%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : RNG001-3 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.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 05:23:51 AM UTC 2025
% Result : Unsatisfiable 0.23s 0.47s
% Output : Proof 0.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.14 % Problem : RNG001-3 : TPTP v9.0.0. Released v1.0.0.
% 0.03/0.14 % Command : run_E %s %d THM
% 0.13/0.35 % Computer : n003.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Jun 20 03:05:09 EDT 2025
% 0.13/0.36 % CPUTime :
% 0.23/0.47 % SZS status Unsatisfiable
% 0.23/0.47 % SZS output start Proof
% 0.23/0.47 tff(sum_type, type, (
% 0.23/0.47 sum: ( $i * $i * $i ) > $o)).
% 0.23/0.47 tff(multiply_type, type, (
% 0.23/0.47 multiply: ( $i * $i ) > $i)).
% 0.23/0.47 tff(additive_identity_type, type, (
% 0.23/0.47 additive_identity: $i)).
% 0.23/0.47 tff(a_type, type, (
% 0.23/0.47 a: $i)).
% 0.23/0.47 tff(product_type, type, (
% 0.23/0.47 product: ( $i * $i * $i ) > $o)).
% 0.23/0.47 tff(additive_inverse_type, type, (
% 0.23/0.47 additive_inverse: $i > $i)).
% 0.23/0.47 tff(1,plain,
% 0.23/0.47 (^[X: $i] : refl(sum(additive_identity, X, X) <=> sum(additive_identity, X, X))),
% 0.23/0.47 inference(bind,[status(th)],[])).
% 0.23/0.47 tff(2,plain,
% 0.23/0.47 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 0.23/0.47 inference(quant_intro,[status(thm)],[1])).
% 0.23/0.47 tff(3,plain,
% 0.23/0.47 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 0.23/0.47 inference(rewrite,[status(thm)],[])).
% 0.23/0.47 tff(4,axiom,(![X: $i] : sum(additive_identity, X, X)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','additive_identity1')).
% 0.23/0.47 tff(5,plain,
% 0.23/0.47 (![X: $i] : sum(additive_identity, X, X)),
% 0.23/0.47 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.23/0.47 tff(6,plain,(
% 0.23/0.47 ![X: $i] : sum(additive_identity, X, X)),
% 0.23/0.47 inference(skolemize,[status(sab)],[5])).
% 0.23/0.47 tff(7,plain,
% 0.23/0.47 (![X: $i] : sum(additive_identity, X, X)),
% 0.23/0.47 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.23/0.47 tff(8,plain,
% 0.23/0.47 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, multiply(a, additive_identity), multiply(a, additive_identity))),
% 0.23/0.47 inference(quant_inst,[status(thm)],[])).
% 0.23/0.47 tff(9,plain,
% 0.23/0.47 (sum(additive_identity, multiply(a, additive_identity), multiply(a, additive_identity))),
% 0.23/0.47 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.23/0.47 tff(10,assumption,(~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))), introduced(assumption)).
% 0.23/0.47 tff(11,plain,
% 0.23/0.47 (^[Y: $i, X: $i] : refl(product(X, Y, multiply(X, Y)) <=> product(X, Y, multiply(X, Y)))),
% 0.23/0.47 inference(bind,[status(th)],[])).
% 0.23/0.47 tff(12,plain,
% 0.23/0.47 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.23/0.47 inference(quant_intro,[status(thm)],[11])).
% 0.23/0.47 tff(13,plain,
% 0.23/0.47 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.23/0.47 inference(rewrite,[status(thm)],[])).
% 0.23/0.47 tff(14,axiom,(![Y: $i, X: $i] : product(X, Y, multiply(X, Y))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','closure_of_multiplication')).
% 0.23/0.47 tff(15,plain,
% 0.23/0.47 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.23/0.47 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.23/0.47 tff(16,plain,(
% 0.23/0.47 ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.23/0.47 inference(skolemize,[status(sab)],[15])).
