%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : BOO006-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n011.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 13.16s 13.40s
% Output : Proof 13.26s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.14 % Problem : BOO006-1 : TPTP v9.0.0. Released v1.0.0.
% 0.08/0.14 % Command : run_E %s %d THM
% 0.14/0.36 % Computer : n011.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Thu Jun 19 11:32:57 EDT 2025
% 0.14/0.36 % CPUTime :
% 13.16/13.40 % SZS status Unsatisfiable
% 13.16/13.40 % SZS output start Proof
% 13.16/13.40 tff(sum_type, type, (
% 13.16/13.40 sum: ( $i * $i * $i ) > $o)).
% 13.16/13.40 tff(x_type, type, (
% 13.16/13.40 x: $i)).
% 13.16/13.40 tff(inverse_type, type, (
% 13.16/13.40 inverse: $i > $i)).
% 13.16/13.40 tff(multiplicative_identity_type, type, (
% 13.16/13.40 multiplicative_identity: $i)).
% 13.16/13.40 tff(multiply_type, type, (
% 13.16/13.40 multiply: ( $i * $i ) > $i)).
% 13.16/13.40 tff(product_type, type, (
% 13.16/13.40 product: ( $i * $i * $i ) > $o)).
% 13.16/13.40 tff(additive_identity_type, type, (
% 13.16/13.40 additive_identity: $i)).
% 13.16/13.40 tff(1,plain,
% 13.16/13.40 (^[X: $i] : refl(product(multiplicative_identity, X, X) <=> product(multiplicative_identity, X, X))),
% 13.16/13.40 inference(bind,[status(th)],[])).
% 13.16/13.40 tff(2,plain,
% 13.16/13.40 (![X: $i] : product(multiplicative_identity, X, X) <=> ![X: $i] : product(multiplicative_identity, X, X)),
% 13.16/13.40 inference(quant_intro,[status(thm)],[1])).
% 13.16/13.40 tff(3,plain,
% 13.16/13.40 (![X: $i] : product(multiplicative_identity, X, X) <=> ![X: $i] : product(multiplicative_identity, X, X)),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(4,axiom,(![X: $i] : product(multiplicative_identity, X, X)), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','multiplicative_identity1')).
% 13.16/13.40 tff(5,plain,
% 13.16/13.40 (![X: $i] : product(multiplicative_identity, X, X)),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[4, 3])).
% 13.16/13.40 tff(6,plain,(
% 13.16/13.40 ![X: $i] : product(multiplicative_identity, X, X)),
% 13.16/13.40 inference(skolemize,[status(sab)],[5])).
% 13.16/13.40 tff(7,plain,
% 13.16/13.40 (![X: $i] : product(multiplicative_identity, X, X)),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[6, 2])).
% 13.16/13.40 tff(8,plain,
% 13.16/13.40 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, inverse(multiplicative_identity), inverse(multiplicative_identity))),
% 13.16/13.40 inference(quant_inst,[status(thm)],[])).
% 13.16/13.40 tff(9,plain,
% 13.16/13.40 (product(multiplicative_identity, inverse(multiplicative_identity), inverse(multiplicative_identity))),
% 13.16/13.40 inference(unit_resolution,[status(thm)],[8, 7])).
% 13.16/13.40 tff(10,plain,
% 13.16/13.40 (^[X: $i] : refl(product(X, inverse(X), additive_identity) <=> product(X, inverse(X), additive_identity))),
% 13.16/13.40 inference(bind,[status(th)],[])).
% 13.16/13.40 tff(11,plain,
% 13.16/13.40 (![X: $i] : product(X, inverse(X), additive_identity) <=> ![X: $i] : product(X, inverse(X), additive_identity)),
% 13.16/13.40 inference(quant_intro,[status(thm)],[10])).
% 13.16/13.40 tff(12,plain,
% 13.16/13.40 (![X: $i] : product(X, inverse(X), additive_identity) <=> ![X: $i] : product(X, inverse(X), additive_identity)),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(13,axiom,(![X: $i] : product(X, inverse(X), additive_identity)), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','multiplicative_inverse2')).
% 13.16/13.40 tff(14,plain,
% 13.16/13.40 (![X: $i] : product(X, inverse(X), additive_identity)),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[13, 12])).
% 13.16/13.40 tff(15,plain,(
% 13.16/13.40 ![X: $i] : product(X, inverse(X), additive_identity)),
% 13.16/13.40 inference(skolemize,[status(sab)],[14])).
% 13.16/13.40 tff(16,plain,
% 13.16/13.40 (![X: $i] : product(X, inverse(X), additive_identity)),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[15, 11])).
% 13.16/13.40 tff(17,plain,
% 13.16/13.40 ((~![X: $i] : product(X, inverse(X), additive_identity)) | product(multiplicative_identity, inverse(multiplicative_identity), additive_identity)),
% 13.16/13.40 inference(quant_inst,[status(thm)],[])).
% 13.16/13.40 tff(18,plain,
% 13.16/13.40 (product(multiplicative_identity, inverse(multiplicative_identity), additive_identity)),
% 13.16/13.40 inference(unit_resolution,[status(thm)],[17, 16])).
% 13.16/13.40 tff(19,plain,
% 13.16/13.40 (^[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))))),
% 13.16/13.40 inference(bind,[status(th)],[])).
% 13.16/13.40 tff(20,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(quant_intro,[status(thm)],[19])).
% 13.16/13.40 tff(21,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(22,plain,
% 13.16/13.40 (^[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))))),
% 13.16/13.40 inference(bind,[status(th)],[])).
% 13.16/13.40 tff(23,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(quant_intro,[status(thm)],[22])).
% 13.16/13.40 tff(24,axiom,(![V: $i, Y: $i, U: $i, X: $i] : (((~product(X, Y, U)) | (~product(X, Y, V))) | (U = V))), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','multiplication_is_well_defined')).
% 13.16/13.40 tff(25,plain,
% 13.16/13.40 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[24, 23])).
% 13.16/13.40 tff(26,plain,
% 13.16/13.40 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[25, 21])).
% 13.16/13.40 tff(27,plain,(
% 13.16/13.40 ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 13.16/13.40 inference(skolemize,[status(sab)],[26])).
% 13.16/13.40 tff(28,plain,
% 13.16/13.40 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[27, 20])).
% 13.16/13.40 tff(29,plain,
% 13.16/13.40 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((~product(multiplicative_identity, inverse(multiplicative_identity), additive_identity)) | (~product(multiplicative_identity, inverse(multiplicative_identity), inverse(multiplicative_identity))) | (inverse(multiplicative_identity) = additive_identity))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | (~product(multiplicative_identity, inverse(multiplicative_identity), additive_identity)) | (~product(multiplicative_identity, inverse(multiplicative_identity), inverse(multiplicative_identity))) | (inverse(multiplicative_identity) = additive_identity))),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(30,plain,
% 13.16/13.40 (((inverse(multiplicative_identity) = additive_identity) | (~product(multiplicative_identity, inverse(multiplicative_identity), additive_identity)) | (~product(multiplicative_identity, inverse(multiplicative_identity), inverse(multiplicative_identity)))) <=> ((~product(multiplicative_identity, inverse(multiplicative_identity), additive_identity)) | (~product(multiplicative_identity, inverse(multiplicative_identity), inverse(multiplicative_identity))) | (inverse(multiplicative_identity) = additive_identity))),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(31,plain,
% 13.16/13.40 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((inverse(multiplicative_identity) = additive_identity) | (~product(multiplicative_identity, inverse(multiplicative_identity), additive_identity)) | (~product(multiplicative_identity, inverse(multiplicative_identity), inverse(multiplicative_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((~product(multiplicative_identity, inverse(multiplicative_identity), additive_identity)) | (~product(multiplicative_identity, inverse(multiplicative_identity), inverse(multiplicative_identity))) | (inverse(multiplicative_identity) = additive_identity)))),
% 13.16/13.40 inference(monotonicity,[status(thm)],[30])).
