%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP123-8.003 : TPTP v9.0.0. Released v1.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:07:03 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.47s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.13 % Problem : GRP123-8.003 : TPTP v9.0.0. Released v1.2.0.
% 0.04/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n015.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Jun 19 15:41:40 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.20/0.47 % SZS status Unsatisfiable
% 0.20/0.47 % SZS output start Proof
% 0.20/0.47 tff(product2_type, type, (
% 0.20/0.47 product2: ( $i * $i * $i ) > $o)).
% 0.20/0.47 tff(e_1_type, type, (
% 0.20/0.47 e_1: $i)).
% 0.20/0.47 tff(e_2_type, type, (
% 0.20/0.47 e_2: $i)).
% 0.20/0.47 tff(product1_type, type, (
% 0.20/0.47 product1: ( $i * $i * $i ) > $o)).
% 0.20/0.47 tff(e_3_type, type, (
% 0.20/0.47 e_3: $i)).
% 0.20/0.47 tff(equalish_type, type, (
% 0.20/0.47 equalish: ( $i * $i ) > $o)).
% 0.20/0.47 tff(group_element_type, type, (
% 0.20/0.47 group_element: $i > $o)).
% 0.20/0.47 tff(1,assumption,(product1(e_3, e_1, e_3)), introduced(assumption)).
% 0.20/0.47 tff(2,plain,
% 0.20/0.47 (^[X: $i] : refl(product1(X, X, X) <=> product1(X, X, X))),
% 0.20/0.47 inference(bind,[status(th)],[])).
% 0.20/0.47 tff(3,plain,
% 0.20/0.47 (![X: $i] : product1(X, X, X) <=> ![X: $i] : product1(X, X, X)),
% 0.20/0.47 inference(quant_intro,[status(thm)],[2])).
% 0.20/0.47 tff(4,plain,
% 0.20/0.47 (![X: $i] : product1(X, X, X) <=> ![X: $i] : product1(X, X, X)),
% 0.20/0.47 inference(rewrite,[status(thm)],[])).
% 0.20/0.47 tff(5,axiom,(![X: $i] : product1(X, X, X)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product1_idempotence')).
% 0.20/0.47 tff(6,plain,
% 0.20/0.47 (![X: $i] : product1(X, X, X)),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.20/0.47 tff(7,plain,(
% 0.20/0.47 ![X: $i] : product1(X, X, X)),
% 0.20/0.47 inference(skolemize,[status(sab)],[6])).
% 0.20/0.47 tff(8,plain,
% 0.20/0.47 (![X: $i] : product1(X, X, X)),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.20/0.47 tff(9,plain,
% 0.20/0.47 ((~![X: $i] : product1(X, X, X)) | product1(e_2, e_2, e_2)),
% 0.20/0.47 inference(quant_inst,[status(thm)],[])).
% 0.20/0.47 tff(10,plain,
% 0.20/0.47 (product1(e_2, e_2, e_2)),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[9, 8])).
% 0.20/0.47 tff(11,assumption,(product1(e_2, e_1, e_2)), introduced(assumption)).
% 0.20/0.47 tff(12,plain,
% 0.20/0.47 (^[W: $i, Z: $i, Y: $i, X: $i] : refl((equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y))) <=> (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y))))),
% 0.20/0.47 inference(bind,[status(th)],[])).
% 0.20/0.47 tff(13,plain,
% 0.20/0.47 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))),
% 0.20/0.47 inference(quant_intro,[status(thm)],[12])).
% 0.20/0.47 tff(14,plain,
% 0.20/0.47 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))),
% 0.20/0.47 inference(rewrite,[status(thm)],[])).
% 0.20/0.47 tff(15,plain,
% 0.20/0.47 (^[W: $i, Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~product1(X, W, Y)) | (~product1(X, Z, Y))) <=> ((~product1(X, Z, Y)) | (~product1(X, W, Y)))), ((((~product1(X, W, Y)) | (~product1(X, Z, Y))) | equalish(W, Z)) <=> (((~product1(X, Z, Y)) | (~product1(X, W, Y))) | equalish(W, Z)))), rewrite((((~product1(X, Z, Y)) | (~product1(X, W, Y))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))), ((((~product1(X, W, Y)) | (~product1(X, Z, Y))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))))),
% 0.20/0.47 inference(bind,[status(th)],[])).
% 0.20/0.47 tff(16,plain,
% 0.20/0.47 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product1(X, W, Y)) | (~product1(X, Z, Y))) | equalish(W, Z)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))),
% 0.20/0.47 inference(quant_intro,[status(thm)],[15])).
