0.22/0.46 tff(174,plain,
% 0.22/0.46 ($false),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[173, 170, 137, 149, 167])).
% 0.22/0.46 tff(175,plain,((~product(e_3, e_2, e_3)) | product(e_3, e_1, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.46 tff(176,plain,
% 0.22/0.46 (~product(e_3, e_2, e_3)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[175, 148])).
% 0.22/0.46 tff(177,assumption,(~product(e_3, e_1, e_3)), introduced(assumption)).
% 0.22/0.46 tff(178,assumption,(~product(e_3, e_2, e_3)), introduced(assumption)).
% 0.22/0.46 tff(179,plain,
% 0.22/0.46 (product(e_3, e_2, e_1)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[96, 178, 42])).
% 0.22/0.46 tff(180,plain,
% 0.22/0.46 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_1)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(181,plain,
% 0.22/0.46 ((equalish(e_3, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_3, e_2, e_1))) <=> (equalish(e_3, e_1) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_1)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(182,plain,
% 0.22/0.46 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_3, e_2, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_1))))),
% 0.22/0.46 inference(monotonicity,[status(thm)],[181])).
% 0.22/0.46 tff(183,plain,
% 0.22/0.46 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_3, e_2, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_1)))),
% 0.22/0.46 inference(transitivity,[status(thm)],[182, 180])).
% 0.22/0.46 tff(184,plain,
% 0.22/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | (equalish(e_3, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_3, e_2, e_1)))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(185,plain,
% 0.22/0.46 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(Z, Y, X)) | (~product(W, Y, X)))) | equalish(e_3, e_1) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_1))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[184, 183])).
% 0.22/0.46 tff(186,plain,
% 0.22/0.46 ((~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_1))),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[185, 14, 11])).
% 0.22/0.46 tff(187,plain,
% 0.22/0.46 (~product(e_1, e_2, e_1)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[186, 179])).
% 0.22/0.46 tff(188,assumption,(~product(e_1, e_2, e_3)), introduced(assumption)).
% 0.22/0.46 tff(189,assumption,(~product(e_1, e_2, e_1)), introduced(assumption)).
% 0.22/0.46 tff(190,plain,
% 0.22/0.46 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_1)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_1)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(191,plain,
% 0.22/0.46 (((~group_element(e_2)) | (~group_element(e_1)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1)) <=> ((~group_element(e_1)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(192,plain,
% 0.22/0.46 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_1)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_1)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1)))),
% 0.22/0.46 inference(monotonicity,[status(thm)],[191])).
% 0.22/0.46 tff(193,plain,
% 0.22/0.46 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_1)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_1)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1))),
% 0.22/0.46 inference(transitivity,[status(thm)],[192, 190])).
% 0.22/0.46 tff(194,plain,
% 0.22/0.46 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_1)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(195,plain,
% 0.22/0.46 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (~group_element(e_1)) | (~group_element(e_2)) | product(e_1, e_2, e_3) | product(e_1, e_2, e_2) | product(e_1, e_2, e_1)),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[194, 193])).
% 0.22/0.46 tff(196,plain,
% 0.22/0.46 (product(e_1, e_2, e_3) | product(e_1, e_2, e_2)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[195, 118, 49, 89, 189])).
% 0.22/0.46 tff(197,plain,
% 0.22/0.46 (product(e_1, e_2, e_2)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[196, 188])).
% 0.22/0.46 tff(198,plain,
% 0.22/0.46 (^[Y: $i, X: $i] : refl((product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) <=> (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))))),
% 0.22/0.46 inference(bind,[status(th)],[])).
% 0.22/0.46 tff(199,plain,
% 0.22/0.46 (![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) <=> ![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.22/0.47 inference(quant_intro,[status(thm)],[198])).
% 0.22/0.47 tff(200,plain,
% 0.22/0.47 (![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) <=> ![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(201,plain,
% 0.22/0.47 (^[Y: $i, X: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) <=> (product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))), (((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) <=> ((product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) | product(e_2, X, Y)))), rewrite(((product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) | product(e_2, X, Y)) <=> (product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))), (((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) <=> (product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))))), ((((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) | product(e_3, X, Y)) <=> ((product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) | product(e_3, X, Y)))), rewrite(((product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) | product(e_3, X, Y)) <=> (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))), ((((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) | product(e_3, X, Y)) <=> (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))))),
% 0.22/0.47 inference(bind,[status(th)],[])).
% 0.22/0.47 tff(202,plain,
% 0.22/0.47 (![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) | product(e_3, X, Y)) <=> ![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.22/0.47 inference(quant_intro,[status(thm)],[201])).
