%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : BOO005-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n006.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 15.50s 15.69s
% Output : Proof 15.50s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11 % Problem : BOO005-1 : TPTP v9.0.0. Released v1.0.0.
% 0.06/0.11 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n006.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.32 % CPULimit : 300
% 0.16/0.32 % WCLimit : 300
% 0.16/0.32 % DateTime : Thu Jun 19 11:32:22 EDT 2025
% 0.16/0.32 % CPUTime :
% 15.50/15.69 % SZS status Unsatisfiable
% 15.50/15.69 % SZS output start Proof
% 15.50/15.69 tff(sum_type, type, (
% 15.50/15.69 sum: ( $i * $i * $i ) > $o)).
% 15.50/15.69 tff(inverse_type, type, (
% 15.50/15.69 inverse: $i > $i)).
% 15.50/15.69 tff(x_type, type, (
% 15.50/15.69 x: $i)).
% 15.50/15.69 tff(additive_identity_type, type, (
% 15.50/15.69 additive_identity: $i)).
% 15.50/15.69 tff(product_type, type, (
% 15.50/15.69 product: ( $i * $i * $i ) > $o)).
% 15.50/15.69 tff(multiply_type, type, (
% 15.50/15.69 multiply: ( $i * $i ) > $i)).
% 15.50/15.69 tff(multiplicative_identity_type, type, (
% 15.50/15.69 multiplicative_identity: $i)).
% 15.50/15.69 tff(add_type, type, (
% 15.50/15.69 add: ( $i * $i ) > $i)).
% 15.50/15.69 tff(1,plain,
% 15.50/15.69 (^[X: $i] : refl(sum(additive_identity, X, X) <=> sum(additive_identity, X, X))),
% 15.50/15.69 inference(bind,[status(th)],[])).
% 15.50/15.69 tff(2,plain,
% 15.50/15.69 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 15.50/15.69 inference(quant_intro,[status(thm)],[1])).
% 15.50/15.69 tff(3,plain,
% 15.50/15.69 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(4,axiom,(![X: $i] : sum(additive_identity, X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_identity1')).
% 15.50/15.69 tff(5,plain,
% 15.50/15.69 (![X: $i] : sum(additive_identity, X, X)),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[4, 3])).
% 15.50/15.69 tff(6,plain,(
% 15.50/15.69 ![X: $i] : sum(additive_identity, X, X)),
% 15.50/15.69 inference(skolemize,[status(sab)],[5])).
% 15.50/15.69 tff(7,plain,
% 15.50/15.69 (![X: $i] : sum(additive_identity, X, X)),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[6, 2])).
% 15.50/15.69 tff(8,plain,
% 15.50/15.69 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, inverse(x), inverse(x))),
% 15.50/15.69 inference(quant_inst,[status(thm)],[])).
% 15.50/15.69 tff(9,plain,
% 15.50/15.69 (sum(additive_identity, inverse(x), inverse(x))),
% 15.50/15.69 inference(unit_resolution,[status(thm)],[8, 7])).
% 15.50/15.69 tff(10,assumption,(~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))), introduced(assumption)).
% 15.50/15.69 tff(11,plain,
% 15.50/15.69 (^[X: $i] : refl(product(X, multiplicative_identity, X) <=> product(X, multiplicative_identity, X))),
% 15.50/15.69 inference(bind,[status(th)],[])).
% 15.50/15.69 tff(12,plain,
% 15.50/15.69 (![X: $i] : product(X, multiplicative_identity, X) <=> ![X: $i] : product(X, multiplicative_identity, X)),
% 15.50/15.69 inference(quant_intro,[status(thm)],[11])).
% 15.50/15.69 tff(13,plain,
% 15.50/15.69 (![X: $i] : product(X, multiplicative_identity, X) <=> ![X: $i] : product(X, multiplicative_identity, X)),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(14,axiom,(![X: $i] : product(X, multiplicative_identity, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_identity2')).
% 15.50/15.69 tff(15,plain,
% 15.50/15.69 (![X: $i] : product(X, multiplicative_identity, X)),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[14, 13])).
% 15.50/15.69 tff(16,plain,(
% 15.50/15.69 ![X: $i] : product(X, multiplicative_identity, X)),
% 15.50/15.69 inference(skolemize,[status(sab)],[15])).
% 15.50/15.69 tff(17,plain,
% 15.50/15.69 (![X: $i] : product(X, multiplicative_identity, X)),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[16, 12])).
% 15.50/15.69 tff(18,plain,
% 15.50/15.69 ((~![X: $i] : product(X, multiplicative_identity, X)) | product(x, multiplicative_identity, x)),
% 15.50/15.69 inference(quant_inst,[status(thm)],[])).
% 15.50/15.69 tff(19,plain,
% 15.50/15.69 (product(x, multiplicative_identity, x)),
% 15.50/15.69 inference(unit_resolution,[status(thm)],[18, 17])).
