%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : BOO004-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.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 05:00:06 AM UTC 2025
% Result : Unsatisfiable 1.84s 2.01s
% Output : Proof 1.84s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : BOO004-1 : TPTP v9.0.0. Released v1.0.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n001.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 11:36:49 EDT 2025
% 0.13/0.34 % CPUTime :
% 1.84/2.01 % SZS status Unsatisfiable
% 1.84/2.01 % SZS output start Proof
% 1.84/2.01 tff(product_type, type, (
% 1.84/2.01 product: ( $i * $i * $i ) > $o)).
% 1.84/2.01 tff(x_type, type, (
% 1.84/2.01 x: $i)).
% 1.84/2.01 tff(add_type, type, (
% 1.84/2.01 add: ( $i * $i ) > $i)).
% 1.84/2.01 tff(multiplicative_identity_type, type, (
% 1.84/2.01 multiplicative_identity: $i)).
% 1.84/2.01 tff(additive_identity_type, type, (
% 1.84/2.01 additive_identity: $i)).
% 1.84/2.01 tff(inverse_type, type, (
% 1.84/2.01 inverse: $i > $i)).
% 1.84/2.01 tff(sum_type, type, (
% 1.84/2.01 sum: ( $i * $i * $i ) > $o)).
% 1.84/2.01 tff(1,assumption,(~product(multiplicative_identity, add(x, x), x)), introduced(assumption)).
% 1.84/2.01 tff(2,plain,
% 1.84/2.01 (^[X: $i] : refl(product(inverse(X), X, additive_identity) <=> product(inverse(X), X, additive_identity))),
% 1.84/2.01 inference(bind,[status(th)],[])).
% 1.84/2.01 tff(3,plain,
% 1.84/2.01 (![X: $i] : product(inverse(X), X, additive_identity) <=> ![X: $i] : product(inverse(X), X, additive_identity)),
% 1.84/2.01 inference(quant_intro,[status(thm)],[2])).
% 1.84/2.01 tff(4,plain,
% 1.84/2.01 (![X: $i] : product(inverse(X), X, additive_identity) <=> ![X: $i] : product(inverse(X), X, additive_identity)),
% 1.84/2.01 inference(rewrite,[status(thm)],[])).
% 1.84/2.01 tff(5,axiom,(![X: $i] : product(inverse(X), X, additive_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_inverse1')).
% 1.84/2.01 tff(6,plain,
% 1.84/2.01 (![X: $i] : product(inverse(X), X, additive_identity)),
% 1.84/2.01 inference(modus_ponens,[status(thm)],[5, 4])).
% 1.84/2.01 tff(7,plain,(
% 1.84/2.01 ![X: $i] : product(inverse(X), X, additive_identity)),
% 1.84/2.01 inference(skolemize,[status(sab)],[6])).
% 1.84/2.01 tff(8,plain,
% 1.84/2.01 (![X: $i] : product(inverse(X), X, additive_identity)),
% 1.84/2.01 inference(modus_ponens,[status(thm)],[7, 3])).
% 1.84/2.01 tff(9,plain,
% 1.84/2.01 ((~![X: $i] : product(inverse(X), X, additive_identity)) | product(inverse(x), x, additive_identity)),
% 1.84/2.01 inference(quant_inst,[status(thm)],[])).
% 1.84/2.01 tff(10,plain,
% 1.84/2.01 (product(inverse(x), x, additive_identity)),
% 1.84/2.01 inference(unit_resolution,[status(thm)],[9, 8])).
% 1.84/2.01 tff(11,plain,
% 1.84/2.01 (^[X: $i] : refl(sum(inverse(X), X, multiplicative_identity) <=> sum(inverse(X), X, multiplicative_identity))),
% 1.84/2.01 inference(bind,[status(th)],[])).
% 1.84/2.01 tff(12,plain,
% 1.84/2.01 (![X: $i] : sum(inverse(X), X, multiplicative_identity) <=> ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 1.84/2.01 inference(quant_intro,[status(thm)],[11])).
% 1.84/2.01 tff(13,plain,
% 1.84/2.01 (![X: $i] : sum(inverse(X), X, multiplicative_identity) <=> ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 1.84/2.01 inference(rewrite,[status(thm)],[])).