% 0.22/0.47 tff(203,axiom,(![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) | product(e_3, X, Y))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','row_surjectivity')).
% 0.22/0.47 tff(204,plain,
% 0.22/0.47 (![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[203, 202])).
% 0.22/0.47 tff(205,plain,
% 0.22/0.47 (![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[204, 200])).
% 0.22/0.47 tff(206,plain,(
% 0.22/0.47 ![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.22/0.47 inference(skolemize,[status(sab)],[205])).
% 0.22/0.47 tff(207,plain,
% 0.22/0.47 (![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[206, 199])).
% 0.22/0.47 tff(208,plain,
% 0.22/0.47 (((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_3) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3))) <=> ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_2, e_3) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(209,plain,
% 0.22/0.47 ((product(e_3, e_2, e_3) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3) | (~group_element(e_3)) | (~group_element(e_2))) <=> (product(e_3, e_2, e_3) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(210,plain,
% 0.22/0.47 (((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_3) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3) | (~group_element(e_3)) | (~group_element(e_2)))) <=> ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_3) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3)))),
% 0.22/0.47 inference(monotonicity,[status(thm)],[209])).
% 0.22/0.47 tff(211,plain,
% 0.22/0.47 (((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_3) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3) | (~group_element(e_3)) | (~group_element(e_2)))) <=> ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_2, e_3) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3))),
% 0.22/0.47 inference(transitivity,[status(thm)],[210, 208])).
% 0.22/0.47 tff(212,plain,
% 0.22/0.47 ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_3) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3) | (~group_element(e_3)) | (~group_element(e_2)))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(213,plain,
% 0.22/0.47 ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_2, e_3) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3)),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[212, 211])).
% 0.22/0.47 tff(214,plain,
% 0.22/0.47 (product(e_2, e_2, e_3) | product(e_1, e_2, e_3)),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[213, 207, 49, 46, 178])).
% 0.22/0.47 tff(215,plain,
% 0.22/0.47 (product(e_2, e_2, e_3)),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[214, 188])).
% 0.22/0.47 tff(216,plain,
% 0.22/0.47 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : refl(((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1))) <=> ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1))))),
% 0.22/0.47 inference(bind,[status(th)],[])).
% 0.22/0.47 tff(217,plain,
% 0.22/0.47 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))),
% 0.22/0.47 inference(quant_intro,[status(thm)],[216])).
% 0.22/0.47 tff(218,plain,
% 0.22/0.47 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(219,plain,
% 0.22/0.47 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : rewrite((((~product(X, Y, Z1)) | (~product(Z1, Y, Z2))) | product(X, Z1, Z2)) <=> ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1))))),
% 0.22/0.47 inference(bind,[status(th)],[])).
% 0.22/0.47 tff(220,plain,
% 0.22/0.47 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (((~product(X, Y, Z1)) | (~product(Z1, Y, Z2))) | product(X, Z1, Z2)) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))),
% 0.22/0.47 inference(quant_intro,[status(thm)],[219])).
% 0.22/0.47 tff(221,axiom,(![Y: $i, Z1: $i, X: $i, Z2: $i] : (((~product(X, Y, Z1)) | (~product(Z1, Y, Z2))) | product(X, Z1, Z2))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','qg3')).
% 0.22/0.47 tff(222,plain,
% 0.22/0.47 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[221, 220])).
% 0.22/0.47 tff(223,plain,
% 0.22/0.47 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[222, 218])).
% 0.22/0.47 tff(224,plain,(
% 0.22/0.47 ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))),
% 0.22/0.47 inference(skolemize,[status(sab)],[223])).
% 0.22/0.47 tff(225,plain,
% 0.22/0.47 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[224, 217])).
% 0.22/0.47 tff(226,plain,
% 0.22/0.47 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))) | ((~product(e_2, e_2, e_3)) | product(e_1, e_2, e_3) | (~product(e_1, e_2, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))) | (~product(e_2, e_2, e_3)) | product(e_1, e_2, e_3) | (~product(e_1, e_2, e_2)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(227,plain,
% 0.22/0.47 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))) | ((~product(e_2, e_2, e_3)) | product(e_1, e_2, e_3) | (~product(e_1, e_2, e_2)))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(228,plain,
% 0.22/0.47 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))) | (~product(e_2, e_2, e_3)) | product(e_1, e_2, e_3) | (~product(e_1, e_2, e_2))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[227, 226])).
% 0.22/0.47 tff(229,plain,
% 0.22/0.47 ($false),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[228, 225, 215, 188, 197])).