% 0.20/0.47 tff(17,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product1(X, W, Y)) | (~product1(X, Z, Y))) | equalish(W, Z))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product1_right_cancellation')).
% 0.20/0.47 tff(18,plain,
% 0.20/0.47 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.20/0.47 tff(19,plain,
% 0.20/0.47 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.20/0.47 tff(20,plain,(
% 0.20/0.47 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))),
% 0.20/0.47 inference(skolemize,[status(sab)],[19])).
% 0.20/0.47 tff(21,plain,
% 0.20/0.47 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[20, 13])).
% 0.20/0.47 tff(22,plain,
% 0.20/0.47 ((~equalish(e_1, e_2)) <=> (~equalish(e_1, e_2))),
% 0.20/0.47 inference(rewrite,[status(thm)],[])).
% 0.20/0.47 tff(23,axiom,(~equalish(e_1, e_2)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_1_is_not_e_2')).
% 0.20/0.47 tff(24,plain,
% 0.20/0.47 (~equalish(e_1, e_2)),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.20/0.47 tff(25,plain,
% 0.20/0.47 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))) | (equalish(e_1, e_2) | (~product1(e_2, e_1, e_2)) | (~product1(e_2, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))) | equalish(e_1, e_2) | (~product1(e_2, e_1, e_2)) | (~product1(e_2, e_2, e_2)))),
% 0.20/0.47 inference(rewrite,[status(thm)],[])).
% 0.20/0.47 tff(26,plain,
% 0.20/0.47 ((equalish(e_1, e_2) | (~product1(e_2, e_2, e_2)) | (~product1(e_2, e_1, e_2))) <=> (equalish(e_1, e_2) | (~product1(e_2, e_1, e_2)) | (~product1(e_2, e_2, e_2)))),
% 0.20/0.47 inference(rewrite,[status(thm)],[])).
% 0.20/0.47 tff(27,plain,
% 0.20/0.47 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))) | (equalish(e_1, e_2) | (~product1(e_2, e_2, e_2)) | (~product1(e_2, e_1, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))) | (equalish(e_1, e_2) | (~product1(e_2, e_1, e_2)) | (~product1(e_2, e_2, e_2))))),
% 0.20/0.47 inference(monotonicity,[status(thm)],[26])).
% 0.20/0.47 tff(28,plain,
% 0.20/0.47 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))) | (equalish(e_1, e_2) | (~product1(e_2, e_2, e_2)) | (~product1(e_2, e_1, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))) | equalish(e_1, e_2) | (~product1(e_2, e_1, e_2)) | (~product1(e_2, e_2, e_2)))),
% 0.20/0.47 inference(transitivity,[status(thm)],[27, 25])).
% 0.20/0.47 tff(29,plain,
% 0.20/0.47 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))) | (equalish(e_1, e_2) | (~product1(e_2, e_2, e_2)) | (~product1(e_2, e_1, e_2)))),
% 0.20/0.47 inference(quant_inst,[status(thm)],[])).
% 0.20/0.47 tff(30,plain,
% 0.20/0.47 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(X, Z, Y)) | (~product1(X, W, Y)))) | equalish(e_1, e_2) | (~product1(e_2, e_1, e_2)) | (~product1(e_2, e_2, e_2))),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.20/0.47 tff(31,plain,
% 0.20/0.47 ($false),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[30, 24, 21, 11, 10])).
% 0.20/0.47 tff(32,plain,(~product1(e_2, e_1, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.47 tff(33,plain,
% 0.20/0.47 ((~![X: $i] : product1(X, X, X)) | product1(e_1, e_1, e_1)),
% 0.20/0.47 inference(quant_inst,[status(thm)],[])).
% 0.20/0.47 tff(34,plain,
% 0.20/0.47 (product1(e_1, e_1, e_1)),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[33, 8])).
% 0.20/0.47 tff(35,assumption,(product1(e_2, e_1, e_1)), introduced(assumption)).
% 0.20/0.47 tff(36,plain,
% 0.20/0.47 (^[W: $i, Z: $i, Y: $i, X: $i] : refl((equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X))) <=> (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X))))),
% 0.20/0.47 inference(bind,[status(th)],[])).
% 0.20/0.47 tff(37,plain,
% 0.20/0.47 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))),
% 0.20/0.48 inference(quant_intro,[status(thm)],[36])).