% 1.84/2.01 tff(14,axiom,(![X: $i] : sum(inverse(X), X, multiplicative_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_inverse1')).
% 1.84/2.01 tff(15,plain,
% 1.84/2.01 (![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 1.84/2.01 inference(modus_ponens,[status(thm)],[14, 13])).
% 1.84/2.01 tff(16,plain,(
% 1.84/2.01 ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 1.84/2.01 inference(skolemize,[status(sab)],[15])).
% 1.84/2.01 tff(17,plain,
% 1.84/2.01 (![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 1.84/2.01 inference(modus_ponens,[status(thm)],[16, 12])).
% 1.84/2.01 tff(18,plain,
% 1.84/2.01 ((~![X: $i] : sum(inverse(X), X, multiplicative_identity)) | sum(inverse(x), x, multiplicative_identity)),
% 1.84/2.01 inference(quant_inst,[status(thm)],[])).
% 1.84/2.01 tff(19,plain,
% 1.84/2.01 (sum(inverse(x), x, multiplicative_identity)),
% 1.84/2.01 inference(unit_resolution,[status(thm)],[18, 17])).
% 1.84/2.01 tff(20,plain,
% 1.84/2.01 (^[X: $i] : refl(sum(additive_identity, X, X) <=> sum(additive_identity, X, X))),
% 1.84/2.01 inference(bind,[status(th)],[])).
% 1.84/2.01 tff(21,plain,
% 1.84/2.01 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 1.84/2.01 inference(quant_intro,[status(thm)],[20])).
% 1.84/2.01 tff(22,plain,
% 1.84/2.01 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 1.84/2.01 inference(rewrite,[status(thm)],[])).
% 1.84/2.01 tff(23,axiom,(![X: $i] : sum(additive_identity, X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_identity1')).
% 1.84/2.01 tff(24,plain,
% 1.84/2.01 (![X: $i] : sum(additive_identity, X, X)),
% 1.84/2.01 inference(modus_ponens,[status(thm)],[23, 22])).
% 1.84/2.01 tff(25,plain,(
% 1.84/2.01 ![X: $i] : sum(additive_identity, X, X)),
% 1.84/2.01 inference(skolemize,[status(sab)],[24])).
% 1.84/2.01 tff(26,plain,
% 1.84/2.01 (![X: $i] : sum(additive_identity, X, X)),
% 1.84/2.01 inference(modus_ponens,[status(thm)],[25, 21])).
% 1.84/2.01 tff(27,plain,
% 1.84/2.01 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, x, x)),
% 1.84/2.01 inference(quant_inst,[status(thm)],[])).
% 1.84/2.01 tff(28,plain,
% 1.84/2.01 (sum(additive_identity, x, x)),
% 1.84/2.01 inference(unit_resolution,[status(thm)],[27, 26])).
% 1.84/2.01 tff(29,plain,
% 1.84/2.01 (^[Y: $i, X: $i] : refl(sum(X, Y, add(X, Y)) <=> sum(X, Y, add(X, Y)))),
% 1.84/2.01 inference(bind,[status(th)],[])).
% 1.84/2.01 tff(30,plain,
% 1.84/2.01 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 1.84/2.01 inference(quant_intro,[status(thm)],[29])).
% 1.84/2.01 tff(31,plain,
% 1.84/2.01 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 1.84/2.01 inference(rewrite,[status(thm)],[])).
% 1.84/2.01 tff(32,axiom,(![Y: $i, X: $i] : sum(X, Y, add(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','closure_of_addition')).
% 1.84/2.01 tff(33,plain,
% 1.84/2.01 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 1.84/2.01 inference(modus_ponens,[status(thm)],[32, 31])).
% 1.84/2.01 tff(34,plain,(
% 1.84/2.01 ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 1.84/2.01 inference(skolemize,[status(sab)],[33])).
% 1.84/2.01 tff(35,plain,
% 1.84/2.01 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 1.84/2.01 inference(modus_ponens,[status(thm)],[34, 30])).
% 1.84/2.01 tff(36,plain,
% 1.84/2.01 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(x, x, add(x, x))),
% 1.84/2.01 inference(quant_inst,[status(thm)],[])).