% 0.22/0.47 tff(230,plain,(product(e_1, e_2, e_3) | product(e_1, e_2, e_1) | product(e_3, e_2, e_3)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.47 tff(231,plain,
% 0.22/0.47 (product(e_1, e_2, e_3)),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[230, 187, 178])).
% 0.22/0.47 tff(232,plain,
% 0.22/0.47 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))) | ((~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3)) | product(e_3, e_1, e_3))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3)) | product(e_3, e_1, e_3))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(233,plain,
% 0.22/0.47 (((~product(e_1, e_2, e_3)) | product(e_3, e_1, e_3) | (~product(e_3, e_2, e_1))) <=> ((~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3)) | product(e_3, e_1, e_3))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(234,plain,
% 0.22/0.47 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))) | ((~product(e_1, e_2, e_3)) | product(e_3, e_1, e_3) | (~product(e_3, e_2, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))) | ((~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3)) | product(e_3, e_1, e_3)))),
% 0.22/0.47 inference(monotonicity,[status(thm)],[233])).
% 0.22/0.47 tff(235,plain,
% 0.22/0.47 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))) | ((~product(e_1, e_2, e_3)) | product(e_3, e_1, e_3) | (~product(e_3, e_2, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3)) | product(e_3, e_1, e_3))),
% 0.22/0.47 inference(transitivity,[status(thm)],[234, 232])).
% 0.22/0.47 tff(236,plain,
% 0.22/0.47 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))) | ((~product(e_1, e_2, e_3)) | product(e_3, e_1, e_3) | (~product(e_3, e_2, e_1)))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(237,plain,
% 0.22/0.47 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(Z1, Y, Z2)) | product(X, Z1, Z2) | (~product(X, Y, Z1)))) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3)) | product(e_3, e_1, e_3)),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[236, 235])).
% 0.22/0.47 tff(238,plain,
% 0.22/0.47 ($false),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[237, 225, 179, 231, 177])).
% 0.22/0.47 tff(239,plain,(product(e_3, e_2, e_3) | product(e_3, e_1, e_3) | product(e_3, e_2, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.47 tff(240,plain,
% 0.22/0.47 (product(e_3, e_1, e_3)),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[239, 176, 42])).
% 0.22/0.47 tff(241,plain,
% 0.22/0.47 ((~equalish(e_3, e_2)) <=> (~equalish(e_3, e_2))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(242,axiom,(~equalish(e_3, e_2)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_3_is_not_e_2')).
% 0.22/0.47 tff(243,plain,
% 0.22/0.47 (~equalish(e_3, e_2)),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[242, 241])).
% 0.22/0.47 tff(244,plain,
% 0.22/0.47 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_3, e_2) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_1, e_3)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_3, e_2) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_1, e_3)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(245,plain,
% 0.22/0.47 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_3, e_2) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_1, e_3)))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(246,plain,
% 0.22/0.47 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_3, e_2) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_1, e_3))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[245, 244])).
% 0.22/0.47 tff(247,plain,
% 0.22/0.47 ($false),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[246, 243, 137, 127, 240])).
% 0.22/0.47 tff(248,plain,(product(e_3, e_2, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.47 tff(249,assumption,(product(e_3, e_2, e_2)), introduced(assumption)).
% 0.22/0.47 tff(250,assumption,(product(e_3, e_1, e_2)), introduced(assumption)).
% 0.22/0.47 tff(251,plain,
% 0.22/0.47 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_1) | (~product(e_3, e_1, e_2)) | (~product(e_3, 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_2, e_1) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_2, e_2)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(252,plain,
% 0.22/0.47 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | (equalish(e_2, e_1) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_2, e_2)))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(253,plain,
% 0.22/0.47 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Z, Y)) | (~product(X, W, Y)))) | equalish(e_2, e_1) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_2, e_2))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[252, 251])).
% 0.22/0.47 tff(254,plain,
% 0.22/0.47 ($false),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[253, 140, 107, 250, 249])).
% 0.22/0.47 tff(255,plain,((~product(e_3, e_2, e_2)) | (~product(e_3, e_1, e_2))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.47 tff(256,plain,
% 0.22/0.47 (~product(e_3, e_1, e_2)),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[255, 248])).
% 0.22/0.47 tff(257,plain,
% 0.22/0.47 (product(e_3, e_1, e_3) | product(e_3, e_1, e_1)),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[124, 118, 46, 89, 256])).
% 0.22/0.47 tff(258,plain,
% 0.22/0.47 (product(e_3, e_1, e_3)),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[257, 41])).