% 0.23/0.47 tff(17,plain,
% 0.23/0.47 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.23/0.47 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.23/0.47 tff(18,plain,
% 0.23/0.47 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(a, additive_identity, multiply(a, additive_identity))),
% 0.23/0.47 inference(quant_inst,[status(thm)],[])).
% 0.23/0.47 tff(19,plain,
% 0.23/0.47 (product(a, additive_identity, multiply(a, additive_identity))),
% 0.23/0.47 inference(unit_resolution,[status(thm)],[18, 17])).
% 0.23/0.47 tff(20,plain,
% 0.23/0.47 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, additive_identity, additive_identity)),
% 0.23/0.47 inference(quant_inst,[status(thm)],[])).
% 0.23/0.47 tff(21,plain,
% 0.23/0.47 (sum(additive_identity, additive_identity, additive_identity)),
% 0.23/0.47 inference(unit_resolution,[status(thm)],[20, 7])).
% 0.23/0.47 tff(22,plain,
% 0.23/0.47 (^[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))))),
% 0.23/0.47 inference(bind,[status(th)],[])).
% 0.23/0.47 tff(23,plain,
% 0.23/0.47 (![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)))),
% 0.23/0.48 inference(quant_intro,[status(thm)],[22])).
% 0.23/0.48 tff(24,plain,
% 0.23/0.48 (![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)))),
% 0.23/0.48 inference(rewrite,[status(thm)],[])).
% 0.23/0.48 tff(25,plain,
% 0.23/0.48 (^[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)))))),
% 0.23/0.48 inference(bind,[status(th)],[])).
% 0.23/0.48 tff(26,plain,
% 0.23/0.48 (![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)))),
% 0.23/0.48 inference(quant_intro,[status(thm)],[25])).
% 0.23/0.48 tff(27,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','distributivity1')).
% 0.23/0.48 tff(28,plain,
% 0.23/0.48 (![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)))),
% 0.23/0.48 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.23/0.48 tff(29,plain,
% 0.23/0.48 (![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)))),
% 0.23/0.48 inference(modus_ponens,[status(thm)],[28, 24])).
% 0.23/0.48 tff(30,plain,(
% 0.23/0.48 ![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)))),
% 0.23/0.48 inference(skolemize,[status(sab)],[29])).
% 0.23/0.48 tff(31,plain,
% 0.23/0.48 (![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)))),
% 0.23/0.48 inference(modus_ponens,[status(thm)],[30, 23])).
% 0.23/0.48 tff(32,plain,
% 0.23/0.48 (((~![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(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, 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)))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)))),
% 0.23/0.48 inference(rewrite,[status(thm)],[])).
% 0.23/0.48 tff(33,plain,
% 0.23/0.48 ((sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, additive_identity, multiply(a, additive_identity)))) <=> ((~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)))),
% 0.23/0.48 inference(rewrite,[status(thm)],[])).
% 0.23/0.48 tff(34,plain,
% 0.23/0.48 (((~![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(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, additive_identity, multiply(a, 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)))) | ((~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))))),
% 0.23/0.48 inference(monotonicity,[status(thm)],[33])).
% 0.23/0.48 tff(35,plain,
% 0.23/0.48 (((~![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(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, additive_identity, multiply(a, 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)))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)))),
% 0.23/0.48 inference(transitivity,[status(thm)],[34, 32])).
% 0.23/0.48 tff(36,plain,
% 0.23/0.48 ((~![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(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, additive_identity, multiply(a, additive_identity))))),
% 0.23/0.48 inference(quant_inst,[status(thm)],[])).
% 0.23/0.48 tff(37,plain,
% 0.23/0.48 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))),
% 0.23/0.48 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.23/0.48 tff(38,plain,
% 0.23/0.48 ($false),
% 0.23/0.48 inference(unit_resolution,[status(thm)],[37, 31, 21, 19, 10])).
% 0.23/0.48 tff(39,plain,(sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))), inference(lemma,lemma(discharge,[]))).