% 0.20/0.48 tff(38,plain,
% 0.20/0.48 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(39,plain,
% 0.20/0.48 (^[W: $i, Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~product1(W, Y, X)) | (~product1(Z, Y, X))) <=> ((~product1(Z, Y, X)) | (~product1(W, Y, X)))), ((((~product1(W, Y, X)) | (~product1(Z, Y, X))) | equalish(W, Z)) <=> (((~product1(Z, Y, X)) | (~product1(W, Y, X))) | equalish(W, Z)))), rewrite((((~product1(Z, Y, X)) | (~product1(W, Y, X))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))), ((((~product1(W, Y, X)) | (~product1(Z, Y, X))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))))),
% 0.20/0.48 inference(bind,[status(th)],[])).
% 0.20/0.48 tff(40,plain,
% 0.20/0.48 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product1(W, Y, X)) | (~product1(Z, Y, X))) | equalish(W, Z)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))),
% 0.20/0.48 inference(quant_intro,[status(thm)],[39])).
% 0.20/0.48 tff(41,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product1(W, Y, X)) | (~product1(Z, Y, X))) | equalish(W, Z))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product1_left_cancellation')).
% 0.20/0.48 tff(42,plain,
% 0.20/0.48 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.20/0.48 tff(43,plain,
% 0.20/0.48 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[42, 38])).
% 0.20/0.48 tff(44,plain,(
% 0.20/0.48 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))),
% 0.20/0.48 inference(skolemize,[status(sab)],[43])).
% 0.20/0.48 tff(45,plain,
% 0.20/0.48 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[44, 37])).
% 0.20/0.48 tff(46,plain,
% 0.20/0.48 ((~equalish(e_2, e_1)) <=> (~equalish(e_2, e_1))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(47,axiom,(~equalish(e_2, e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_2_is_not_e_1')).
% 0.20/0.48 tff(48,plain,
% 0.20/0.48 (~equalish(e_2, e_1)),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.20/0.48 tff(49,plain,
% 0.20/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))) | (equalish(e_2, e_1) | (~product1(e_1, e_1, e_1)) | (~product1(e_2, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))) | equalish(e_2, e_1) | (~product1(e_1, e_1, e_1)) | (~product1(e_2, e_1, e_1)))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(50,plain,
% 0.20/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))) | (equalish(e_2, e_1) | (~product1(e_1, e_1, e_1)) | (~product1(e_2, e_1, e_1)))),
% 0.20/0.48 inference(quant_inst,[status(thm)],[])).
% 0.20/0.48 tff(51,plain,
% 0.20/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))) | equalish(e_2, e_1) | (~product1(e_1, e_1, e_1)) | (~product1(e_2, e_1, e_1))),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.20/0.48 tff(52,plain,
% 0.20/0.48 ($false),
% 0.20/0.48 inference(unit_resolution,[status(thm)],[51, 48, 45, 35, 34])).
% 0.20/0.48 tff(53,plain,(~product1(e_2, e_1, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.48 tff(54,plain,
% 0.20/0.48 (^[Y: $i, X: $i] : refl(((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y))) <=> ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y))))),
% 0.20/0.48 inference(bind,[status(th)],[])).
% 0.20/0.48 tff(55,plain,
% 0.20/0.48 (![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y))) <=> ![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))),
% 0.20/0.48 inference(quant_intro,[status(thm)],[54])).
% 0.20/0.48 tff(56,plain,
% 0.20/0.48 (![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y))) <=> ![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(57,plain,
% 0.20/0.48 (^[Y: $i, X: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~group_element(X)) | (~group_element(Y))) | product1(X, Y, e_1)) <=> ((~group_element(X)) | product1(X, Y, e_1) | (~group_element(Y)))), (((((~group_element(X)) | (~group_element(Y))) | product1(X, Y, e_1)) | product1(X, Y, e_2)) <=> (((~group_element(X)) | product1(X, Y, e_1) | (~group_element(Y))) | product1(X, Y, e_2)))), rewrite((((~group_element(X)) | product1(X, Y, e_1) | (~group_element(Y))) | product1(X, Y, e_2)) <=> ((~group_element(X)) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))), (((((~group_element(X)) | (~group_element(Y))) | product1(X, Y, e_1)) | product1(X, Y, e_2)) <=> ((~group_element(X)) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y))))), ((((((~group_element(X)) | (~group_element(Y))) | product1(X, Y, e_1)) | product1(X, Y, e_2)) | product1(X, Y, e_3)) <=> (((~group_element(X)) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y))) | product1(X, Y, e_3)))), rewrite((((~group_element(X)) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y))) | product1(X, Y, e_3)) <=> ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))), ((((((~group_element(X)) | (~group_element(Y))) | product1(X, Y, e_1)) | product1(X, Y, e_2)) | product1(X, Y, e_3)) <=> ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))))),
% 0.20/0.48 inference(bind,[status(th)],[])).
