%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : BOO003-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n007.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 05:00:05 AM UTC 2025
% Result : Unsatisfiable 0.86s 1.01s
% Output : Proof 0.86s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : BOO003-1 : TPTP v9.0.0. Released v1.0.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n007.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Jun 19 11:34:05 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.86/1.01 % SZS status Unsatisfiable
% 0.86/1.01 % SZS output start Proof
% 0.86/1.01 tff(product_type, type, (
% 0.86/1.01 product: ( $i * $i * $i ) > $o)).
% 0.86/1.01 tff(multiply_type, type, (
% 0.86/1.01 multiply: ( $i * $i ) > $i)).
% 0.86/1.01 tff(x_type, type, (
% 0.86/1.01 x: $i)).
% 0.86/1.01 tff(multiplicative_identity_type, type, (
% 0.86/1.01 multiplicative_identity: $i)).
% 0.86/1.01 tff(sum_type, type, (
% 0.86/1.01 sum: ( $i * $i * $i ) > $o)).
% 0.86/1.01 tff(additive_identity_type, type, (
% 0.86/1.01 additive_identity: $i)).
% 0.86/1.01 tff(inverse_type, type, (
% 0.86/1.01 inverse: $i > $i)).
% 0.86/1.01 tff(1,assumption,(~product(x, multiplicative_identity, multiply(x, x))), introduced(assumption)).
% 0.86/1.01 tff(2,assumption,(~sum(additive_identity, multiply(x, x), multiply(x, x))), introduced(assumption)).
% 0.86/1.01 tff(3,plain,
% 0.86/1.01 (^[X: $i] : refl(sum(additive_identity, X, X) <=> sum(additive_identity, X, X))),
% 0.86/1.01 inference(bind,[status(th)],[])).
% 0.86/1.01 tff(4,plain,
% 0.86/1.01 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 0.86/1.01 inference(quant_intro,[status(thm)],[3])).
% 0.86/1.01 tff(5,plain,
% 0.86/1.01 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 0.86/1.01 inference(rewrite,[status(thm)],[])).
% 0.86/1.01 tff(6,axiom,(![X: $i] : sum(additive_identity, X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_identity1')).
% 0.86/1.01 tff(7,plain,
% 0.86/1.01 (![X: $i] : sum(additive_identity, X, X)),
% 0.86/1.01 inference(modus_ponens,[status(thm)],[6, 5])).
% 0.86/1.01 tff(8,plain,(
% 0.86/1.01 ![X: $i] : sum(additive_identity, X, X)),
% 0.86/1.01 inference(skolemize,[status(sab)],[7])).
% 0.86/1.01 tff(9,plain,
% 0.86/1.01 (![X: $i] : sum(additive_identity, X, X)),
% 0.86/1.01 inference(modus_ponens,[status(thm)],[8, 4])).
% 0.86/1.01 tff(10,plain,
% 0.86/1.01 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, x, x)),
% 0.86/1.01 inference(quant_inst,[status(thm)],[])).
% 0.86/1.01 tff(11,plain,
% 0.86/1.01 (sum(additive_identity, x, x)),
% 0.86/1.01 inference(unit_resolution,[status(thm)],[10, 9])).
% 0.86/1.01 tff(12,plain,
% 0.86/1.01 (^[Y: $i, X: $i] : refl(product(X, Y, multiply(X, Y)) <=> product(X, Y, multiply(X, Y)))),
% 0.86/1.01 inference(bind,[status(th)],[])).
% 0.86/1.01 tff(13,plain,
% 0.86/1.01 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.86/1.01 inference(quant_intro,[status(thm)],[12])).
% 0.86/1.01 tff(14,plain,
% 0.86/1.01 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.86/1.01 inference(rewrite,[status(thm)],[])).
% 0.86/1.01 tff(15,axiom,(![Y: $i, X: $i] : product(X, Y, multiply(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','closure_of_multiplication')).