% 1.84/2.01 tff(37,plain,
% 1.84/2.01 (sum(x, x, add(x, x))),
% 1.84/2.01 inference(unit_resolution,[status(thm)],[36, 35])).
% 1.84/2.01 tff(38,plain,
% 1.84/2.01 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))))),
% 1.84/2.01 inference(bind,[status(th)],[])).
% 1.84/2.01 tff(39,plain,
% 1.84/2.01 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 1.84/2.01 inference(quant_intro,[status(thm)],[38])).
% 1.84/2.01 tff(40,plain,
% 1.84/2.01 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 1.84/2.01 inference(rewrite,[status(thm)],[])).
% 1.84/2.01 tff(41,plain,
% 1.84/2.01 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) <=> ((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) <=> (((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | (~sum(V3, X, V4))))), rewrite((((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | (~sum(V3, X, V4))) <=> ((~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) <=> ((~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))))), ((((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) | product(V1, V2, V4)) <=> (((~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | product(V1, V2, V4)))), rewrite((((~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | product(V1, V2, V4)) <=> (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), ((((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) | product(V1, V2, V4)) <=> (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))))),
% 1.84/2.01 inference(bind,[status(th)],[])).
% 1.84/2.01 tff(42,plain,
% 1.84/2.01 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) | product(V1, V2, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 1.84/2.01 inference(quant_intro,[status(thm)],[41])).
% 1.84/2.01 tff(43,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) | product(V1, V2, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity7')).
% 1.84/2.01 tff(44,plain,
% 1.84/2.01 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 1.84/2.01 inference(modus_ponens,[status(thm)],[43, 42])).
% 1.84/2.01 tff(45,plain,
% 1.84/2.01 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 1.84/2.01 inference(modus_ponens,[status(thm)],[44, 40])).
% 1.84/2.01 tff(46,plain,(
% 1.84/2.01 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 1.84/2.01 inference(skolemize,[status(sab)],[45])).
% 1.84/2.01 tff(47,plain,
% 1.84/2.01 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 1.84/2.01 inference(modus_ponens,[status(thm)],[46, 39])).
% 1.84/2.01 tff(48,plain,
% 1.84/2.01 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~sum(additive_identity, x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(inverse(x), x, additive_identity)) | (~sum(x, x, add(x, x))) | product(multiplicative_identity, add(x, x), x))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~sum(additive_identity, x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(inverse(x), x, additive_identity)) | (~sum(x, x, add(x, x))) | product(multiplicative_identity, add(x, x), x))),
% 1.84/2.01 inference(rewrite,[status(thm)],[])).
% 1.84/2.01 tff(49,plain,
% 1.84/2.01 ((product(multiplicative_identity, add(x, x), x) | (~product(inverse(x), x, additive_identity)) | (~sum(additive_identity, x, x)) | (~sum(x, x, add(x, x))) | (~sum(inverse(x), x, multiplicative_identity))) <=> ((~sum(additive_identity, x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(inverse(x), x, additive_identity)) | (~sum(x, x, add(x, x))) | product(multiplicative_identity, add(x, x), x))),
% 1.84/2.01 inference(rewrite,[status(thm)],[])).
% 1.84/2.01 tff(50,plain,
% 1.84/2.01 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (product(multiplicative_identity, add(x, x), x) | (~product(inverse(x), x, additive_identity)) | (~sum(additive_identity, x, x)) | (~sum(x, x, add(x, x))) | (~sum(inverse(x), x, multiplicative_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~sum(additive_identity, x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(inverse(x), x, additive_identity)) | (~sum(x, x, add(x, x))) | product(multiplicative_identity, add(x, x), x)))),
% 1.84/2.01 inference(monotonicity,[status(thm)],[49])).
% 1.84/2.01 tff(51,plain,
% 1.84/2.01 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (product(multiplicative_identity, add(x, x), x) | (~product(inverse(x), x, additive_identity)) | (~sum(additive_identity, x, x)) | (~sum(x, x, add(x, x))) | (~sum(inverse(x), x, multiplicative_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~sum(additive_identity, x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(inverse(x), x, additive_identity)) | (~sum(x, x, add(x, x))) | product(multiplicative_identity, add(x, x), x))),
% 1.84/2.02 inference(transitivity,[status(thm)],[50, 48])).