% 0.20/0.48 tff(58,plain,
% 0.20/0.48 (![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product1(X, Y, e_1)) | product1(X, Y, e_2)) | product1(X, Y, e_3)) <=> ![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))),
% 0.20/0.48 inference(quant_intro,[status(thm)],[57])).
% 0.20/0.48 tff(59,axiom,(![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product1(X, Y, e_1)) | product1(X, Y, e_2)) | product1(X, Y, e_3))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product1_total_function1')).
% 0.20/0.48 tff(60,plain,
% 0.20/0.48 (![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.20/0.48 tff(61,plain,
% 0.20/0.48 (![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[60, 56])).
% 0.20/0.48 tff(62,plain,(
% 0.20/0.48 ![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))),
% 0.20/0.48 inference(skolemize,[status(sab)],[61])).
% 0.20/0.48 tff(63,plain,
% 0.20/0.48 (![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[62, 55])).
% 0.20/0.48 tff(64,plain,
% 0.20/0.48 (group_element(e_2) <=> group_element(e_2)),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(65,axiom,(group_element(e_2)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_2')).
% 0.20/0.48 tff(66,plain,
% 0.20/0.48 (group_element(e_2)),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.20/0.48 tff(67,plain,
% 0.20/0.48 (group_element(e_1) <=> group_element(e_1)),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(68,axiom,(group_element(e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_1')).
% 0.20/0.48 tff(69,plain,
% 0.20/0.48 (group_element(e_1)),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[68, 67])).
% 0.20/0.48 tff(70,plain,
% 0.20/0.48 (((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | ((~group_element(e_1)) | (~group_element(e_2)) | product1(e_2, e_1, e_1) | product1(e_2, e_1, e_2) | product1(e_2, e_1, e_3))) <=> ((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | (~group_element(e_1)) | (~group_element(e_2)) | product1(e_2, e_1, e_1) | product1(e_2, e_1, e_2) | product1(e_2, e_1, e_3))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(71,plain,
% 0.20/0.48 (((~group_element(e_2)) | product1(e_2, e_1, e_3) | product1(e_2, e_1, e_2) | product1(e_2, e_1, e_1) | (~group_element(e_1))) <=> ((~group_element(e_1)) | (~group_element(e_2)) | product1(e_2, e_1, e_1) | product1(e_2, e_1, e_2) | product1(e_2, e_1, e_3))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(72,plain,
% 0.20/0.48 (((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | ((~group_element(e_2)) | product1(e_2, e_1, e_3) | product1(e_2, e_1, e_2) | product1(e_2, e_1, e_1) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | ((~group_element(e_1)) | (~group_element(e_2)) | product1(e_2, e_1, e_1) | product1(e_2, e_1, e_2) | product1(e_2, e_1, e_3)))),
% 0.20/0.48 inference(monotonicity,[status(thm)],[71])).
% 0.20/0.48 tff(73,plain,
% 0.20/0.48 (((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | ((~group_element(e_2)) | product1(e_2, e_1, e_3) | product1(e_2, e_1, e_2) | product1(e_2, e_1, e_1) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | (~group_element(e_1)) | (~group_element(e_2)) | product1(e_2, e_1, e_1) | product1(e_2, e_1, e_2) | product1(e_2, e_1, e_3))),
% 0.20/0.48 inference(transitivity,[status(thm)],[72, 70])).
% 0.20/0.48 tff(74,plain,
% 0.20/0.48 ((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | ((~group_element(e_2)) | product1(e_2, e_1, e_3) | product1(e_2, e_1, e_2) | product1(e_2, e_1, e_1) | (~group_element(e_1)))),
% 0.20/0.48 inference(quant_inst,[status(thm)],[])).
% 0.20/0.48 tff(75,plain,
% 0.20/0.48 ((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | (~group_element(e_1)) | (~group_element(e_2)) | product1(e_2, e_1, e_1) | product1(e_2, e_1, e_2) | product1(e_2, e_1, e_3)),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[74, 73])).
% 0.20/0.48 tff(76,plain,
% 0.20/0.48 (product1(e_2, e_1, e_2) | product1(e_2, e_1, e_3)),
% 0.20/0.48 inference(unit_resolution,[status(thm)],[75, 69, 66, 63, 53])).