% 0.86/1.01 tff(16,plain,
% 0.86/1.01 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.86/1.01 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.86/1.01 tff(17,plain,(
% 0.86/1.01 ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.86/1.01 inference(skolemize,[status(sab)],[16])).
% 0.86/1.01 tff(18,plain,
% 0.86/1.01 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.86/1.01 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.86/1.01 tff(19,plain,
% 0.86/1.01 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(x, x, multiply(x, x))),
% 0.86/1.01 inference(quant_inst,[status(thm)],[])).
% 0.86/1.01 tff(20,plain,
% 0.86/1.01 (product(x, x, multiply(x, x))),
% 0.86/1.01 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.86/1.01 tff(21,plain,
% 0.86/1.01 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4))) <=> (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4))))),
% 0.86/1.01 inference(bind,[status(th)],[])).
% 0.86/1.01 tff(22,plain,
% 0.86/1.01 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))),
% 0.86/1.01 inference(quant_intro,[status(thm)],[21])).
% 0.86/1.01 tff(23,plain,
% 0.86/1.01 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))),
% 0.86/1.02 inference(rewrite,[status(thm)],[])).
% 0.86/1.02 tff(24,plain,
% 0.86/1.02 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) <=> ((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))), (((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) <=> (((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) | (~product(V1, V2, V4))))), rewrite((((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) | (~product(V1, V2, V4))) <=> ((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))), (((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) <=> ((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4))))), ((((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(X, V3, V4)) <=> (((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4))) | sum(X, V3, V4)))), rewrite((((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4))) | sum(X, V3, V4)) <=> (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))), ((((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(X, V3, V4)) <=> (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))))),
% 0.86/1.02 inference(bind,[status(th)],[])).
% 0.86/1.02 tff(25,plain,
% 0.86/1.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(X, V3, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))),
% 0.86/1.02 inference(quant_intro,[status(thm)],[24])).
% 0.86/1.02 tff(26,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(X, V3, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity6')).
% 0.86/1.02 tff(27,plain,
% 0.86/1.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.86/1.02 tff(28,plain,
% 0.86/1.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[27, 23])).
% 0.86/1.02 tff(29,plain,(
% 0.86/1.02 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))),
% 0.86/1.02 inference(skolemize,[status(sab)],[28])).
% 0.86/1.02 tff(30,plain,
% 0.86/1.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[29, 22])).
% 0.86/1.02 tff(31,plain,
% 0.86/1.02 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | ((~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x))) | sum(additive_identity, multiply(x, x), multiply(x, x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x))) | sum(additive_identity, multiply(x, x), multiply(x, x)))),
% 0.86/1.02 inference(rewrite,[status(thm)],[])).
% 0.86/1.02 tff(32,plain,
% 0.86/1.02 ((sum(additive_identity, multiply(x, x), multiply(x, x)) | (~product(x, x, multiply(x, x))) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x)))) <=> ((~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x))) | sum(additive_identity, multiply(x, x), multiply(x, x)))),
% 0.86/1.02 inference(rewrite,[status(thm)],[])).
% 0.86/1.02 tff(33,plain,
% 0.86/1.02 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (sum(additive_identity, multiply(x, x), multiply(x, x)) | (~product(x, x, multiply(x, x))) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | ((~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x))) | sum(additive_identity, multiply(x, x), multiply(x, x))))),
% 0.86/1.02 inference(monotonicity,[status(thm)],[32])).
% 0.86/1.02 tff(34,plain,
% 0.86/1.02 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (sum(additive_identity, multiply(x, x), multiply(x, x)) | (~product(x, x, multiply(x, x))) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x))) | sum(additive_identity, multiply(x, x), multiply(x, x)))),
% 0.86/1.02 inference(transitivity,[status(thm)],[33, 31])).
% 0.86/1.02 tff(35,plain,
% 0.86/1.02 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (sum(additive_identity, multiply(x, x), multiply(x, x)) | (~product(x, x, multiply(x, x))) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x))))),
% 0.86/1.02 inference(quant_inst,[status(thm)],[])).