% 13.16/13.40 tff(32,plain,
% 13.16/13.40 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((inverse(multiplicative_identity) = additive_identity) | (~product(multiplicative_identity, inverse(multiplicative_identity), additive_identity)) | (~product(multiplicative_identity, inverse(multiplicative_identity), inverse(multiplicative_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | (~product(multiplicative_identity, inverse(multiplicative_identity), additive_identity)) | (~product(multiplicative_identity, inverse(multiplicative_identity), inverse(multiplicative_identity))) | (inverse(multiplicative_identity) = additive_identity))),
% 13.16/13.40 inference(transitivity,[status(thm)],[31, 29])).
% 13.16/13.40 tff(33,plain,
% 13.16/13.40 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((inverse(multiplicative_identity) = additive_identity) | (~product(multiplicative_identity, inverse(multiplicative_identity), additive_identity)) | (~product(multiplicative_identity, inverse(multiplicative_identity), inverse(multiplicative_identity))))),
% 13.16/13.40 inference(quant_inst,[status(thm)],[])).
% 13.16/13.40 tff(34,plain,
% 13.16/13.40 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | (~product(multiplicative_identity, inverse(multiplicative_identity), additive_identity)) | (~product(multiplicative_identity, inverse(multiplicative_identity), inverse(multiplicative_identity))) | (inverse(multiplicative_identity) = additive_identity)),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[33, 32])).
% 13.16/13.40 tff(35,plain,
% 13.16/13.40 (inverse(multiplicative_identity) = additive_identity),
% 13.16/13.40 inference(unit_resolution,[status(thm)],[34, 28, 18, 9])).
% 13.16/13.40 tff(36,plain,
% 13.16/13.40 (product(multiply(x, x), inverse(multiplicative_identity), multiply(multiply(x, x), additive_identity)) <=> product(multiply(x, x), additive_identity, multiply(multiply(x, x), additive_identity))),
% 13.16/13.40 inference(monotonicity,[status(thm)],[35])).
% 13.16/13.40 tff(37,plain,
% 13.16/13.40 (product(multiply(x, x), additive_identity, multiply(multiply(x, x), additive_identity)) <=> product(multiply(x, x), inverse(multiplicative_identity), multiply(multiply(x, x), additive_identity))),
% 13.16/13.40 inference(symmetry,[status(thm)],[36])).
% 13.16/13.40 tff(38,plain,
% 13.16/13.40 (^[Y: $i, X: $i] : refl(product(X, Y, multiply(X, Y)) <=> product(X, Y, multiply(X, Y)))),
% 13.16/13.40 inference(bind,[status(th)],[])).
% 13.16/13.40 tff(39,plain,
% 13.16/13.40 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 13.16/13.40 inference(quant_intro,[status(thm)],[38])).
% 13.16/13.40 tff(40,plain,
% 13.16/13.40 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(41,axiom,(![Y: $i, X: $i] : product(X, Y, multiply(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','closure_of_multiplication')).
% 13.16/13.40 tff(42,plain,
% 13.16/13.40 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[41, 40])).
% 13.16/13.40 tff(43,plain,(
% 13.16/13.40 ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 13.16/13.40 inference(skolemize,[status(sab)],[42])).
% 13.16/13.40 tff(44,plain,
% 13.16/13.40 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[43, 39])).
% 13.16/13.40 tff(45,plain,
% 13.16/13.40 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(multiply(x, x), additive_identity, multiply(multiply(x, x), additive_identity))),
% 13.16/13.40 inference(quant_inst,[status(thm)],[])).
% 13.16/13.40 tff(46,plain,
% 13.16/13.40 (product(multiply(x, x), additive_identity, multiply(multiply(x, x), additive_identity))),
% 13.16/13.40 inference(unit_resolution,[status(thm)],[45, 44])).
% 13.16/13.40 tff(47,plain,
% 13.16/13.40 (product(multiply(x, x), inverse(multiplicative_identity), multiply(multiply(x, x), additive_identity))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[46, 37])).
% 13.16/13.40 tff(48,plain,
% 13.16/13.40 (sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity)) <=> sum(multiply(multiply(x, x), additive_identity), additive_identity, multiply(x, additive_identity))),
% 13.16/13.40 inference(monotonicity,[status(thm)],[35])).
% 13.16/13.40 tff(49,plain,
% 13.16/13.40 (sum(multiply(multiply(x, x), additive_identity), additive_identity, multiply(x, additive_identity)) <=> sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity))),
% 13.16/13.40 inference(symmetry,[status(thm)],[48])).
% 13.16/13.40 tff(50,plain,
% 13.16/13.40 ((~sum(multiply(multiply(x, x), additive_identity), additive_identity, multiply(x, additive_identity))) <=> (~sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity)))),
% 13.16/13.40 inference(monotonicity,[status(thm)],[49])).
% 13.16/13.40 tff(51,assumption,(~sum(inverse(multiply(x, x)), additive_identity, inverse(multiply(x, x)))), introduced(assumption)).
% 13.16/13.40 tff(52,plain,
% 13.16/13.40 (^[X: $i] : refl(sum(X, additive_identity, X) <=> sum(X, additive_identity, X))),
% 13.16/13.40 inference(bind,[status(th)],[])).
% 13.16/13.40 tff(53,plain,
% 13.16/13.40 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 13.16/13.40 inference(quant_intro,[status(thm)],[52])).
% 13.16/13.40 tff(54,plain,
% 13.16/13.40 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(55,axiom,(![X: $i] : sum(X, additive_identity, X)), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','additive_identity2')).
% 13.16/13.40 tff(56,plain,
% 13.16/13.40 (![X: $i] : sum(X, additive_identity, X)),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[55, 54])).
% 13.16/13.40 tff(57,plain,(
% 13.16/13.40 ![X: $i] : sum(X, additive_identity, X)),
% 13.16/13.40 inference(skolemize,[status(sab)],[56])).
% 13.16/13.40 tff(58,plain,
% 13.16/13.40 (![X: $i] : sum(X, additive_identity, X)),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[57, 53])).
% 13.16/13.40 tff(59,plain,
% 13.16/13.40 ((~![X: $i] : sum(X, additive_identity, X)) | sum(inverse(multiply(x, x)), additive_identity, inverse(multiply(x, x)))),
% 13.16/13.40 inference(quant_inst,[status(thm)],[])).
% 13.16/13.40 tff(60,plain,
% 13.16/13.40 ($false),
% 13.16/13.40 inference(unit_resolution,[status(thm)],[59, 58, 51])).
% 13.16/13.40 tff(61,plain,(sum(inverse(multiply(x, x)), additive_identity, inverse(multiply(x, x)))), inference(lemma,lemma(discharge,[]))).
% 13.16/13.40 tff(62,plain,
% 13.16/13.40 (^[Z: $i, Y: $i, X: $i] : refl(((~sum(X, Y, Z)) | sum(Y, X, Z)) <=> ((~sum(X, Y, Z)) | sum(Y, X, Z)))),
% 13.16/13.40 inference(bind,[status(th)],[])).
% 13.16/13.40 tff(63,plain,
% 13.16/13.40 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z)) <=> ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 13.16/13.40 inference(quant_intro,[status(thm)],[62])).
% 13.16/13.40 tff(64,plain,
% 13.16/13.40 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z)) <=> ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(65,axiom,(![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','commutativity_of_addition')).