% 15.50/15.69 tff(20,plain,
% 15.50/15.69 (^[X: $i] : refl(product(multiplicative_identity, X, X) <=> product(multiplicative_identity, X, X))),
% 15.50/15.69 inference(bind,[status(th)],[])).
% 15.50/15.69 tff(21,plain,
% 15.50/15.69 (![X: $i] : product(multiplicative_identity, X, X) <=> ![X: $i] : product(multiplicative_identity, X, X)),
% 15.50/15.69 inference(quant_intro,[status(thm)],[20])).
% 15.50/15.69 tff(22,plain,
% 15.50/15.69 (![X: $i] : product(multiplicative_identity, X, X) <=> ![X: $i] : product(multiplicative_identity, X, X)),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(23,axiom,(![X: $i] : product(multiplicative_identity, X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_identity1')).
% 15.50/15.69 tff(24,plain,
% 15.50/15.69 (![X: $i] : product(multiplicative_identity, X, X)),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[23, 22])).
% 15.50/15.69 tff(25,plain,(
% 15.50/15.69 ![X: $i] : product(multiplicative_identity, X, X)),
% 15.50/15.69 inference(skolemize,[status(sab)],[24])).
% 15.50/15.69 tff(26,plain,
% 15.50/15.69 (![X: $i] : product(multiplicative_identity, X, X)),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[25, 21])).
% 15.50/15.69 tff(27,plain,
% 15.50/15.69 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, multiplicative_identity, multiplicative_identity)),
% 15.50/15.69 inference(quant_inst,[status(thm)],[])).
% 15.50/15.69 tff(28,plain,
% 15.50/15.69 (product(multiplicative_identity, multiplicative_identity, multiplicative_identity)),
% 15.50/15.69 inference(unit_resolution,[status(thm)],[27, 26])).
% 15.50/15.69 tff(29,plain,
% 15.50/15.69 (^[Y: $i, X: $i] : refl(sum(X, Y, add(X, Y)) <=> sum(X, Y, add(X, Y)))),
% 15.50/15.69 inference(bind,[status(th)],[])).
% 15.50/15.69 tff(30,plain,
% 15.50/15.69 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 15.50/15.69 inference(quant_intro,[status(thm)],[29])).
% 15.50/15.69 tff(31,plain,
% 15.50/15.69 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(32,axiom,(![Y: $i, X: $i] : sum(X, Y, add(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','closure_of_addition')).
% 15.50/15.69 tff(33,plain,
% 15.50/15.69 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[32, 31])).
% 15.50/15.69 tff(34,plain,(
% 15.50/15.69 ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 15.50/15.69 inference(skolemize,[status(sab)],[33])).
% 15.50/15.69 tff(35,plain,
% 15.50/15.69 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[34, 30])).
% 15.50/15.69 tff(36,plain,
% 15.50/15.69 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(x, multiplicative_identity, add(x, multiplicative_identity))),
% 15.50/15.69 inference(quant_inst,[status(thm)],[])).
% 15.50/15.69 tff(37,plain,
% 15.50/15.69 (sum(x, multiplicative_identity, add(x, multiplicative_identity))),
% 15.50/15.69 inference(unit_resolution,[status(thm)],[36, 35])).
% 15.50/15.69 tff(38,plain,
% 15.50/15.69 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl(((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1))) <=> ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1))))),
% 15.50/15.69 inference(bind,[status(th)],[])).
% 15.50/15.69 tff(39,plain,
% 15.50/15.69 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | 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(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 15.50/15.69 inference(quant_intro,[status(thm)],[38])).
% 15.50/15.69 tff(40,plain,
% 15.50/15.69 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | 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(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(41,plain,
% 15.50/15.69 (^[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))) | (~sum(V1, V2, V4))) <=> (((~sum(Y, Z, V3)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | (~sum(V1, V2, V4))))), rewrite((((~sum(Y, Z, V3)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | (~sum(V1, V2, V4))) <=> ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))), (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) <=> ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))))), ((((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(V3, X, V4)) <=> (((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | product(V3, X, V4)))), rewrite((((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | product(V3, X, V4)) <=> ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))), ((((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(V3, X, V4)) <=> ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))))),
% 15.50/15.69 inference(bind,[status(th)],[])).
% 15.50/15.69 tff(42,plain,
% 15.50/15.69 (![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))) | (~sum(V1, V2, V4))) | product(V3, X, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 15.50/15.69 inference(quant_intro,[status(thm)],[41])).
% 15.50/15.69 tff(43,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))) | (~sum(V1, V2, V4))) | product(V3, X, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity4')).
% 15.50/15.69 tff(44,plain,
% 15.50/15.69 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[43, 42])).
% 15.50/15.69 tff(45,plain,
% 15.50/15.69 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[44, 40])).