% 0.86/1.02 tff(36,plain,
% 0.86/1.02 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x))) | sum(additive_identity, multiply(x, x), multiply(x, x))),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.86/1.02 tff(37,plain,
% 0.86/1.02 ($false),
% 0.86/1.02 inference(unit_resolution,[status(thm)],[36, 30, 20, 11, 2])).
% 0.86/1.02 tff(38,plain,(sum(additive_identity, multiply(x, x), multiply(x, x))), inference(lemma,lemma(discharge,[]))).
% 0.86/1.02 tff(39,plain,
% 0.86/1.02 (^[X: $i] : refl(product(X, inverse(X), additive_identity) <=> product(X, inverse(X), additive_identity))),
% 0.86/1.02 inference(bind,[status(th)],[])).
% 0.86/1.02 tff(40,plain,
% 0.86/1.02 (![X: $i] : product(X, inverse(X), additive_identity) <=> ![X: $i] : product(X, inverse(X), additive_identity)),
% 0.86/1.02 inference(quant_intro,[status(thm)],[39])).
% 0.86/1.02 tff(41,plain,
% 0.86/1.02 (![X: $i] : product(X, inverse(X), additive_identity) <=> ![X: $i] : product(X, inverse(X), additive_identity)),
% 0.86/1.02 inference(rewrite,[status(thm)],[])).
% 0.86/1.02 tff(42,axiom,(![X: $i] : product(X, inverse(X), additive_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_inverse2')).
% 0.86/1.02 tff(43,plain,
% 0.86/1.02 (![X: $i] : product(X, inverse(X), additive_identity)),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.86/1.02 tff(44,plain,(
% 0.86/1.02 ![X: $i] : product(X, inverse(X), additive_identity)),
% 0.86/1.02 inference(skolemize,[status(sab)],[43])).
% 0.86/1.02 tff(45,plain,
% 0.86/1.02 (![X: $i] : product(X, inverse(X), additive_identity)),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[44, 40])).
% 0.86/1.02 tff(46,plain,
% 0.86/1.02 ((~![X: $i] : product(X, inverse(X), additive_identity)) | product(x, inverse(x), additive_identity)),
% 0.86/1.02 inference(quant_inst,[status(thm)],[])).
% 0.86/1.02 tff(47,plain,
% 0.86/1.02 (product(x, inverse(x), additive_identity)),
% 0.86/1.02 inference(unit_resolution,[status(thm)],[46, 45])).
% 0.86/1.02 tff(48,plain,
% 0.86/1.02 (^[X: $i] : refl(sum(inverse(X), X, multiplicative_identity) <=> sum(inverse(X), X, multiplicative_identity))),
% 0.86/1.02 inference(bind,[status(th)],[])).
% 0.86/1.02 tff(49,plain,
% 0.86/1.02 (![X: $i] : sum(inverse(X), X, multiplicative_identity) <=> ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 0.86/1.02 inference(quant_intro,[status(thm)],[48])).
% 0.86/1.02 tff(50,plain,
% 0.86/1.02 (![X: $i] : sum(inverse(X), X, multiplicative_identity) <=> ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 0.86/1.02 inference(rewrite,[status(thm)],[])).
% 0.86/1.02 tff(51,axiom,(![X: $i] : sum(inverse(X), X, multiplicative_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_inverse1')).
% 0.86/1.02 tff(52,plain,
% 0.86/1.02 (![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.86/1.02 tff(53,plain,(
% 0.86/1.02 ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 0.86/1.02 inference(skolemize,[status(sab)],[52])).
% 0.86/1.02 tff(54,plain,
% 0.86/1.02 (![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[53, 49])).
% 0.86/1.02 tff(55,plain,
% 0.86/1.02 ((~![X: $i] : sum(inverse(X), X, multiplicative_identity)) | sum(inverse(x), x, multiplicative_identity)),
% 0.86/1.02 inference(quant_inst,[status(thm)],[])).
% 0.86/1.02 tff(56,plain,
% 0.86/1.02 (sum(inverse(x), x, multiplicative_identity)),
% 0.86/1.02 inference(unit_resolution,[status(thm)],[55, 54])).