% 13.16/13.40 tff(66,plain,
% 13.16/13.40 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[65, 64])).
% 13.16/13.40 tff(67,plain,(
% 13.16/13.40 ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 13.16/13.40 inference(skolemize,[status(sab)],[66])).
% 13.16/13.40 tff(68,plain,
% 13.16/13.40 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[67, 63])).
% 13.16/13.40 tff(69,plain,
% 13.16/13.40 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(inverse(multiply(x, x)), additive_identity, inverse(multiply(x, x)))) | sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(inverse(multiply(x, x)), additive_identity, inverse(multiply(x, x)))) | sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x))))),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(70,plain,
% 13.16/13.40 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(inverse(multiply(x, x)), additive_identity, inverse(multiply(x, x)))) | sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x))))),
% 13.16/13.40 inference(quant_inst,[status(thm)],[])).
% 13.16/13.40 tff(71,plain,
% 13.16/13.40 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(inverse(multiply(x, x)), additive_identity, inverse(multiply(x, x)))) | sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[70, 69])).
% 13.16/13.40 tff(72,plain,
% 13.16/13.40 (sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))),
% 13.16/13.40 inference(unit_resolution,[status(thm)],[71, 68, 61])).
% 13.16/13.40 tff(73,plain,
% 13.16/13.40 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(x, additive_identity, multiply(x, additive_identity))),
% 13.16/13.40 inference(quant_inst,[status(thm)],[])).
% 13.16/13.40 tff(74,plain,
% 13.16/13.40 (product(x, additive_identity, multiply(x, additive_identity))),
% 13.16/13.40 inference(unit_resolution,[status(thm)],[73, 44])).
% 13.16/13.40 tff(75,plain,
% 13.16/13.40 ((~![X: $i] : sum(X, additive_identity, X)) | sum(additive_identity, additive_identity, additive_identity)),
% 13.16/13.40 inference(quant_inst,[status(thm)],[])).
% 13.16/13.40 tff(76,plain,
% 13.16/13.40 (sum(additive_identity, additive_identity, additive_identity)),
% 13.16/13.40 inference(unit_resolution,[status(thm)],[75, 58])).
% 13.16/13.40 tff(77,plain,
% 13.16/13.40 ((~product(x, additive_identity, additive_identity)) <=> (~product(x, additive_identity, additive_identity))),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(78,axiom,(~product(x, additive_identity, additive_identity)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_equations')).
% 13.16/13.40 tff(79,plain,
% 13.16/13.40 (~product(x, additive_identity, additive_identity)),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[78, 77])).
% 13.16/13.40 tff(80,plain,
% 13.16/13.40 (^[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))))),
% 13.16/13.40 inference(bind,[status(th)],[])).
% 13.16/13.40 tff(81,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(quant_intro,[status(thm)],[80])).
% 13.16/13.40 tff(82,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(83,plain,
% 13.16/13.40 (^[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)))))),
% 13.16/13.40 inference(bind,[status(th)],[])).
% 13.16/13.40 tff(84,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(quant_intro,[status(thm)],[83])).
% 13.16/13.40 tff(85,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/sandbox/benchmark/Axioms/BOO002-0.ax','distributivity2')).
% 13.16/13.40 tff(86,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[85, 84])).
% 13.16/13.40 tff(87,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[86, 82])).
% 13.16/13.40 tff(88,plain,(
% 13.16/13.40 ![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)))),
% 13.16/13.40 inference(skolemize,[status(sab)],[87])).
% 13.16/13.40 tff(89,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[88, 81])).
% 13.16/13.40 tff(90,plain,
% 13.16/13.40 (((~![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, additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity)) | (~product(x, additive_identity, multiply(x, additive_identity))))) <=> ((~![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, additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity)) | (~product(x, additive_identity, multiply(x, additive_identity))))),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(91,plain,
% 13.16/13.40 ((product(x, additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(x, additive_identity, multiply(x, additive_identity))) | (~product(x, additive_identity, multiply(x, additive_identity))) | (~sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity))) <=> (product(x, additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity)) | (~product(x, additive_identity, multiply(x, additive_identity))))),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(92,plain,
% 13.16/13.40 (((~![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, additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(x, additive_identity, multiply(x, additive_identity))) | (~product(x, additive_identity, multiply(x, additive_identity))) | (~sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity)))) <=> ((~![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, additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity)) | (~product(x, additive_identity, multiply(x, additive_identity)))))),
% 13.16/13.40 inference(monotonicity,[status(thm)],[91])).
% 13.16/13.40 tff(93,plain,
% 13.16/13.40 (((~![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, additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(x, additive_identity, multiply(x, additive_identity))) | (~product(x, additive_identity, multiply(x, additive_identity))) | (~sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity)))) <=> ((~![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, additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity)) | (~product(x, additive_identity, multiply(x, additive_identity))))),
% 13.16/13.40 inference(transitivity,[status(thm)],[92, 90])).
% 13.16/13.40 tff(94,plain,
% 13.16/13.40 ((~![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, additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(x, additive_identity, multiply(x, additive_identity))) | (~product(x, additive_identity, multiply(x, additive_identity))) | (~sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity)))),
% 13.16/13.40 inference(quant_inst,[status(thm)],[])).
% 13.16/13.40 tff(95,plain,
% 13.16/13.40 ((~![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, additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity)) | (~product(x, additive_identity, multiply(x, additive_identity)))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[94, 93])).
% 13.16/13.40 tff(96,plain,
% 13.16/13.40 (~sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity)),
% 13.16/13.40 inference(unit_resolution,[status(thm)],[95, 89, 79, 76, 74])).
% 13.16/13.40 tff(97,plain,
% 13.16/13.40 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, additive_identity, additive_identity)),
% 13.16/13.40 inference(quant_inst,[status(thm)],[])).
% 13.16/13.40 tff(98,plain,
% 13.16/13.40 (product(multiplicative_identity, additive_identity, additive_identity)),
% 13.16/13.40 inference(unit_resolution,[status(thm)],[97, 7])).
% 13.16/13.40 tff(99,plain,
% 13.16/13.40 (^[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))))),
% 13.16/13.40 inference(bind,[status(th)],[])).
% 13.16/13.40 tff(100,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(quant_intro,[status(thm)],[99])).
% 13.16/13.40 tff(101,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(102,plain,
% 13.16/13.40 (^[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)))))),
% 13.16/13.40 inference(bind,[status(th)],[])).
% 13.16/13.40 tff(103,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(quant_intro,[status(thm)],[102])).
% 13.16/13.40 tff(104,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/sandbox/benchmark/Axioms/BOO002-0.ax','distributivity1')).
% 13.16/13.40 tff(105,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[104, 103])).
% 13.16/13.40 tff(106,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[105, 101])).
% 13.16/13.40 tff(107,plain,(
% 13.16/13.40 ![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)))),
% 13.16/13.40 inference(skolemize,[status(sab)],[106])).
% 13.16/13.40 tff(108,plain,
% 13.16/13.40 (![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)))),
% 13.16/13.40 inference(modus_ponens,[status(thm)],[107, 100])).
% 13.16/13.40 tff(109,plain,
% 13.16/13.40 (((~![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)))) | ((~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity) | (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity))))) <=> ((~![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)))) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity) | (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity))))),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(110,plain,
% 13.16/13.40 ((sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity))) | (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity)))) <=> ((~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity) | (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity))))),
% 13.16/13.40 inference(rewrite,[status(thm)],[])).
% 13.16/13.40 tff(111,plain,
% 13.16/13.40 (((~![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(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity))) | (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity))))) <=> ((~![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)))) | ((~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity) | (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity)))))),
% 13.16/13.41 inference(monotonicity,[status(thm)],[110])).