% 0.23/0.48 tff(40,plain,
% 0.23/0.48 (^[X: $i] : refl(sum(additive_inverse(X), X, additive_identity) <=> sum(additive_inverse(X), X, additive_identity))),
% 0.23/0.48 inference(bind,[status(th)],[])).
% 0.23/0.48 tff(41,plain,
% 0.23/0.48 (![X: $i] : sum(additive_inverse(X), X, additive_identity) <=> ![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 0.23/0.48 inference(quant_intro,[status(thm)],[40])).
% 0.23/0.48 tff(42,plain,
% 0.23/0.48 (![X: $i] : sum(additive_inverse(X), X, additive_identity) <=> ![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 0.23/0.48 inference(rewrite,[status(thm)],[])).
% 0.23/0.48 tff(43,axiom,(![X: $i] : sum(additive_inverse(X), X, additive_identity)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','additive_inverse1')).
% 0.23/0.48 tff(44,plain,
% 0.23/0.48 (![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 0.23/0.48 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.23/0.48 tff(45,plain,(
% 0.23/0.48 ![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 0.23/0.48 inference(skolemize,[status(sab)],[44])).
% 0.23/0.48 tff(46,plain,
% 0.23/0.48 (![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 0.23/0.48 inference(modus_ponens,[status(thm)],[45, 41])).
% 0.23/0.48 tff(47,plain,
% 0.23/0.48 ((~![X: $i] : sum(additive_inverse(X), X, additive_identity)) | sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)),
% 0.23/0.48 inference(quant_inst,[status(thm)],[])).
% 0.23/0.48 tff(48,plain,
% 0.23/0.48 (sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)),
% 0.23/0.48 inference(unit_resolution,[status(thm)],[47, 46])).
% 0.23/0.48 tff(49,plain,
% 0.23/0.48 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : refl((sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) <=> (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))))),
% 0.23/0.48 inference(bind,[status(th)],[])).
% 0.23/0.48 tff(50,plain,
% 0.23/0.48 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 0.23/0.48 inference(quant_intro,[status(thm)],[49])).
% 0.23/0.48 tff(51,plain,
% 0.23/0.48 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 0.23/0.48 inference(rewrite,[status(thm)],[])).
% 0.23/0.48 tff(52,plain,
% 0.23/0.48 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : trans(monotonicity(rewrite((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) <=> ((~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))), (((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) | sum(U, Z, W)) <=> (((~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) | sum(U, Z, W)))), rewrite((((~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) | sum(U, Z, W)) <=> (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))), (((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) | sum(U, Z, W)) <=> (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))))),
% 0.23/0.48 inference(bind,[status(th)],[])).
% 0.23/0.48 tff(53,plain,
% 0.23/0.48 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) | sum(U, Z, W)) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 0.23/0.48 inference(quant_intro,[status(thm)],[52])).
% 0.23/0.48 tff(54,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) | sum(U, Z, W))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','associativity_of_addition2')).
% 0.23/0.48 tff(55,plain,
% 0.23/0.48 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 0.23/0.48 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.23/0.49 tff(56,plain,
% 0.23/0.49 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 0.23/0.49 inference(modus_ponens,[status(thm)],[55, 51])).
% 0.23/0.49 tff(57,plain,(
% 0.23/0.49 ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 0.23/0.49 inference(skolemize,[status(sab)],[56])).
% 0.23/0.49 tff(58,plain,
% 0.23/0.49 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 0.23/0.49 inference(modus_ponens,[status(thm)],[57, 50])).
% 0.23/0.49 tff(59,plain,
% 0.23/0.49 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | ((~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | sum(additive_identity, multiply(a, additive_identity), additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | sum(additive_identity, multiply(a, additive_identity), additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))))),
% 0.23/0.49 inference(rewrite,[status(thm)],[])).