% 0.86/1.02 tff(57,plain,
% 0.86/1.02 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((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)) | (~sum(V1, V2, V4))))),
% 0.86/1.02 inference(bind,[status(th)],[])).
% 0.86/1.02 tff(58,plain,
% 0.86/1.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.86/1.02 inference(quant_intro,[status(thm)],[57])).
% 0.86/1.02 tff(59,plain,
% 0.86/1.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.86/1.02 inference(rewrite,[status(thm)],[])).
% 0.86/1.02 tff(60,plain,
% 0.86/1.02 (^[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))) | (~sum(V1, V2, V4))) <=> (((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~sum(V1, V2, V4))))), rewrite((((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~sum(V1, V2, V4))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~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)))), rewrite((((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))))),
% 0.86/1.02 inference(bind,[status(th)],[])).
% 0.86/1.02 tff(61,plain,
% 0.86/1.02 (![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))) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.86/1.02 inference(quant_intro,[status(thm)],[60])).
% 0.86/1.02 tff(62,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))) | (~sum(V1, V2, V4))) | product(X, V3, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity2')).
% 0.86/1.02 tff(63,plain,
% 0.86/1.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.86/1.02 tff(64,plain,
% 0.86/1.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[63, 59])).
% 0.86/1.02 tff(65,plain,(
% 0.86/1.02 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.86/1.02 inference(skolemize,[status(sab)],[64])).
% 0.86/1.02 tff(66,plain,
% 0.86/1.02 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[65, 58])).
% 0.86/1.02 tff(67,plain,
% 0.86/1.02 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | ((~product(x, inverse(x), additive_identity)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | product(x, multiplicative_identity, multiply(x, x)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~product(x, inverse(x), additive_identity)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | product(x, multiplicative_identity, multiply(x, x)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))))),
% 0.86/1.02 inference(rewrite,[status(thm)],[])).
% 0.86/1.02 tff(68,plain,
% 0.86/1.02 ((product(x, multiplicative_identity, multiply(x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | (~product(x, inverse(x), additive_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x)))) <=> ((~product(x, inverse(x), additive_identity)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | product(x, multiplicative_identity, multiply(x, x)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))))),
% 0.86/1.02 inference(rewrite,[status(thm)],[])).
% 0.86/1.02 tff(69,plain,
% 0.86/1.02 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(x, multiplicative_identity, multiply(x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | (~product(x, inverse(x), additive_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | ((~product(x, inverse(x), additive_identity)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | product(x, multiplicative_identity, multiply(x, x)) | (~sum(additive_identity, multiply(x, x), multiply(x, x)))))),
% 0.86/1.02 inference(monotonicity,[status(thm)],[68])).
% 0.86/1.02 tff(70,plain,
% 0.86/1.02 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(x, multiplicative_identity, multiply(x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | (~product(x, inverse(x), additive_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~product(x, inverse(x), additive_identity)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | product(x, multiplicative_identity, multiply(x, x)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))))),
% 0.86/1.02 inference(transitivity,[status(thm)],[69, 67])).
% 0.86/1.02 tff(71,plain,
% 0.86/1.02 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(x, multiplicative_identity, multiply(x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | (~product(x, inverse(x), additive_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))))),
% 0.86/1.02 inference(quant_inst,[status(thm)],[])).
% 0.86/1.02 tff(72,plain,
% 0.86/1.02 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~product(x, inverse(x), additive_identity)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | product(x, multiplicative_identity, multiply(x, x)) | (~sum(additive_identity, multiply(x, x), multiply(x, x)))),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.86/1.02 tff(73,plain,
% 0.86/1.02 ($false),
% 0.86/1.02 inference(unit_resolution,[status(thm)],[72, 66, 20, 56, 47, 38, 1])).
% 0.86/1.02 tff(74,plain,(product(x, multiplicative_identity, multiply(x, x))), inference(lemma,lemma(discharge,[]))).