% 15.50/15.69 tff(46,plain,(
% 15.50/15.69 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 15.50/15.69 inference(skolemize,[status(sab)],[45])).
% 15.50/15.69 tff(47,plain,
% 15.50/15.69 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[46, 39])).
% 15.50/15.69 tff(48,plain,
% 15.50/15.69 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(x, multiplicative_identity, x)) | product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(x, multiplicative_identity, x)) | product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity)))),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(49,plain,
% 15.50/15.69 (((~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity)) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~product(x, multiplicative_identity, x))) <=> ((~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(x, multiplicative_identity, x)) | product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity)))),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(50,plain,
% 15.50/15.69 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity)) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~product(x, multiplicative_identity, x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(x, multiplicative_identity, x)) | product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))))),
% 15.50/15.69 inference(monotonicity,[status(thm)],[49])).
% 15.50/15.69 tff(51,plain,
% 15.50/15.69 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity)) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~product(x, multiplicative_identity, x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(x, multiplicative_identity, x)) | product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity)))),
% 15.50/15.69 inference(transitivity,[status(thm)],[50, 48])).
% 15.50/15.69 tff(52,plain,
% 15.50/15.69 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity)) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~product(x, multiplicative_identity, x)))),
% 15.50/15.69 inference(quant_inst,[status(thm)],[])).
% 15.50/15.69 tff(53,plain,
% 15.50/15.69 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(x, multiplicative_identity, x)) | product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[52, 51])).
% 15.50/15.69 tff(54,plain,
% 15.50/15.69 ($false),
% 15.50/15.69 inference(unit_resolution,[status(thm)],[53, 47, 37, 28, 19, 10])).
% 15.50/15.69 tff(55,plain,(product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))), inference(lemma,lemma(discharge,[]))).
% 15.50/15.69 tff(56,plain,
% 15.50/15.69 (^[Y: $i, X: $i] : refl(product(X, Y, multiply(X, Y)) <=> product(X, Y, multiply(X, Y)))),
% 15.50/15.69 inference(bind,[status(th)],[])).
% 15.50/15.69 tff(57,plain,
% 15.50/15.69 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 15.50/15.69 inference(quant_intro,[status(thm)],[56])).
% 15.50/15.69 tff(58,plain,
% 15.50/15.69 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(59,axiom,(![Y: $i, X: $i] : product(X, Y, multiply(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','closure_of_multiplication')).
% 15.50/15.69 tff(60,plain,
% 15.50/15.69 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[59, 58])).
% 15.50/15.69 tff(61,plain,(
% 15.50/15.69 ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 15.50/15.69 inference(skolemize,[status(sab)],[60])).
% 15.50/15.69 tff(62,plain,
% 15.50/15.69 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[61, 57])).
% 15.50/15.69 tff(63,plain,
% 15.50/15.69 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(add(x, multiplicative_identity), multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))),
% 15.50/15.69 inference(quant_inst,[status(thm)],[])).
% 15.50/15.69 tff(64,plain,
% 15.50/15.69 (product(add(x, multiplicative_identity), multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))),
% 15.50/15.69 inference(unit_resolution,[status(thm)],[63, 62])).
% 15.50/15.69 tff(65,plain,
% 15.50/15.69 (^[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))))),
% 15.50/15.69 inference(bind,[status(th)],[])).
% 15.50/15.69 tff(66,plain,
% 15.50/15.69 (![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)))),
% 15.50/15.69 inference(quant_intro,[status(thm)],[65])).
% 15.50/15.69 tff(67,plain,
% 15.50/15.69 (![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)))),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(68,plain,
% 15.50/15.69 (^[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))))),
% 15.50/15.69 inference(bind,[status(th)],[])).
% 15.50/15.69 tff(69,plain,
% 15.50/15.69 (![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)))),
% 15.50/15.69 inference(quant_intro,[status(thm)],[68])).
% 15.50/15.69 tff(70,axiom,(![V: $i, Y: $i, U: $i, X: $i] : (((~product(X, Y, U)) | (~product(X, Y, V))) | (U = V))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplication_is_well_defined')).
% 15.50/15.69 tff(71,plain,
% 15.50/15.69 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[70, 69])).
% 15.50/15.69 tff(72,plain,
% 15.50/15.69 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[71, 67])).
% 15.50/15.69 tff(73,plain,(
% 15.50/15.69 ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 15.50/15.69 inference(skolemize,[status(sab)],[72])).
% 15.50/15.69 tff(74,plain,
% 15.50/15.69 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[73, 66])).