% 0.23/0.49 tff(60,plain,
% 0.23/0.49 ((sum(additive_identity, multiply(a, additive_identity), additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity))) <=> ((~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | sum(additive_identity, multiply(a, additive_identity), additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))))),
% 0.23/0.49 inference(rewrite,[status(thm)],[])).
% 0.23/0.49 tff(61,plain,
% 0.23/0.49 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(additive_identity, multiply(a, additive_identity), additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | ((~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | sum(additive_identity, multiply(a, additive_identity), additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)))))),
% 0.23/0.49 inference(monotonicity,[status(thm)],[60])).
% 0.23/0.49 tff(62,plain,
% 0.23/0.49 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(additive_identity, multiply(a, additive_identity), additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | sum(additive_identity, multiply(a, additive_identity), additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))))),
% 0.23/0.49 inference(transitivity,[status(thm)],[61, 59])).
% 0.23/0.49 tff(63,plain,
% 0.23/0.49 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(additive_identity, multiply(a, additive_identity), additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)))),
% 0.23/0.49 inference(quant_inst,[status(thm)],[])).
% 0.23/0.49 tff(64,plain,
% 0.23/0.49 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | sum(additive_identity, multiply(a, additive_identity), additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)))),
% 0.23/0.49 inference(modus_ponens,[status(thm)],[63, 62])).
% 0.23/0.49 tff(65,plain,
% 0.23/0.49 (sum(additive_identity, multiply(a, additive_identity), additive_identity)),
% 0.23/0.49 inference(unit_resolution,[status(thm)],[64, 58, 39, 48])).
% 0.23/0.49 tff(66,plain,
% 0.23/0.49 ((~product(a, additive_identity, additive_identity)) <=> (~product(a, additive_identity, additive_identity))),
% 0.23/0.49 inference(rewrite,[status(thm)],[])).
% 0.23/0.49 tff(67,axiom,(~product(a, additive_identity, additive_identity)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_a_times_additive_id_is_additive_id')).
% 0.23/0.49 tff(68,plain,
% 0.23/0.49 (~product(a, additive_identity, additive_identity)),
% 0.23/0.49 inference(modus_ponens,[status(thm)],[67, 66])).
% 0.23/0.49 tff(69,plain,
% 0.23/0.49 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) <=> (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))))),
% 0.23/0.49 inference(bind,[status(th)],[])).
% 0.23/0.49 tff(70,plain,
% 0.23/0.49 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.23/0.49 inference(quant_intro,[status(thm)],[69])).
% 0.23/0.49 tff(71,plain,
% 0.23/0.49 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.23/0.49 inference(rewrite,[status(thm)],[])).
% 0.23/0.49 tff(72,plain,
% 0.23/0.49 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) <=> (((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~sum(V1, V2, V4))))), rewrite((((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~sum(V1, V2, V4))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> (((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) | product(X, V3, V4)))), rewrite((((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))))),
% 0.23/0.49 inference(bind,[status(th)],[])).
% 0.23/0.49 tff(73,plain,
% 0.23/0.49 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.23/0.49 inference(quant_intro,[status(thm)],[72])).
% 0.23/0.49 tff(74,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','distributivity2')).
% 0.23/0.49 tff(75,plain,
% 0.23/0.49 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.23/0.49 inference(modus_ponens,[status(thm)],[74, 73])).
% 0.23/0.49 tff(76,plain,
% 0.23/0.49 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.23/0.49 inference(modus_ponens,[status(thm)],[75, 71])).
% 0.23/0.49 tff(77,plain,(
% 0.23/0.49 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.23/0.49 inference(skolemize,[status(sab)],[76])).
% 0.23/0.49 tff(78,plain,
% 0.23/0.49 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.23/0.49 inference(modus_ponens,[status(thm)],[77, 70])).
% 0.23/0.49 tff(79,plain,
% 0.23/0.49 (((~![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(a, additive_identity, additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, 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(a, additive_identity, additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))))),
% 0.23/0.49 inference(rewrite,[status(thm)],[])).