% 0.86/1.02 tff(75,plain,
% 0.86/1.02 (^[X: $i] : refl(product(X, multiplicative_identity, X) <=> product(X, multiplicative_identity, X))),
% 0.86/1.02 inference(bind,[status(th)],[])).
% 0.86/1.02 tff(76,plain,
% 0.86/1.02 (![X: $i] : product(X, multiplicative_identity, X) <=> ![X: $i] : product(X, multiplicative_identity, X)),
% 0.86/1.02 inference(quant_intro,[status(thm)],[75])).
% 0.86/1.02 tff(77,plain,
% 0.86/1.02 (![X: $i] : product(X, multiplicative_identity, X) <=> ![X: $i] : product(X, multiplicative_identity, X)),
% 0.86/1.02 inference(rewrite,[status(thm)],[])).
% 0.86/1.02 tff(78,axiom,(![X: $i] : product(X, multiplicative_identity, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_identity2')).
% 0.86/1.02 tff(79,plain,
% 0.86/1.02 (![X: $i] : product(X, multiplicative_identity, X)),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[78, 77])).
% 0.86/1.02 tff(80,plain,(
% 0.86/1.02 ![X: $i] : product(X, multiplicative_identity, X)),
% 0.86/1.02 inference(skolemize,[status(sab)],[79])).
% 0.86/1.02 tff(81,plain,
% 0.86/1.02 (![X: $i] : product(X, multiplicative_identity, X)),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[80, 76])).
% 0.86/1.02 tff(82,plain,
% 0.86/1.02 ((~![X: $i] : product(X, multiplicative_identity, X)) | product(x, multiplicative_identity, x)),
% 0.86/1.02 inference(quant_inst,[status(thm)],[])).
% 0.86/1.02 tff(83,plain,
% 0.86/1.02 (product(x, multiplicative_identity, x)),
% 0.86/1.02 inference(unit_resolution,[status(thm)],[82, 81])).
% 0.86/1.02 tff(84,plain,
% 0.86/1.02 (^[V: $i, Y: $i, U: $i, X: $i] : refl(((U = V) | (~product(X, Y, V)) | (~product(X, Y, U))) <=> ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U))))),
% 0.86/1.02 inference(bind,[status(th)],[])).
% 0.86/1.02 tff(85,plain,
% 0.86/1.02 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U))) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 0.86/1.02 inference(quant_intro,[status(thm)],[84])).
% 0.86/1.02 tff(86,plain,
% 0.86/1.02 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U))) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 0.86/1.02 inference(rewrite,[status(thm)],[])).
% 0.86/1.02 tff(87,plain,
% 0.86/1.02 (^[V: $i, Y: $i, U: $i, X: $i] : rewrite((((~product(X, Y, U)) | (~product(X, Y, V))) | (U = V)) <=> ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U))))),
% 0.86/1.02 inference(bind,[status(th)],[])).
% 0.86/1.02 tff(88,plain,
% 0.86/1.02 (![V: $i, Y: $i, U: $i, X: $i] : (((~product(X, Y, U)) | (~product(X, Y, V))) | (U = V)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 0.86/1.02 inference(quant_intro,[status(thm)],[87])).
% 0.86/1.02 tff(89,axiom,(![V: $i, Y: $i, U: $i, X: $i] : (((~product(X, Y, U)) | (~product(X, Y, V))) | (U = V))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplication_is_well_defined')).
% 0.86/1.02 tff(90,plain,
% 0.86/1.02 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[89, 88])).
% 0.86/1.02 tff(91,plain,
% 0.86/1.02 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[90, 86])).
% 0.86/1.02 tff(92,plain,(
% 0.86/1.02 ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 0.86/1.02 inference(skolemize,[status(sab)],[91])).
% 0.86/1.02 tff(93,plain,
% 0.86/1.02 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[92, 85])).