% 13.16/13.41 tff(112,plain,
% 13.16/13.41 (((~![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(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity))) | (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity))))) <=> ((~![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)))) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity) | (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity))))),
% 13.16/13.41 inference(transitivity,[status(thm)],[111, 109])).
% 13.16/13.41 tff(113,plain,
% 13.16/13.41 ((~![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(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity))) | (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity))))),
% 13.16/13.41 inference(quant_inst,[status(thm)],[])).
% 13.16/13.41 tff(114,plain,
% 13.16/13.41 ((~![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)))) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | sum(multiply(x, additive_identity), multiply(x, additive_identity), additive_identity) | (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity)))),
% 13.16/13.41 inference(modus_ponens,[status(thm)],[113, 112])).
% 13.16/13.41 tff(115,plain,
% 13.16/13.41 (~product(multiplicative_identity, additive_identity, multiply(x, additive_identity))),
% 13.16/13.41 inference(unit_resolution,[status(thm)],[114, 108, 76, 98, 96])).
% 13.16/13.41 tff(116,plain,
% 13.16/13.41 (((~![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(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | product(multiplicative_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, additive_identity, multiply(x, additive_identity))))) <=> ((~![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(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | product(multiplicative_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, additive_identity, multiply(x, additive_identity))))),
% 13.16/13.41 inference(rewrite,[status(thm)],[])).
% 13.16/13.41 tff(117,plain,
% 13.16/13.41 ((product(multiplicative_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, multiply(x, additive_identity)))) <=> ((~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | product(multiplicative_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, additive_identity, multiply(x, additive_identity))))),
% 13.16/13.41 inference(rewrite,[status(thm)],[])).
% 13.16/13.41 tff(118,plain,
% 13.16/13.41 (((~![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(multiplicative_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, multiply(x, additive_identity))))) <=> ((~![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(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | product(multiplicative_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, additive_identity, multiply(x, additive_identity)))))),
% 13.16/13.41 inference(monotonicity,[status(thm)],[117])).
% 13.16/13.41 tff(119,plain,
% 13.16/13.41 (((~![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(multiplicative_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, multiply(x, additive_identity))))) <=> ((~![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(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | product(multiplicative_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, additive_identity, multiply(x, additive_identity))))),
% 13.16/13.41 inference(transitivity,[status(thm)],[118, 116])).
% 13.16/13.41 tff(120,plain,
% 13.16/13.41 ((~![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(multiplicative_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, multiply(x, additive_identity))))),
% 13.16/13.41 inference(quant_inst,[status(thm)],[])).
% 13.16/13.41 tff(121,plain,
% 13.16/13.41 ((~![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(multiplicative_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | product(multiplicative_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, additive_identity, multiply(x, additive_identity)))),
% 13.16/13.41 inference(modus_ponens,[status(thm)],[120, 119])).
% 13.16/13.41 tff(122,plain,
% 13.16/13.41 (~sum(additive_identity, additive_identity, multiply(x, additive_identity))),
% 13.16/13.41 inference(unit_resolution,[status(thm)],[121, 89, 76, 98, 115])).
% 13.16/13.41 tff(123,plain,
% 13.16/13.41 (^[X: $i] : refl(product(inverse(X), X, additive_identity) <=> product(inverse(X), X, additive_identity))),
% 13.16/13.41 inference(bind,[status(th)],[])).
% 13.16/13.41 tff(124,plain,
% 13.16/13.41 (![X: $i] : product(inverse(X), X, additive_identity) <=> ![X: $i] : product(inverse(X), X, additive_identity)),
% 13.16/13.41 inference(quant_intro,[status(thm)],[123])).
% 13.16/13.41 tff(125,plain,
% 13.16/13.41 (![X: $i] : product(inverse(X), X, additive_identity) <=> ![X: $i] : product(inverse(X), X, additive_identity)),
% 13.16/13.41 inference(rewrite,[status(thm)],[])).
% 13.16/13.41 tff(126,axiom,(![X: $i] : product(inverse(X), X, additive_identity)), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','multiplicative_inverse1')).
% 13.16/13.41 tff(127,plain,
% 13.16/13.41 (![X: $i] : product(inverse(X), X, additive_identity)),
% 13.16/13.41 inference(modus_ponens,[status(thm)],[126, 125])).
% 13.16/13.41 tff(128,plain,(
% 13.16/13.41 ![X: $i] : product(inverse(X), X, additive_identity)),
% 13.16/13.41 inference(skolemize,[status(sab)],[127])).
% 13.16/13.41 tff(129,plain,
% 13.16/13.41 (![X: $i] : product(inverse(X), X, additive_identity)),
% 13.16/13.41 inference(modus_ponens,[status(thm)],[128, 124])).
% 13.16/13.41 tff(130,plain,
% 13.16/13.41 ((~![X: $i] : product(inverse(X), X, additive_identity)) | product(inverse(multiply(x, x)), multiply(x, x), additive_identity)),
% 13.16/13.41 inference(quant_inst,[status(thm)],[])).
% 13.16/13.41 tff(131,plain,
% 13.16/13.41 (product(inverse(multiply(x, x)), multiply(x, x), additive_identity)),
% 13.16/13.41 inference(unit_resolution,[status(thm)],[130, 129])).
% 13.16/13.41 tff(132,plain,
% 13.16/13.41 (^[Z: $i, Y: $i, X: $i] : refl(((~product(X, Y, Z)) | product(Y, X, Z)) <=> ((~product(X, Y, Z)) | product(Y, X, Z)))),
% 13.16/13.41 inference(bind,[status(th)],[])).
% 13.16/13.41 tff(133,plain,
% 13.16/13.41 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z)) <=> ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 13.16/13.41 inference(quant_intro,[status(thm)],[132])).
% 13.16/13.41 tff(134,plain,
% 13.16/13.41 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z)) <=> ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 13.16/13.41 inference(rewrite,[status(thm)],[])).
% 13.16/13.41 tff(135,axiom,(![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','commutativity_of_multiplication')).
% 13.16/13.41 tff(136,plain,
% 13.16/13.41 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 13.16/13.41 inference(modus_ponens,[status(thm)],[135, 134])).
% 13.16/13.41 tff(137,plain,(
% 13.16/13.41 ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 13.16/13.41 inference(skolemize,[status(sab)],[136])).
% 13.16/13.41 tff(138,plain,
% 13.16/13.41 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 13.16/13.41 inference(modus_ponens,[status(thm)],[137, 133])).
% 13.16/13.41 tff(139,plain,
% 13.16/13.41 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | ((~product(inverse(multiply(x, x)), multiply(x, x), additive_identity)) | product(multiply(x, x), inverse(multiply(x, x)), additive_identity))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | (~product(inverse(multiply(x, x)), multiply(x, x), additive_identity)) | product(multiply(x, x), inverse(multiply(x, x)), additive_identity))),
% 13.16/13.41 inference(rewrite,[status(thm)],[])).
% 13.16/13.41 tff(140,plain,
% 13.16/13.41 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | ((~product(inverse(multiply(x, x)), multiply(x, x), additive_identity)) | product(multiply(x, x), inverse(multiply(x, x)), additive_identity))),
% 13.16/13.41 inference(quant_inst,[status(thm)],[])).
% 13.16/13.41 tff(141,plain,
% 13.16/13.41 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | (~product(inverse(multiply(x, x)), multiply(x, x), additive_identity)) | product(multiply(x, x), inverse(multiply(x, x)), additive_identity)),
% 13.16/13.41 inference(modus_ponens,[status(thm)],[140, 139])).