% 1.84/2.02 tff(52,plain,
% 1.84/2.02 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (product(multiplicative_identity, add(x, x), x) | (~product(inverse(x), x, additive_identity)) | (~sum(additive_identity, x, x)) | (~sum(x, x, add(x, x))) | (~sum(inverse(x), x, multiplicative_identity)))),
% 1.84/2.02 inference(quant_inst,[status(thm)],[])).
% 1.84/2.02 tff(53,plain,
% 1.84/2.02 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~sum(additive_identity, x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(inverse(x), x, additive_identity)) | (~sum(x, x, add(x, x))) | product(multiplicative_identity, add(x, x), x)),
% 1.84/2.02 inference(modus_ponens,[status(thm)],[52, 51])).
% 1.84/2.02 tff(54,plain,
% 1.84/2.02 ($false),
% 1.84/2.02 inference(unit_resolution,[status(thm)],[53, 47, 37, 28, 19, 10, 1])).
% 1.84/2.02 tff(55,plain,(product(multiplicative_identity, add(x, x), x)), inference(lemma,lemma(discharge,[]))).
% 1.84/2.02 tff(56,plain,
% 1.84/2.02 (^[X: $i] : refl(product(multiplicative_identity, X, X) <=> product(multiplicative_identity, X, X))),
% 1.84/2.02 inference(bind,[status(th)],[])).
% 1.84/2.02 tff(57,plain,
% 1.84/2.02 (![X: $i] : product(multiplicative_identity, X, X) <=> ![X: $i] : product(multiplicative_identity, X, X)),
% 1.84/2.02 inference(quant_intro,[status(thm)],[56])).
% 1.84/2.02 tff(58,plain,
% 1.84/2.02 (![X: $i] : product(multiplicative_identity, X, X) <=> ![X: $i] : product(multiplicative_identity, X, X)),
% 1.84/2.02 inference(rewrite,[status(thm)],[])).
% 1.84/2.02 tff(59,axiom,(![X: $i] : product(multiplicative_identity, X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_identity1')).
% 1.84/2.02 tff(60,plain,
% 1.84/2.02 (![X: $i] : product(multiplicative_identity, X, X)),
% 1.84/2.02 inference(modus_ponens,[status(thm)],[59, 58])).
% 1.84/2.02 tff(61,plain,(
% 1.84/2.02 ![X: $i] : product(multiplicative_identity, X, X)),
% 1.84/2.02 inference(skolemize,[status(sab)],[60])).
% 1.84/2.02 tff(62,plain,
% 1.84/2.02 (![X: $i] : product(multiplicative_identity, X, X)),
% 1.84/2.02 inference(modus_ponens,[status(thm)],[61, 57])).
% 1.84/2.02 tff(63,plain,
% 1.84/2.02 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, x, x)),
% 1.84/2.02 inference(quant_inst,[status(thm)],[])).
% 1.84/2.02 tff(64,plain,
% 1.84/2.02 (product(multiplicative_identity, x, x)),
% 1.84/2.02 inference(unit_resolution,[status(thm)],[63, 62])).
% 1.84/2.02 tff(65,plain,
% 1.84/2.02 ((~sum(x, x, x)) <=> (~sum(x, x, x))),
% 1.84/2.02 inference(rewrite,[status(thm)],[])).
% 1.84/2.02 tff(66,axiom,(~sum(x, x, x)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_both_equalities')).
% 1.84/2.02 tff(67,plain,
% 1.84/2.02 (~sum(x, x, x)),
% 1.84/2.02 inference(modus_ponens,[status(thm)],[66, 65])).
% 1.84/2.02 tff(68,plain,
% 1.84/2.02 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) <=> (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))))),
% 1.84/2.02 inference(bind,[status(th)],[])).
% 1.84/2.02 tff(69,plain,
% 1.84/2.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 1.84/2.02 inference(quant_intro,[status(thm)],[68])).
% 1.84/2.02 tff(70,plain,
% 1.84/2.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 1.84/2.02 inference(rewrite,[status(thm)],[])).