% 0.86/1.02 tff(94,plain,
% 0.86/1.02 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((~product(x, multiplicative_identity, x)) | (~product(x, multiplicative_identity, multiply(x, x))) | (multiply(x, x) = x))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | (~product(x, multiplicative_identity, x)) | (~product(x, multiplicative_identity, multiply(x, x))) | (multiply(x, x) = x))),
% 0.86/1.02 inference(rewrite,[status(thm)],[])).
% 0.86/1.02 tff(95,plain,
% 0.86/1.02 (((multiply(x, x) = x) | (~product(x, multiplicative_identity, x)) | (~product(x, multiplicative_identity, multiply(x, x)))) <=> ((~product(x, multiplicative_identity, x)) | (~product(x, multiplicative_identity, multiply(x, x))) | (multiply(x, x) = x))),
% 0.86/1.02 inference(rewrite,[status(thm)],[])).
% 0.86/1.02 tff(96,plain,
% 0.86/1.02 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((multiply(x, x) = x) | (~product(x, multiplicative_identity, x)) | (~product(x, multiplicative_identity, multiply(x, x))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((~product(x, multiplicative_identity, x)) | (~product(x, multiplicative_identity, multiply(x, x))) | (multiply(x, x) = x)))),
% 0.86/1.02 inference(monotonicity,[status(thm)],[95])).
% 0.86/1.02 tff(97,plain,
% 0.86/1.02 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((multiply(x, x) = x) | (~product(x, multiplicative_identity, x)) | (~product(x, multiplicative_identity, multiply(x, x))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | (~product(x, multiplicative_identity, x)) | (~product(x, multiplicative_identity, multiply(x, x))) | (multiply(x, x) = x))),
% 0.86/1.02 inference(transitivity,[status(thm)],[96, 94])).
% 0.86/1.02 tff(98,plain,
% 0.86/1.02 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((multiply(x, x) = x) | (~product(x, multiplicative_identity, x)) | (~product(x, multiplicative_identity, multiply(x, x))))),
% 0.86/1.02 inference(quant_inst,[status(thm)],[])).
% 0.86/1.02 tff(99,plain,
% 0.86/1.02 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | (~product(x, multiplicative_identity, x)) | (~product(x, multiplicative_identity, multiply(x, x))) | (multiply(x, x) = x)),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[98, 97])).
% 0.86/1.02 tff(100,plain,
% 0.86/1.02 (multiply(x, x) = x),
% 0.86/1.02 inference(unit_resolution,[status(thm)],[99, 93, 83, 74])).
% 0.86/1.02 tff(101,plain,
% 0.86/1.02 (product(x, x, multiply(x, x)) <=> product(x, x, x)),
% 0.86/1.02 inference(monotonicity,[status(thm)],[100])).
% 0.86/1.02 tff(102,plain,
% 0.86/1.02 (product(x, x, x) <=> product(x, x, multiply(x, x))),
% 0.86/1.02 inference(symmetry,[status(thm)],[101])).
% 0.86/1.02 tff(103,plain,
% 0.86/1.02 ((~product(x, x, x)) <=> (~product(x, x, multiply(x, x)))),
% 0.86/1.02 inference(monotonicity,[status(thm)],[102])).
% 0.86/1.02 tff(104,plain,
% 0.86/1.02 ((~product(x, x, x)) <=> (~product(x, x, x))),
% 0.86/1.02 inference(rewrite,[status(thm)],[])).
% 0.86/1.02 tff(105,axiom,(~product(x, x, x)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_both_equalities')).
% 0.86/1.02 tff(106,plain,
% 0.86/1.02 (~product(x, x, x)),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[105, 104])).
% 0.86/1.02 tff(107,plain,
% 0.86/1.02 (~product(x, x, multiply(x, x))),
% 0.86/1.02 inference(modus_ponens,[status(thm)],[106, 103])).
% 0.86/1.02 tff(108,plain,
% 0.86/1.02 ($false),
% 0.86/1.02 inference(unit_resolution,[status(thm)],[20, 107])).
% 0.86/1.02 % SZS output end Proof
% 0.86/1.02 % E exiting
%------------------------------------------------------------------------------