% 13.16/13.41 tff(142,plain,
% 13.16/13.41 (product(multiply(x, x), inverse(multiply(x, x)), additive_identity)),
% 13.16/13.41 inference(unit_resolution,[status(thm)],[141, 138, 131])).
% 13.16/13.41 tff(143,plain,
% 13.16/13.41 (^[X: $i] : refl(sum(additive_identity, X, X) <=> sum(additive_identity, X, X))),
% 13.16/13.41 inference(bind,[status(th)],[])).
% 13.16/13.41 tff(144,plain,
% 13.16/13.41 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 13.16/13.41 inference(quant_intro,[status(thm)],[143])).
% 13.16/13.41 tff(145,plain,
% 13.16/13.41 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 13.16/13.41 inference(rewrite,[status(thm)],[])).
% 13.16/13.41 tff(146,axiom,(![X: $i] : sum(additive_identity, X, X)), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','additive_identity1')).
% 13.16/13.41 tff(147,plain,
% 13.16/13.41 (![X: $i] : sum(additive_identity, X, X)),
% 13.16/13.41 inference(modus_ponens,[status(thm)],[146, 145])).
% 13.16/13.41 tff(148,plain,(
% 13.16/13.41 ![X: $i] : sum(additive_identity, X, X)),
% 13.16/13.41 inference(skolemize,[status(sab)],[147])).
% 13.16/13.41 tff(149,plain,
% 13.16/13.41 (![X: $i] : sum(additive_identity, X, X)),
% 13.16/13.41 inference(modus_ponens,[status(thm)],[148, 144])).
% 13.16/13.41 tff(150,plain,
% 13.16/13.41 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, x, x)),
% 13.16/13.41 inference(quant_inst,[status(thm)],[])).
% 13.16/13.41 tff(151,plain,
% 13.16/13.41 (sum(additive_identity, x, x)),
% 13.16/13.41 inference(unit_resolution,[status(thm)],[150, 149])).
% 13.16/13.41 tff(152,plain,
% 13.16/13.41 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(x, x, multiply(x, x))),
% 13.16/13.41 inference(quant_inst,[status(thm)],[])).
% 13.16/13.41 tff(153,plain,
% 13.16/13.41 (product(x, x, multiply(x, x))),
% 13.16/13.41 inference(unit_resolution,[status(thm)],[152, 44])).
% 13.16/13.41 tff(154,plain,
% 13.16/13.41 (^[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))))),
% 13.16/13.41 inference(bind,[status(th)],[])).
% 13.16/13.41 tff(155,plain,
% 13.16/13.41 (![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)))),
% 13.16/13.41 inference(quant_intro,[status(thm)],[154])).
% 13.16/13.41 tff(156,plain,
% 13.16/13.41 (![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)))),
% 13.16/13.41 inference(rewrite,[status(thm)],[])).
% 13.16/13.41 tff(157,plain,
% 13.16/13.41 (^[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)))))),
% 13.26/13.41 inference(bind,[status(th)],[])).
% 13.26/13.41 tff(158,plain,
% 13.26/13.41 (![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)))),
% 13.26/13.41 inference(quant_intro,[status(thm)],[157])).
% 13.26/13.41 tff(159,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/sandbox/benchmark/Axioms/BOO002-0.ax','distributivity6')).
% 13.26/13.41 tff(160,plain,
% 13.26/13.41 (![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)))),
% 13.26/13.41 inference(modus_ponens,[status(thm)],[159, 158])).
% 13.26/13.41 tff(161,plain,
% 13.26/13.41 (![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)))),
% 13.26/13.41 inference(modus_ponens,[status(thm)],[160, 156])).
% 13.26/13.41 tff(162,plain,(
% 13.26/13.41 ![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)))),
% 13.26/13.41 inference(skolemize,[status(sab)],[161])).
% 13.26/13.41 tff(163,plain,
% 13.26/13.41 (![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)))),
% 13.26/13.41 inference(modus_ponens,[status(thm)],[162, 155])).
% 13.26/13.41 tff(164,plain,
% 13.26/13.41 (((~![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)) | sum(additive_identity, multiply(x, x), multiply(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)) | sum(additive_identity, multiply(x, x), multiply(x, x)) | (~product(x, x, multiply(x, x))))),
% 13.26/13.41 inference(rewrite,[status(thm)],[])).
% 13.26/13.41 tff(165,plain,
% 13.26/13.41 ((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)) | sum(additive_identity, multiply(x, x), multiply(x, x)) | (~product(x, x, multiply(x, x))))),
% 13.26/13.41 inference(rewrite,[status(thm)],[])).
% 13.26/13.41 tff(166,plain,
% 13.26/13.41 (((~![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)) | sum(additive_identity, multiply(x, x), multiply(x, x)) | (~product(x, x, multiply(x, x)))))),
% 13.26/13.41 inference(monotonicity,[status(thm)],[165])).
% 13.26/13.41 tff(167,plain,
% 13.26/13.41 (((~![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)) | sum(additive_identity, multiply(x, x), multiply(x, x)) | (~product(x, x, multiply(x, x))))),
% 13.26/13.41 inference(transitivity,[status(thm)],[166, 164])).
% 13.26/13.41 tff(168,plain,
% 13.26/13.41 ((~![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))))),
% 13.26/13.41 inference(quant_inst,[status(thm)],[])).
% 13.26/13.41 tff(169,plain,
% 13.26/13.41 ((~![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)) | sum(additive_identity, multiply(x, x), multiply(x, x)) | (~product(x, x, multiply(x, x)))),
% 13.26/13.41 inference(modus_ponens,[status(thm)],[168, 167])).
% 13.26/13.41 tff(170,plain,
% 13.26/13.41 (sum(additive_identity, multiply(x, x), multiply(x, x))),
% 13.26/13.41 inference(unit_resolution,[status(thm)],[169, 163, 153, 151])).
% 13.26/13.41 tff(171,plain,
% 13.26/13.41 (((~![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)))) | ((~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | (~product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity))) | sum(additive_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(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)))) | (~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | (~product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity))) | sum(additive_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))))),
% 13.26/13.41 inference(rewrite,[status(thm)],[])).
% 13.26/13.41 tff(172,plain,
% 13.26/13.41 ((sum(additive_identity, additive_identity, multiply(x, additive_identity)) | (~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | (~product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity)))) <=> ((~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | (~product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity))) | sum(additive_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))))),
% 13.26/13.41 inference(rewrite,[status(thm)],[])).
% 13.26/13.41 tff(173,plain,
% 13.26/13.41 (((~![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, additive_identity, multiply(x, additive_identity)) | (~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | (~product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity))))) <=> ((~![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)))) | ((~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | (~product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity))) | sum(additive_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x))))))),
% 13.26/13.41 inference(monotonicity,[status(thm)],[172])).
% 13.26/13.41 tff(174,plain,
% 13.26/13.41 (((~![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, additive_identity, multiply(x, additive_identity)) | (~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | (~product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity))))) <=> ((~![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)))) | (~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | (~product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity))) | sum(additive_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))))),
% 13.26/13.41 inference(transitivity,[status(thm)],[173, 171])).
% 13.26/13.41 tff(175,plain,
% 13.26/13.41 ((~![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, additive_identity, multiply(x, additive_identity)) | (~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | (~product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity))))),
% 13.26/13.41 inference(quant_inst,[status(thm)],[])).
% 13.26/13.41 tff(176,plain,
% 13.26/13.41 ((~![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)))) | (~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | (~product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity))) | sum(additive_identity, additive_identity, multiply(x, additive_identity)) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x))))),
% 13.26/13.41 inference(modus_ponens,[status(thm)],[175, 174])).