% 0.23/0.49 tff(80,plain,
% 0.23/0.49 ((product(a, additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity))) <=> (product(a, additive_identity, additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))))),
% 0.23/0.49 inference(rewrite,[status(thm)],[])).
% 0.23/0.49 tff(81,plain,
% 0.23/0.49 (((~![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(a, additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~sum(multiply(a, additive_identity), multiply(a, 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(a, additive_identity, additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity)))))),
% 0.23/0.49 inference(monotonicity,[status(thm)],[80])).
% 0.23/0.49 tff(82,plain,
% 0.23/0.49 (((~![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(a, additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~sum(multiply(a, additive_identity), multiply(a, 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(a, additive_identity, additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))))),
% 0.23/0.49 inference(transitivity,[status(thm)],[81, 79])).
% 0.23/0.49 tff(83,plain,
% 0.23/0.49 ((~![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(a, additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity)))),
% 0.23/0.49 inference(quant_inst,[status(thm)],[])).
% 0.23/0.49 tff(84,plain,
% 0.23/0.49 ((~![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(a, additive_identity, additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity)))),
% 0.23/0.49 inference(modus_ponens,[status(thm)],[83, 82])).
% 0.23/0.49 tff(85,plain,
% 0.23/0.49 (~sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity)),
% 0.23/0.49 inference(unit_resolution,[status(thm)],[84, 78, 68, 21, 19])).
% 0.23/0.49 tff(86,plain,
% 0.23/0.49 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity) | (~sum(additive_identity, multiply(a, additive_identity), additive_identity)) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_identity, multiply(a, additive_identity), multiply(a, additive_identity))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity) | (~sum(additive_identity, multiply(a, additive_identity), additive_identity)) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_identity, multiply(a, additive_identity), multiply(a, additive_identity))))),
% 0.23/0.49 inference(rewrite,[status(thm)],[])).
% 0.23/0.49 tff(87,plain,
% 0.23/0.49 ((sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_identity, multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_identity, multiply(a, additive_identity), additive_identity))) <=> (sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity) | (~sum(additive_identity, multiply(a, additive_identity), additive_identity)) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_identity, multiply(a, additive_identity), multiply(a, additive_identity))))),
% 0.23/0.50 inference(rewrite,[status(thm)],[])).
% 0.23/0.50 tff(88,plain,
% 0.23/0.50 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_identity, multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_identity, multiply(a, additive_identity), additive_identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity) | (~sum(additive_identity, multiply(a, additive_identity), additive_identity)) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_identity, multiply(a, additive_identity), multiply(a, additive_identity)))))),
% 0.23/0.50 inference(monotonicity,[status(thm)],[87])).
% 0.23/0.50 tff(89,plain,
% 0.23/0.50 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_identity, multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_identity, multiply(a, additive_identity), additive_identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity) | (~sum(additive_identity, multiply(a, additive_identity), additive_identity)) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_identity, multiply(a, additive_identity), multiply(a, additive_identity))))),
% 0.23/0.50 inference(transitivity,[status(thm)],[88, 86])).
% 0.23/0.50 tff(90,plain,
% 0.23/0.50 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_identity, multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_identity, multiply(a, additive_identity), additive_identity)))),
% 0.23/0.50 inference(quant_inst,[status(thm)],[])).
% 0.23/0.50 tff(91,plain,
% 0.23/0.50 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity) | (~sum(additive_identity, multiply(a, additive_identity), additive_identity)) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (~sum(additive_identity, multiply(a, additive_identity), multiply(a, additive_identity)))),
% 0.23/0.50 inference(modus_ponens,[status(thm)],[90, 89])).
% 0.23/0.50 tff(92,plain,
% 0.23/0.50 ($false),
% 0.23/0.50 inference(unit_resolution,[status(thm)],[91, 58, 85, 65, 39, 9])).
% 0.23/0.50 % SZS output end Proof
% 0.23/0.50 % E exiting
%------------------------------------------------------------------------------