% 0.20/0.48 tff(77,plain,
% 0.20/0.48 (product1(e_2, e_1, e_3)),
% 0.20/0.48 inference(unit_resolution,[status(thm)],[76, 32])).
% 0.20/0.48 tff(78,plain,
% 0.20/0.48 ((~equalish(e_3, e_2)) <=> (~equalish(e_3, e_2))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(79,axiom,(~equalish(e_3, e_2)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_3_is_not_e_2')).
% 0.20/0.48 tff(80,plain,
% 0.20/0.48 (~equalish(e_3, e_2)),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[79, 78])).
% 0.20/0.48 tff(81,plain,
% 0.20/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))) | (equalish(e_3, e_2) | (~product1(e_2, e_1, e_3)) | (~product1(e_3, e_1, e_3)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))) | equalish(e_3, e_2) | (~product1(e_2, e_1, e_3)) | (~product1(e_3, e_1, e_3)))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(82,plain,
% 0.20/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))) | (equalish(e_3, e_2) | (~product1(e_2, e_1, e_3)) | (~product1(e_3, e_1, e_3)))),
% 0.20/0.48 inference(quant_inst,[status(thm)],[])).
% 0.20/0.48 tff(83,plain,
% 0.20/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))) | equalish(e_3, e_2) | (~product1(e_2, e_1, e_3)) | (~product1(e_3, e_1, e_3))),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.20/0.48 tff(84,plain,
% 0.20/0.48 ($false),
% 0.20/0.48 inference(unit_resolution,[status(thm)],[83, 80, 45, 77, 1])).
% 0.20/0.48 tff(85,plain,(~product1(e_3, e_1, e_3)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.48 tff(86,assumption,(product1(e_3, e_1, e_1)), introduced(assumption)).
% 0.20/0.48 tff(87,plain,
% 0.20/0.48 ((~equalish(e_3, e_1)) <=> (~equalish(e_3, e_1))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(88,axiom,(~equalish(e_3, e_1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','e_3_is_not_e_1')).
% 0.20/0.48 tff(89,plain,
% 0.20/0.48 (~equalish(e_3, e_1)),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.20/0.48 tff(90,plain,
% 0.20/0.48 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))) | (equalish(e_3, e_1) | (~product1(e_1, e_1, e_1)) | (~product1(e_3, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))) | equalish(e_3, e_1) | (~product1(e_1, e_1, e_1)) | (~product1(e_3, e_1, e_1)))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(91,plain,
% 0.20/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))) | (equalish(e_3, e_1) | (~product1(e_1, e_1, e_1)) | (~product1(e_3, e_1, e_1)))),
% 0.20/0.48 inference(quant_inst,[status(thm)],[])).
% 0.20/0.48 tff(92,plain,
% 0.20/0.48 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product1(Z, Y, X)) | (~product1(W, Y, X)))) | equalish(e_3, e_1) | (~product1(e_1, e_1, e_1)) | (~product1(e_3, e_1, e_1))),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.20/0.48 tff(93,plain,
% 0.20/0.48 ($false),
% 0.20/0.48 inference(unit_resolution,[status(thm)],[92, 89, 45, 86, 34])).
% 0.20/0.48 tff(94,plain,(~product1(e_3, e_1, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.48 tff(95,plain,
% 0.20/0.48 (group_element(e_3) <=> group_element(e_3)),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(96,axiom,(group_element(e_3)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','element_3')).
% 0.20/0.48 tff(97,plain,
% 0.20/0.48 (group_element(e_3)),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[96, 95])).
% 0.20/0.48 tff(98,plain,
% 0.20/0.48 (((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | ((~group_element(e_1)) | product1(e_3, e_1, e_2) | (~group_element(e_3)) | product1(e_3, e_1, e_1) | product1(e_3, e_1, e_3))) <=> ((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | (~group_element(e_1)) | product1(e_3, e_1, e_2) | (~group_element(e_3)) | product1(e_3, e_1, e_1) | product1(e_3, e_1, e_3))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(99,plain,
% 0.20/0.48 (((~group_element(e_3)) | product1(e_3, e_1, e_3) | product1(e_3, e_1, e_2) | product1(e_3, e_1, e_1) | (~group_element(e_1))) <=> ((~group_element(e_1)) | product1(e_3, e_1, e_2) | (~group_element(e_3)) | product1(e_3, e_1, e_1) | product1(e_3, e_1, e_3))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(100,plain,
% 0.20/0.48 (((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | ((~group_element(e_3)) | product1(e_3, e_1, e_3) | product1(e_3, e_1, e_2) | product1(e_3, e_1, e_1) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | ((~group_element(e_1)) | product1(e_3, e_1, e_2) | (~group_element(e_3)) | product1(e_3, e_1, e_1) | product1(e_3, e_1, e_3)))),
% 0.20/0.48 inference(monotonicity,[status(thm)],[99])).