% 13.26/13.41 tff(177,plain,
% 13.26/13.41 (~product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity))),
% 13.26/13.41 inference(unit_resolution,[status(thm)],[176, 163, 170, 142, 122, 72])).
% 13.26/13.41 tff(178,plain,
% 13.26/13.41 (((~![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(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity)) | (~product(multiply(x, x), additive_identity, multiply(multiply(x, x), additive_identity))) | (~sum(multiply(multiply(x, x), additive_identity), additive_identity, multiply(x, additive_identity))) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(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(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity)) | (~product(multiply(x, x), additive_identity, multiply(multiply(x, x), additive_identity))) | (~sum(multiply(multiply(x, x), additive_identity), additive_identity, multiply(x, additive_identity))) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))))),
% 13.26/13.42 inference(rewrite,[status(thm)],[])).
% 13.26/13.42 tff(179,plain,
% 13.26/13.42 ((product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity)) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))) | (~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | (~product(multiply(x, x), additive_identity, multiply(multiply(x, x), additive_identity))) | (~sum(multiply(multiply(x, x), additive_identity), additive_identity, multiply(x, additive_identity)))) <=> ((~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity)) | (~product(multiply(x, x), additive_identity, multiply(multiply(x, x), additive_identity))) | (~sum(multiply(multiply(x, x), additive_identity), additive_identity, multiply(x, additive_identity))) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))))),
% 13.26/13.42 inference(rewrite,[status(thm)],[])).
% 13.26/13.42 tff(180,plain,
% 13.26/13.42 (((~![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(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity)) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))) | (~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | (~product(multiply(x, x), additive_identity, multiply(multiply(x, x), additive_identity))) | (~sum(multiply(multiply(x, x), additive_identity), additive_identity, multiply(x, additive_identity))))) <=> ((~![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(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity)) | (~product(multiply(x, x), additive_identity, multiply(multiply(x, x), additive_identity))) | (~sum(multiply(multiply(x, x), additive_identity), additive_identity, multiply(x, additive_identity))) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x))))))),
% 13.26/13.42 inference(monotonicity,[status(thm)],[179])).
% 13.26/13.42 tff(181,plain,
% 13.26/13.42 (((~![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(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity)) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))) | (~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | (~product(multiply(x, x), additive_identity, multiply(multiply(x, x), additive_identity))) | (~sum(multiply(multiply(x, x), additive_identity), additive_identity, multiply(x, additive_identity))))) <=> ((~![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(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity)) | (~product(multiply(x, x), additive_identity, multiply(multiply(x, x), additive_identity))) | (~sum(multiply(multiply(x, x), additive_identity), additive_identity, multiply(x, additive_identity))) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))))),
% 13.26/13.42 inference(transitivity,[status(thm)],[180, 178])).
% 13.26/13.42 tff(182,plain,
% 13.26/13.42 ((~![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(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity)) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x)))) | (~product(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | (~product(multiply(x, x), additive_identity, multiply(multiply(x, x), additive_identity))) | (~sum(multiply(multiply(x, x), additive_identity), additive_identity, multiply(x, additive_identity))))),
% 13.26/13.42 inference(quant_inst,[status(thm)],[])).
% 13.26/13.42 tff(183,plain,
% 13.26/13.42 ((~![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(multiply(x, x), inverse(multiply(x, x)), additive_identity)) | product(multiply(x, x), inverse(multiply(x, x)), multiply(x, additive_identity)) | (~product(multiply(x, x), additive_identity, multiply(multiply(x, x), additive_identity))) | (~sum(multiply(multiply(x, x), additive_identity), additive_identity, multiply(x, additive_identity))) | (~sum(additive_identity, inverse(multiply(x, x)), inverse(multiply(x, x))))),
% 13.26/13.42 inference(modus_ponens,[status(thm)],[182, 181])).
% 13.26/13.42 tff(184,plain,
% 13.26/13.42 (~sum(multiply(multiply(x, x), additive_identity), additive_identity, multiply(x, additive_identity))),
% 13.26/13.42 inference(unit_resolution,[status(thm)],[183, 89, 142, 177, 72, 46])).
% 13.26/13.42 tff(185,plain,
% 13.26/13.42 (~sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity))),
% 13.26/13.42 inference(modus_ponens,[status(thm)],[184, 50])).
% 13.26/13.42 tff(186,plain,
% 13.26/13.42 (sum(inverse(multiplicative_identity), inverse(multiplicative_identity), inverse(multiplicative_identity)) <=> sum(additive_identity, additive_identity, additive_identity)),
% 13.26/13.42 inference(monotonicity,[status(thm)],[35, 35, 35])).
% 13.26/13.42 tff(187,plain,
% 13.26/13.42 (sum(additive_identity, additive_identity, additive_identity) <=> sum(inverse(multiplicative_identity), inverse(multiplicative_identity), inverse(multiplicative_identity))),
% 13.26/13.42 inference(symmetry,[status(thm)],[186])).
% 13.26/13.42 tff(188,plain,
% 13.26/13.42 (sum(inverse(multiplicative_identity), inverse(multiplicative_identity), inverse(multiplicative_identity))),
% 13.26/13.42 inference(modus_ponens,[status(thm)],[76, 187])).
% 13.26/13.42 tff(189,plain,
% 13.26/13.42 (additive_identity = inverse(multiplicative_identity)),
% 13.26/13.42 inference(symmetry,[status(thm)],[35])).
% 13.26/13.42 tff(190,plain,
% 13.26/13.42 (product(x, additive_identity, multiply(x, additive_identity)) <=> product(x, inverse(multiplicative_identity), multiply(x, additive_identity))),
% 13.26/13.42 inference(monotonicity,[status(thm)],[189])).
% 13.26/13.42 tff(191,plain,
% 13.26/13.42 (product(x, inverse(multiplicative_identity), multiply(x, additive_identity))),
% 13.26/13.42 inference(modus_ponens,[status(thm)],[74, 190])).
% 13.26/13.42 tff(192,plain,
% 13.26/13.42 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl(((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))))),
% 13.26/13.42 inference(bind,[status(th)],[])).
% 13.26/13.42 tff(193,plain,
% 13.26/13.42 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | 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(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 13.26/13.42 inference(quant_intro,[status(thm)],[192])).
% 13.26/13.42 tff(194,plain,
% 13.26/13.42 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | 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(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 13.26/13.42 inference(rewrite,[status(thm)],[])).
% 13.26/13.42 tff(195,plain,
% 13.26/13.42 (^[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))) | (~product(V1, V2, V4))) <=> (((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | (~product(V1, V2, V4))))), rewrite((((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | (~product(V1, V2, V4))) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))))), ((((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(V3, X, V4)) <=> (((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | sum(V3, X, V4)))), rewrite((((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | sum(V3, X, V4)) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), ((((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(V3, X, V4)) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))))),
% 13.26/13.42 inference(bind,[status(th)],[])).
% 13.26/13.42 tff(196,plain,
% 13.26/13.42 (![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))) | (~product(V1, V2, V4))) | sum(V3, X, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 13.26/13.42 inference(quant_intro,[status(thm)],[195])).
% 13.26/13.42 tff(197,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))) | (~product(V1, V2, V4))) | sum(V3, X, V4))), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','distributivity8')).
% 13.26/13.42 tff(198,plain,
% 13.26/13.42 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 13.26/13.42 inference(modus_ponens,[status(thm)],[197, 196])).
% 13.26/13.42 tff(199,plain,
% 13.26/13.42 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 13.26/13.42 inference(modus_ponens,[status(thm)],[198, 194])).
% 13.26/13.42 tff(200,plain,(
% 13.26/13.42 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 13.26/13.42 inference(skolemize,[status(sab)],[199])).