% 0.22/0.47 tff(259,plain,
% 0.22/0.47 (product(e_3, e_3, e_1) | product(e_3, e_1, e_1)),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[156, 59, 118, 46, 150])).
% 0.22/0.47 tff(260,plain,
% 0.22/0.47 (product(e_3, e_3, e_1)),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[259, 41])).
% 0.22/0.47 tff(261,plain,
% 0.22/0.47 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | ((~product(e_3, e_3, e_1)) | product(e_3, e_1, e_1) | (~product(e_3, e_1, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (~product(e_3, e_3, e_1)) | product(e_3, e_1, e_1) | (~product(e_3, e_1, e_3)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(262,plain,
% 0.22/0.47 ((product(e_3, e_1, e_1) | (~product(e_3, e_3, e_1)) | (~product(e_3, e_1, e_3))) <=> ((~product(e_3, e_3, e_1)) | product(e_3, e_1, e_1) | (~product(e_3, e_1, e_3)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(263,plain,
% 0.22/0.47 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_1, e_1) | (~product(e_3, e_3, e_1)) | (~product(e_3, e_1, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | ((~product(e_3, e_3, e_1)) | product(e_3, e_1, e_1) | (~product(e_3, e_1, e_3))))),
% 0.22/0.48 inference(monotonicity,[status(thm)],[262])).
% 0.22/0.48 tff(264,plain,
% 0.22/0.48 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_1, e_1) | (~product(e_3, e_3, e_1)) | (~product(e_3, e_1, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (~product(e_3, e_3, e_1)) | product(e_3, e_1, e_1) | (~product(e_3, e_1, e_3)))),
% 0.22/0.48 inference(transitivity,[status(thm)],[263, 261])).
% 0.22/0.48 tff(265,plain,
% 0.22/0.48 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (product(e_3, e_1, e_1) | (~product(e_3, e_3, e_1)) | (~product(e_3, e_1, e_3)))),
% 0.22/0.48 inference(quant_inst,[status(thm)],[])).
% 0.22/0.48 tff(266,plain,
% 0.22/0.48 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(Z1, Y, Z2) | (~product(X, Z1, Z2)) | (~product(X, Y, Z1)))) | (~product(e_3, e_3, e_1)) | product(e_3, e_1, e_1) | (~product(e_3, e_1, e_3))),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[265, 264])).
% 0.22/0.48 tff(267,plain,
% 0.22/0.48 ($false),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[266, 32, 260, 41, 258])).
% 0.22/0.48 tff(268,plain,(product(e_3, e_1, e_1) | product(e_3, e_2, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.48 tff(269,plain,
% 0.22/0.48 (product(e_3, e_2, e_1)),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[268, 40])).
% 0.22/0.48 tff(270,plain,
% 0.22/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_1, e_2) | (~product(e_3, e_2, e_1)) | (~product(e_3, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_1, e_2) | (~product(e_3, e_2, e_1)) | (~product(e_3, e_2, e_2)))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(271,plain,
% 0.22/0.48 ((equalish(e_1, e_2) | (~product(e_3, e_2, e_2)) | (~product(e_3, e_2, e_1))) <=> (equalish(e_1, e_2) | (~product(e_3, e_2, e_1)) | (~product(e_3, e_2, e_2)))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(272,plain,
% 0.22/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_1, e_2) | (~product(e_3, e_2, e_2)) | (~product(e_3, e_2, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_1, e_2) | (~product(e_3, e_2, e_1)) | (~product(e_3, e_2, e_2))))),
% 0.22/0.48 inference(monotonicity,[status(thm)],[271])).
% 0.22/0.48 tff(273,plain,
% 0.22/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_1, e_2) | (~product(e_3, e_2, e_2)) | (~product(e_3, e_2, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_1, e_2) | (~product(e_3, e_2, e_1)) | (~product(e_3, e_2, e_2)))),
% 0.22/0.48 inference(transitivity,[status(thm)],[272, 270])).
% 0.22/0.48 tff(274,plain,
% 0.22/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_1, e_2) | (~product(e_3, e_2, e_2)) | (~product(e_3, e_2, e_1)))),
% 0.22/0.48 inference(quant_inst,[status(thm)],[])).
% 0.22/0.48 tff(275,plain,
% 0.22/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_1, e_2) | (~product(e_3, e_2, e_1)) | (~product(e_3, e_2, e_2))),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[274, 273])).
% 0.22/0.48 tff(276,plain,
% 0.22/0.48 ($false),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[275, 110, 137, 248, 269])).
% 0.22/0.48 % SZS output end Proof
% 0.22/0.48 % E exiting
%------------------------------------------------------------------------------