% 0.20/0.48 tff(101,plain,
% 0.20/0.48 (((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | ((~group_element(e_3)) | product1(e_3, e_1, e_3) | product1(e_3, e_1, e_2) | product1(e_3, e_1, e_1) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | (~group_element(e_1)) | product1(e_3, e_1, e_2) | (~group_element(e_3)) | product1(e_3, e_1, e_1) | product1(e_3, e_1, e_3))),
% 0.20/0.48 inference(transitivity,[status(thm)],[100, 98])).
% 0.20/0.48 tff(102,plain,
% 0.20/0.48 ((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | ((~group_element(e_3)) | product1(e_3, e_1, e_3) | product1(e_3, e_1, e_2) | product1(e_3, e_1, e_1) | (~group_element(e_1)))),
% 0.20/0.48 inference(quant_inst,[status(thm)],[])).
% 0.20/0.48 tff(103,plain,
% 0.20/0.48 ((~![Y: $i, X: $i] : ((~group_element(X)) | product1(X, Y, e_3) | product1(X, Y, e_2) | product1(X, Y, e_1) | (~group_element(Y)))) | (~group_element(e_1)) | product1(e_3, e_1, e_2) | (~group_element(e_3)) | product1(e_3, e_1, e_1) | product1(e_3, e_1, e_3)),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.20/0.48 tff(104,plain,
% 0.20/0.48 (product1(e_3, e_1, e_2) | product1(e_3, e_1, e_1) | product1(e_3, e_1, e_3)),
% 0.20/0.48 inference(unit_resolution,[status(thm)],[103, 69, 97, 63])).
% 0.20/0.48 tff(105,plain,
% 0.20/0.48 (product1(e_3, e_1, e_2) | product1(e_3, e_1, e_3)),
% 0.20/0.48 inference(unit_resolution,[status(thm)],[104, 94])).
% 0.20/0.48 tff(106,plain,
% 0.20/0.48 (product1(e_3, e_1, e_2)),
% 0.20/0.48 inference(unit_resolution,[status(thm)],[105, 85])).
% 0.20/0.48 tff(107,assumption,(~product2(e_2, e_2, e_1)), introduced(assumption)).
% 0.20/0.48 tff(108,assumption,(product1(e_3, e_1, e_2)), introduced(assumption)).
% 0.20/0.48 tff(109,plain,
% 0.20/0.48 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : refl((product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1))) <=> (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1))))),
% 0.20/0.48 inference(bind,[status(th)],[])).
% 0.20/0.48 tff(110,plain,
% 0.20/0.48 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))),
% 0.20/0.48 inference(quant_intro,[status(thm)],[109])).
% 0.20/0.48 tff(111,plain,
% 0.20/0.48 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(112,plain,
% 0.20/0.48 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : rewrite((((~product1(X, Y, Z1)) | (~product1(Z1, Y, Z2))) | product2(Z2, X, Y)) <=> (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1))))),
% 0.20/0.48 inference(bind,[status(th)],[])).
% 0.20/0.48 tff(113,plain,
% 0.20/0.48 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (((~product1(X, Y, Z1)) | (~product1(Z1, Y, Z2))) | product2(Z2, X, Y)) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))),
% 0.20/0.48 inference(quant_intro,[status(thm)],[112])).
% 0.20/0.48 tff(114,axiom,(![Y: $i, Z1: $i, X: $i, Z2: $i] : (((~product1(X, Y, Z1)) | (~product1(Z1, Y, Z2))) | product2(Z2, X, Y))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','qg1a')).
% 0.20/0.48 tff(115,plain,
% 0.20/0.48 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[114, 113])).
% 0.20/0.48 tff(116,plain,
% 0.20/0.48 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[115, 111])).
% 0.20/0.48 tff(117,plain,(
% 0.20/0.48 ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))),
% 0.20/0.48 inference(skolemize,[status(sab)],[116])).
% 0.20/0.48 tff(118,plain,
% 0.20/0.48 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))),
% 0.20/0.48 inference(modus_ponens,[status(thm)],[117, 110])).