% 13.26/13.42 tff(201,plain,
% 13.26/13.42 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 13.26/13.42 inference(modus_ponens,[status(thm)],[200, 193])).
% 13.26/13.42 tff(202,plain,
% 13.26/13.42 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(x, inverse(multiplicative_identity), multiply(x, additive_identity))) | (~sum(multiply(x, x), inverse(multiplicative_identity), x)) | (~sum(inverse(multiplicative_identity), inverse(multiplicative_identity), inverse(multiplicative_identity))) | sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity)) | (~product(multiply(x, x), inverse(multiplicative_identity), multiply(multiply(x, x), additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(x, inverse(multiplicative_identity), multiply(x, additive_identity))) | (~sum(multiply(x, x), inverse(multiplicative_identity), x)) | (~sum(inverse(multiplicative_identity), inverse(multiplicative_identity), inverse(multiplicative_identity))) | sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity)) | (~product(multiply(x, x), inverse(multiplicative_identity), multiply(multiply(x, x), additive_identity))))),
% 13.26/13.42 inference(rewrite,[status(thm)],[])).
% 13.26/13.42 tff(203,plain,
% 13.26/13.42 (((~product(multiply(x, x), inverse(multiplicative_identity), multiply(multiply(x, x), additive_identity))) | (~product(x, inverse(multiplicative_identity), multiply(x, additive_identity))) | sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity)) | (~sum(inverse(multiplicative_identity), inverse(multiplicative_identity), inverse(multiplicative_identity))) | (~sum(multiply(x, x), inverse(multiplicative_identity), x))) <=> ((~product(x, inverse(multiplicative_identity), multiply(x, additive_identity))) | (~sum(multiply(x, x), inverse(multiplicative_identity), x)) | (~sum(inverse(multiplicative_identity), inverse(multiplicative_identity), inverse(multiplicative_identity))) | sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity)) | (~product(multiply(x, x), inverse(multiplicative_identity), multiply(multiply(x, x), additive_identity))))),
% 13.26/13.42 inference(rewrite,[status(thm)],[])).
% 13.26/13.42 tff(204,plain,
% 13.26/13.42 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(multiply(x, x), inverse(multiplicative_identity), multiply(multiply(x, x), additive_identity))) | (~product(x, inverse(multiplicative_identity), multiply(x, additive_identity))) | sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity)) | (~sum(inverse(multiplicative_identity), inverse(multiplicative_identity), inverse(multiplicative_identity))) | (~sum(multiply(x, x), inverse(multiplicative_identity), x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(x, inverse(multiplicative_identity), multiply(x, additive_identity))) | (~sum(multiply(x, x), inverse(multiplicative_identity), x)) | (~sum(inverse(multiplicative_identity), inverse(multiplicative_identity), inverse(multiplicative_identity))) | sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity)) | (~product(multiply(x, x), inverse(multiplicative_identity), multiply(multiply(x, x), additive_identity)))))),
% 13.26/13.42 inference(monotonicity,[status(thm)],[203])).
% 13.26/13.42 tff(205,plain,
% 13.26/13.42 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(multiply(x, x), inverse(multiplicative_identity), multiply(multiply(x, x), additive_identity))) | (~product(x, inverse(multiplicative_identity), multiply(x, additive_identity))) | sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity)) | (~sum(inverse(multiplicative_identity), inverse(multiplicative_identity), inverse(multiplicative_identity))) | (~sum(multiply(x, x), inverse(multiplicative_identity), x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(x, inverse(multiplicative_identity), multiply(x, additive_identity))) | (~sum(multiply(x, x), inverse(multiplicative_identity), x)) | (~sum(inverse(multiplicative_identity), inverse(multiplicative_identity), inverse(multiplicative_identity))) | sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity)) | (~product(multiply(x, x), inverse(multiplicative_identity), multiply(multiply(x, x), additive_identity))))),
% 13.26/13.42 inference(transitivity,[status(thm)],[204, 202])).
% 13.26/13.42 tff(206,plain,
% 13.26/13.42 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(multiply(x, x), inverse(multiplicative_identity), multiply(multiply(x, x), additive_identity))) | (~product(x, inverse(multiplicative_identity), multiply(x, additive_identity))) | sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity)) | (~sum(inverse(multiplicative_identity), inverse(multiplicative_identity), inverse(multiplicative_identity))) | (~sum(multiply(x, x), inverse(multiplicative_identity), x)))),
% 13.26/13.42 inference(quant_inst,[status(thm)],[])).
% 13.26/13.42 tff(207,plain,
% 13.26/13.42 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(x, inverse(multiplicative_identity), multiply(x, additive_identity))) | (~sum(multiply(x, x), inverse(multiplicative_identity), x)) | (~sum(inverse(multiplicative_identity), inverse(multiplicative_identity), inverse(multiplicative_identity))) | sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity)) | (~product(multiply(x, x), inverse(multiplicative_identity), multiply(multiply(x, x), additive_identity)))),
% 13.26/13.42 inference(modus_ponens,[status(thm)],[206, 205])).
% 13.26/13.42 tff(208,plain,
% 13.26/13.42 ((~sum(multiply(x, x), inverse(multiplicative_identity), x)) | (~sum(inverse(multiplicative_identity), inverse(multiplicative_identity), inverse(multiplicative_identity))) | sum(multiply(multiply(x, x), additive_identity), inverse(multiplicative_identity), multiply(x, additive_identity)) | (~product(multiply(x, x), inverse(multiplicative_identity), multiply(multiply(x, x), additive_identity)))),
% 13.26/13.42 inference(unit_resolution,[status(thm)],[207, 201, 191])).
% 13.26/13.42 tff(209,plain,
% 13.26/13.42 (~sum(multiply(x, x), inverse(multiplicative_identity), x)),
% 13.26/13.42 inference(unit_resolution,[status(thm)],[208, 188, 185, 47])).
% 13.26/13.42 tff(210,plain,
% 13.26/13.42 (product(inverse(x), x, inverse(multiplicative_identity)) <=> product(inverse(x), x, additive_identity)),
% 13.26/13.42 inference(monotonicity,[status(thm)],[35])).
% 13.26/13.42 tff(211,plain,
% 13.26/13.42 (product(inverse(x), x, additive_identity) <=> product(inverse(x), x, inverse(multiplicative_identity))),
% 13.26/13.42 inference(symmetry,[status(thm)],[210])).
% 13.26/13.42 tff(212,plain,
% 13.26/13.42 ((~![X: $i] : product(inverse(X), X, additive_identity)) | product(inverse(x), x, additive_identity)),
% 13.26/13.42 inference(quant_inst,[status(thm)],[])).
% 13.26/13.42 tff(213,plain,
% 13.26/13.42 (product(inverse(x), x, additive_identity)),
% 13.26/13.42 inference(unit_resolution,[status(thm)],[212, 129])).
% 13.26/13.42 tff(214,plain,
% 13.26/13.42 (product(inverse(x), x, inverse(multiplicative_identity))),
% 13.26/13.42 inference(modus_ponens,[status(thm)],[213, 211])).
% 13.26/13.42 tff(215,plain,
% 13.26/13.42 (^[X: $i] : refl(sum(X, inverse(X), multiplicative_identity) <=> sum(X, inverse(X), multiplicative_identity))),
% 13.26/13.42 inference(bind,[status(th)],[])).
% 13.26/13.42 tff(216,plain,
% 13.26/13.42 (![X: $i] : sum(X, inverse(X), multiplicative_identity) <=> ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 13.26/13.42 inference(quant_intro,[status(thm)],[215])).
% 13.26/13.42 tff(217,plain,
% 13.26/13.42 (![X: $i] : sum(X, inverse(X), multiplicative_identity) <=> ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 13.26/13.42 inference(rewrite,[status(thm)],[])).