% 1.84/2.02 tff(71,plain,
% 1.84/2.02 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) <=> (((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~product(X, V3, V4))))), rewrite((((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~product(X, V3, V4))) <=> ((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) <=> ((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4)) <=> (((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | sum(V1, V2, V4)))), rewrite((((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | sum(V1, V2, V4)) <=> (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4)) <=> (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))))),
% 1.84/2.02 inference(bind,[status(th)],[])).
% 1.84/2.02 tff(72,plain,
% 1.84/2.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 1.84/2.02 inference(quant_intro,[status(thm)],[71])).
% 1.84/2.02 tff(73,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity1')).
% 1.84/2.02 tff(74,plain,
% 1.84/2.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 1.84/2.02 inference(modus_ponens,[status(thm)],[73, 72])).
% 1.84/2.02 tff(75,plain,
% 1.84/2.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 1.84/2.02 inference(modus_ponens,[status(thm)],[74, 70])).
% 1.84/2.02 tff(76,plain,(
% 1.84/2.02 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 1.84/2.02 inference(skolemize,[status(sab)],[75])).
% 1.84/2.02 tff(77,plain,
% 1.84/2.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 1.84/2.02 inference(modus_ponens,[status(thm)],[76, 69])).
% 1.84/2.02 tff(78,plain,
% 1.84/2.02 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(x, x, x) | (~product(multiplicative_identity, x, x)) | (~sum(x, x, add(x, x))) | (~product(multiplicative_identity, add(x, x), x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | sum(x, x, x) | (~product(multiplicative_identity, x, x)) | (~sum(x, x, add(x, x))) | (~product(multiplicative_identity, add(x, x), x)))),
% 1.84/2.02 inference(rewrite,[status(thm)],[])).
% 1.84/2.02 tff(79,plain,
% 1.84/2.02 ((sum(x, x, x) | (~product(multiplicative_identity, add(x, x), x)) | (~sum(x, x, add(x, x))) | (~product(multiplicative_identity, x, x)) | (~product(multiplicative_identity, x, x))) <=> (sum(x, x, x) | (~product(multiplicative_identity, x, x)) | (~sum(x, x, add(x, x))) | (~product(multiplicative_identity, add(x, x), x)))),
% 1.84/2.02 inference(rewrite,[status(thm)],[])).
% 1.84/2.02 tff(80,plain,
% 1.84/2.02 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(x, x, x) | (~product(multiplicative_identity, add(x, x), x)) | (~sum(x, x, add(x, x))) | (~product(multiplicative_identity, x, x)) | (~product(multiplicative_identity, x, x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(x, x, x) | (~product(multiplicative_identity, x, x)) | (~sum(x, x, add(x, x))) | (~product(multiplicative_identity, add(x, x), x))))),
% 1.84/2.02 inference(monotonicity,[status(thm)],[79])).
% 1.84/2.02 tff(81,plain,
% 1.84/2.02 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(x, x, x) | (~product(multiplicative_identity, add(x, x), x)) | (~sum(x, x, add(x, x))) | (~product(multiplicative_identity, x, x)) | (~product(multiplicative_identity, x, x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | sum(x, x, x) | (~product(multiplicative_identity, x, x)) | (~sum(x, x, add(x, x))) | (~product(multiplicative_identity, add(x, x), x)))),
% 1.84/2.02 inference(transitivity,[status(thm)],[80, 78])).
% 1.84/2.02 tff(82,plain,
% 1.84/2.02 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(x, x, x) | (~product(multiplicative_identity, add(x, x), x)) | (~sum(x, x, add(x, x))) | (~product(multiplicative_identity, x, x)) | (~product(multiplicative_identity, x, x)))),
% 1.84/2.02 inference(quant_inst,[status(thm)],[])).
% 1.84/2.02 tff(83,plain,
% 1.84/2.02 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | sum(x, x, x) | (~product(multiplicative_identity, x, x)) | (~sum(x, x, add(x, x))) | (~product(multiplicative_identity, add(x, x), x))),
% 1.84/2.02 inference(modus_ponens,[status(thm)],[82, 81])).
% 1.84/2.02 tff(84,plain,
% 1.84/2.02 ($false),
% 1.84/2.02 inference(unit_resolution,[status(thm)],[83, 77, 67, 37, 64, 55])).
% 1.84/2.02 % SZS output end Proof
% 1.84/2.02 % E exiting
%------------------------------------------------------------------------------