% 0.20/0.48 tff(119,plain,
% 0.20/0.48 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))) | ((~product1(e_3, e_1, e_2)) | (~product1(e_2, e_1, e_3)) | product2(e_2, e_2, e_1))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))) | (~product1(e_3, e_1, e_2)) | (~product1(e_2, e_1, e_3)) | product2(e_2, e_2, e_1))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(120,plain,
% 0.20/0.48 ((product2(e_2, e_2, e_1) | (~product1(e_3, e_1, e_2)) | (~product1(e_2, e_1, e_3))) <=> ((~product1(e_3, e_1, e_2)) | (~product1(e_2, e_1, e_3)) | product2(e_2, e_2, e_1))),
% 0.20/0.48 inference(rewrite,[status(thm)],[])).
% 0.20/0.48 tff(121,plain,
% 0.20/0.48 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))) | (product2(e_2, e_2, e_1) | (~product1(e_3, e_1, e_2)) | (~product1(e_2, e_1, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))) | ((~product1(e_3, e_1, e_2)) | (~product1(e_2, e_1, e_3)) | product2(e_2, e_2, e_1)))),
% 0.20/0.48 inference(monotonicity,[status(thm)],[120])).
% 0.20/0.48 tff(122,plain,
% 0.20/0.48 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))) | (product2(e_2, e_2, e_1) | (~product1(e_3, e_1, e_2)) | (~product1(e_2, e_1, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))) | (~product1(e_3, e_1, e_2)) | (~product1(e_2, e_1, e_3)) | product2(e_2, e_2, e_1))),
% 0.20/0.48 inference(transitivity,[status(thm)],[121, 119])).
% 0.20/0.49 tff(123,plain,
% 0.20/0.49 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))) | (product2(e_2, e_2, e_1) | (~product1(e_3, e_1, e_2)) | (~product1(e_2, e_1, e_3)))),
% 0.20/0.49 inference(quant_inst,[status(thm)],[])).
% 0.20/0.49 tff(124,plain,
% 0.20/0.49 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product2(Z2, X, Y) | (~product1(Z1, Y, Z2)) | (~product1(X, Y, Z1)))) | (~product1(e_3, e_1, e_2)) | (~product1(e_2, e_1, e_3)) | product2(e_2, e_2, e_1)),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[123, 122])).
% 0.20/0.49 tff(125,plain,
% 0.20/0.49 ($false),
% 0.20/0.49 inference(unit_resolution,[status(thm)],[124, 118, 77, 108, 107])).
% 0.20/0.49 tff(126,plain,((~product1(e_3, e_1, e_2)) | product2(e_2, e_2, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.49 tff(127,plain,
% 0.20/0.49 (product2(e_2, e_2, e_1)),
% 0.20/0.49 inference(unit_resolution,[status(thm)],[126, 106])).
% 0.20/0.49 tff(128,plain,
% 0.20/0.49 (^[X: $i] : refl(product2(X, X, X) <=> product2(X, X, X))),
% 0.20/0.49 inference(bind,[status(th)],[])).
% 0.20/0.49 tff(129,plain,
% 0.20/0.49 (![X: $i] : product2(X, X, X) <=> ![X: $i] : product2(X, X, X)),
% 0.20/0.49 inference(quant_intro,[status(thm)],[128])).
% 0.20/0.49 tff(130,plain,
% 0.20/0.49 (![X: $i] : product2(X, X, X) <=> ![X: $i] : product2(X, X, X)),
% 0.20/0.49 inference(rewrite,[status(thm)],[])).
% 0.20/0.49 tff(131,axiom,(![X: $i] : product2(X, X, X)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product2_idempotence')).
% 0.20/0.49 tff(132,plain,
% 0.20/0.49 (![X: $i] : product2(X, X, X)),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[131, 130])).
% 0.20/0.49 tff(133,plain,(
% 0.20/0.49 ![X: $i] : product2(X, X, X)),
% 0.20/0.49 inference(skolemize,[status(sab)],[132])).
% 0.20/0.49 tff(134,plain,
% 0.20/0.49 (![X: $i] : product2(X, X, X)),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[133, 129])).
% 0.20/0.49 tff(135,plain,
% 0.20/0.49 ((~![X: $i] : product2(X, X, X)) | product2(e_2, e_2, e_2)),
% 0.20/0.49 inference(quant_inst,[status(thm)],[])).
% 0.20/0.49 tff(136,plain,
% 0.20/0.49 (product2(e_2, e_2, e_2)),
% 0.20/0.49 inference(unit_resolution,[status(thm)],[135, 134])).
% 0.20/0.49 tff(137,plain,
% 0.20/0.49 (^[W: $i, Z: $i, Y: $i, X: $i] : refl((equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W))) <=> (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W))))),
% 0.20/0.49 inference(bind,[status(th)],[])).