% 13.26/13.42 tff(218,axiom,(![X: $i] : sum(X, inverse(X), multiplicative_identity)), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','additive_inverse2')).
% 13.26/13.42 tff(219,plain,
% 13.26/13.42 (![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 13.26/13.42 inference(modus_ponens,[status(thm)],[218, 217])).
% 13.26/13.42 tff(220,plain,(
% 13.26/13.42 ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 13.26/13.42 inference(skolemize,[status(sab)],[219])).
% 13.26/13.42 tff(221,plain,
% 13.26/13.42 (![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 13.26/13.42 inference(modus_ponens,[status(thm)],[220, 216])).
% 13.26/13.42 tff(222,plain,
% 13.26/13.42 ((~![X: $i] : sum(X, inverse(X), multiplicative_identity)) | sum(x, inverse(x), multiplicative_identity)),
% 13.26/13.42 inference(quant_inst,[status(thm)],[])).
% 13.26/13.42 tff(223,plain,
% 13.26/13.42 (sum(x, inverse(x), multiplicative_identity)),
% 13.26/13.42 inference(unit_resolution,[status(thm)],[222, 221])).
% 13.26/13.42 tff(224,plain,
% 13.26/13.42 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, x, x)),
% 13.26/13.42 inference(quant_inst,[status(thm)],[])).
% 13.26/13.42 tff(225,plain,
% 13.26/13.42 (product(multiplicative_identity, x, x)),
% 13.26/13.42 inference(unit_resolution,[status(thm)],[224, 7])).
% 13.26/13.42 tff(226,plain,
% 13.26/13.42 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) <=> (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))))),
% 13.26/13.42 inference(bind,[status(th)],[])).
% 13.26/13.42 tff(227,plain,
% 13.26/13.42 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 13.26/13.42 inference(quant_intro,[status(thm)],[226])).
% 13.26/13.42 tff(228,plain,
% 13.26/13.42 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 13.26/13.42 inference(rewrite,[status(thm)],[])).
% 13.26/13.42 tff(229,plain,
% 13.26/13.42 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) <=> ((~sum(Y, Z, V3)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))), (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) <=> (((~sum(Y, Z, V3)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | (~product(V3, X, V4))))), rewrite((((~sum(Y, Z, V3)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | (~product(V3, X, V4))) <=> ((~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))), (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) <=> ((~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))))), ((((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) | sum(V1, V2, V4)) <=> (((~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | sum(V1, V2, V4)))), rewrite((((~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | sum(V1, V2, V4)) <=> (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))), ((((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) | sum(V1, V2, V4)) <=> (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))))),
% 13.26/13.42 inference(bind,[status(th)],[])).
% 13.26/13.42 tff(230,plain,
% 13.26/13.42 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) | sum(V1, V2, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 13.26/13.42 inference(quant_intro,[status(thm)],[229])).
% 13.26/13.42 tff(231,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) | sum(V1, V2, V4))), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','distributivity3')).
% 13.26/13.42 tff(232,plain,
% 13.26/13.42 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 13.26/13.42 inference(modus_ponens,[status(thm)],[231, 230])).
% 13.26/13.42 tff(233,plain,
% 13.26/13.42 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 13.26/13.42 inference(modus_ponens,[status(thm)],[232, 228])).
% 13.26/13.42 tff(234,plain,(
% 13.26/13.42 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 13.26/13.42 inference(skolemize,[status(sab)],[233])).
% 13.26/13.42 tff(235,plain,
% 13.26/13.42 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 13.26/13.42 inference(modus_ponens,[status(thm)],[234, 227])).
% 13.26/13.42 tff(236,plain,
% 13.26/13.42 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~product(x, x, multiply(x, x))) | (~sum(x, inverse(x), multiplicative_identity)) | (~product(multiplicative_identity, x, x)) | sum(multiply(x, x), inverse(multiplicative_identity), x) | (~product(inverse(x), x, inverse(multiplicative_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(x, x, multiply(x, x))) | (~sum(x, inverse(x), multiplicative_identity)) | (~product(multiplicative_identity, x, x)) | sum(multiply(x, x), inverse(multiplicative_identity), x) | (~product(inverse(x), x, inverse(multiplicative_identity))))),
% 13.26/13.42 inference(rewrite,[status(thm)],[])).
% 13.26/13.42 tff(237,plain,
% 13.26/13.42 ((sum(multiply(x, x), inverse(multiplicative_identity), x) | (~sum(x, inverse(x), multiplicative_identity)) | (~product(multiplicative_identity, x, x)) | (~product(inverse(x), x, inverse(multiplicative_identity))) | (~product(x, x, multiply(x, x)))) <=> ((~product(x, x, multiply(x, x))) | (~sum(x, inverse(x), multiplicative_identity)) | (~product(multiplicative_identity, x, x)) | sum(multiply(x, x), inverse(multiplicative_identity), x) | (~product(inverse(x), x, inverse(multiplicative_identity))))),
% 13.26/13.42 inference(rewrite,[status(thm)],[])).
% 13.26/13.42 tff(238,plain,
% 13.26/13.42 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(x, x), inverse(multiplicative_identity), x) | (~sum(x, inverse(x), multiplicative_identity)) | (~product(multiplicative_identity, x, x)) | (~product(inverse(x), x, inverse(multiplicative_identity))) | (~product(x, x, multiply(x, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~product(x, x, multiply(x, x))) | (~sum(x, inverse(x), multiplicative_identity)) | (~product(multiplicative_identity, x, x)) | sum(multiply(x, x), inverse(multiplicative_identity), x) | (~product(inverse(x), x, inverse(multiplicative_identity)))))),
% 13.26/13.42 inference(monotonicity,[status(thm)],[237])).
% 13.26/13.42 tff(239,plain,
% 13.26/13.42 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(x, x), inverse(multiplicative_identity), x) | (~sum(x, inverse(x), multiplicative_identity)) | (~product(multiplicative_identity, x, x)) | (~product(inverse(x), x, inverse(multiplicative_identity))) | (~product(x, x, multiply(x, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(x, x, multiply(x, x))) | (~sum(x, inverse(x), multiplicative_identity)) | (~product(multiplicative_identity, x, x)) | sum(multiply(x, x), inverse(multiplicative_identity), x) | (~product(inverse(x), x, inverse(multiplicative_identity))))),
% 13.26/13.42 inference(transitivity,[status(thm)],[238, 236])).
% 13.26/13.42 tff(240,plain,
% 13.26/13.42 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(x, x), inverse(multiplicative_identity), x) | (~sum(x, inverse(x), multiplicative_identity)) | (~product(multiplicative_identity, x, x)) | (~product(inverse(x), x, inverse(multiplicative_identity))) | (~product(x, x, multiply(x, x))))),
% 13.26/13.42 inference(quant_inst,[status(thm)],[])).
% 13.26/13.42 tff(241,plain,
% 13.26/13.42 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(x, x, multiply(x, x))) | (~sum(x, inverse(x), multiplicative_identity)) | (~product(multiplicative_identity, x, x)) | sum(multiply(x, x), inverse(multiplicative_identity), x) | (~product(inverse(x), x, inverse(multiplicative_identity)))),
% 13.26/13.43 inference(modus_ponens,[status(thm)],[240, 239])).
% 13.26/13.43 tff(242,plain,
% 13.26/13.43 ($false),
% 13.26/13.43 inference(unit_resolution,[status(thm)],[241, 235, 153, 225, 223, 214, 209])).
% 13.26/13.43 % SZS output end Proof
% 13.26/13.44 % E exiting
%------------------------------------------------------------------------------