% 0.20/0.49 tff(138,plain,
% 0.20/0.49 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))),
% 0.20/0.49 inference(quant_intro,[status(thm)],[137])).
% 0.20/0.49 tff(139,plain,
% 0.20/0.49 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))),
% 0.20/0.49 inference(rewrite,[status(thm)],[])).
% 0.20/0.49 tff(140,plain,
% 0.20/0.49 (^[W: $i, Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~product2(X, Y, W)) | (~product2(X, Y, Z))) <=> ((~product2(X, Y, Z)) | (~product2(X, Y, W)))), ((((~product2(X, Y, W)) | (~product2(X, Y, Z))) | equalish(W, Z)) <=> (((~product2(X, Y, Z)) | (~product2(X, Y, W))) | equalish(W, Z)))), rewrite((((~product2(X, Y, Z)) | (~product2(X, Y, W))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))), ((((~product2(X, Y, W)) | (~product2(X, Y, Z))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))))),
% 0.20/0.49 inference(bind,[status(th)],[])).
% 0.20/0.49 tff(141,plain,
% 0.20/0.49 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product2(X, Y, W)) | (~product2(X, Y, Z))) | equalish(W, Z)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))),
% 0.20/0.49 inference(quant_intro,[status(thm)],[140])).
% 0.20/0.49 tff(142,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product2(X, Y, W)) | (~product2(X, Y, Z))) | equalish(W, Z))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product2_total_function2')).
% 0.20/0.49 tff(143,plain,
% 0.20/0.49 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[142, 141])).
% 0.20/0.49 tff(144,plain,
% 0.20/0.49 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[143, 139])).
% 0.20/0.49 tff(145,plain,(
% 0.20/0.49 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))),
% 0.20/0.49 inference(skolemize,[status(sab)],[144])).
% 0.20/0.49 tff(146,plain,
% 0.20/0.49 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[145, 138])).
% 0.20/0.49 tff(147,plain,
% 0.20/0.49 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))) | (equalish(e_2, e_1) | (~product2(e_2, e_2, e_2)) | (~product2(e_2, e_2, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))) | equalish(e_2, e_1) | (~product2(e_2, e_2, e_2)) | (~product2(e_2, e_2, e_1)))),
% 0.20/0.49 inference(rewrite,[status(thm)],[])).
% 0.20/0.49 tff(148,plain,
% 0.20/0.49 ((equalish(e_2, e_1) | (~product2(e_2, e_2, e_1)) | (~product2(e_2, e_2, e_2))) <=> (equalish(e_2, e_1) | (~product2(e_2, e_2, e_2)) | (~product2(e_2, e_2, e_1)))),
% 0.20/0.49 inference(rewrite,[status(thm)],[])).
% 0.20/0.49 tff(149,plain,
% 0.20/0.49 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))) | (equalish(e_2, e_1) | (~product2(e_2, e_2, e_1)) | (~product2(e_2, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))) | (equalish(e_2, e_1) | (~product2(e_2, e_2, e_2)) | (~product2(e_2, e_2, e_1))))),
% 0.20/0.49 inference(monotonicity,[status(thm)],[148])).
% 0.20/0.49 tff(150,plain,
% 0.20/0.49 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))) | (equalish(e_2, e_1) | (~product2(e_2, e_2, e_1)) | (~product2(e_2, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))) | equalish(e_2, e_1) | (~product2(e_2, e_2, e_2)) | (~product2(e_2, e_2, e_1)))),
% 0.20/0.49 inference(transitivity,[status(thm)],[149, 147])).
% 0.20/0.49 tff(151,plain,
% 0.20/0.49 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))) | (equalish(e_2, e_1) | (~product2(e_2, e_2, e_1)) | (~product2(e_2, e_2, e_2)))),
% 0.20/0.49 inference(quant_inst,[status(thm)],[])).
% 0.20/0.49 tff(152,plain,
% 0.20/0.49 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product2(X, Y, Z)) | (~product2(X, Y, W)))) | equalish(e_2, e_1) | (~product2(e_2, e_2, e_2)) | (~product2(e_2, e_2, e_1))),
% 0.20/0.49 inference(modus_ponens,[status(thm)],[151, 150])).
% 0.20/0.49 tff(153,plain,
% 0.20/0.49 ($false),
% 0.20/0.49 inference(unit_resolution,[status(thm)],[152, 48, 146, 136, 127])).
% 0.20/0.49 % SZS output end Proof
% 0.20/0.49 % E exiting
%------------------------------------------------------------------------------