% 15.50/15.69 tff(75,plain,
% 15.50/15.69 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((~product(add(x, multiplicative_identity), multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (add(x, multiplicative_identity) = multiply(add(x, multiplicative_identity), multiplicative_identity)) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | (~product(add(x, multiplicative_identity), multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (add(x, multiplicative_identity) = multiply(add(x, multiplicative_identity), multiplicative_identity)) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))))),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(76,plain,
% 15.50/15.69 (((add(x, multiplicative_identity) = multiply(add(x, multiplicative_identity), multiplicative_identity)) | (~product(add(x, multiplicative_identity), multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity)))) <=> ((~product(add(x, multiplicative_identity), multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (add(x, multiplicative_identity) = multiply(add(x, multiplicative_identity), multiplicative_identity)) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))))),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(77,plain,
% 15.50/15.69 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((add(x, multiplicative_identity) = multiply(add(x, multiplicative_identity), multiplicative_identity)) | (~product(add(x, multiplicative_identity), multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((~product(add(x, multiplicative_identity), multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (add(x, multiplicative_identity) = multiply(add(x, multiplicative_identity), multiplicative_identity)) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity)))))),
% 15.50/15.69 inference(monotonicity,[status(thm)],[76])).
% 15.50/15.69 tff(78,plain,
% 15.50/15.69 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((add(x, multiplicative_identity) = multiply(add(x, multiplicative_identity), multiplicative_identity)) | (~product(add(x, multiplicative_identity), multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | (~product(add(x, multiplicative_identity), multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (add(x, multiplicative_identity) = multiply(add(x, multiplicative_identity), multiplicative_identity)) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))))),
% 15.50/15.69 inference(transitivity,[status(thm)],[77, 75])).
% 15.50/15.69 tff(79,plain,
% 15.50/15.69 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((add(x, multiplicative_identity) = multiply(add(x, multiplicative_identity), multiplicative_identity)) | (~product(add(x, multiplicative_identity), multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))))),
% 15.50/15.69 inference(quant_inst,[status(thm)],[])).
% 15.50/15.69 tff(80,plain,
% 15.50/15.69 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | (~product(add(x, multiplicative_identity), multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (add(x, multiplicative_identity) = multiply(add(x, multiplicative_identity), multiplicative_identity)) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity)))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[79, 78])).
% 15.50/15.69 tff(81,plain,
% 15.50/15.69 (add(x, multiplicative_identity) = multiply(add(x, multiplicative_identity), multiplicative_identity)),
% 15.50/15.69 inference(unit_resolution,[status(thm)],[80, 74, 64, 55])).
% 15.50/15.69 tff(82,plain,
% 15.50/15.69 (multiply(add(x, multiplicative_identity), multiplicative_identity) = add(x, multiplicative_identity)),
% 15.50/15.69 inference(symmetry,[status(thm)],[81])).
% 15.50/15.69 tff(83,plain,
% 15.50/15.69 (product(multiply(add(x, multiplicative_identity), multiplicative_identity), inverse(x), inverse(x)) <=> product(add(x, multiplicative_identity), inverse(x), inverse(x))),
% 15.50/15.69 inference(monotonicity,[status(thm)],[82])).
% 15.50/15.69 tff(84,plain,
% 15.50/15.69 (product(add(x, multiplicative_identity), inverse(x), inverse(x)) <=> product(multiply(add(x, multiplicative_identity), multiplicative_identity), inverse(x), inverse(x))),
% 15.50/15.69 inference(symmetry,[status(thm)],[83])).
% 15.50/15.69 tff(85,plain,
% 15.50/15.69 ((~product(add(x, multiplicative_identity), inverse(x), inverse(x))) <=> (~product(multiply(add(x, multiplicative_identity), multiplicative_identity), inverse(x), inverse(x)))),
% 15.50/15.69 inference(monotonicity,[status(thm)],[84])).
% 15.50/15.69 tff(86,assumption,(~product(multiplicative_identity, x, multiply(x, x))), introduced(assumption)).
% 15.50/15.69 tff(87,plain,
% 15.50/15.69 (^[X: $i] : refl(product(inverse(X), X, additive_identity) <=> product(inverse(X), X, additive_identity))),
% 15.50/15.69 inference(bind,[status(th)],[])).
% 15.50/15.69 tff(88,plain,
% 15.50/15.69 (![X: $i] : product(inverse(X), X, additive_identity) <=> ![X: $i] : product(inverse(X), X, additive_identity)),
% 15.50/15.69 inference(quant_intro,[status(thm)],[87])).
% 15.50/15.69 tff(89,plain,
% 15.50/15.69 (![X: $i] : product(inverse(X), X, additive_identity) <=> ![X: $i] : product(inverse(X), X, additive_identity)),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(90,axiom,(![X: $i] : product(inverse(X), X, additive_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_inverse1')).
% 15.50/15.69 tff(91,plain,
% 15.50/15.69 (![X: $i] : product(inverse(X), X, additive_identity)),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[90, 89])).
% 15.50/15.69 tff(92,plain,(
% 15.50/15.69 ![X: $i] : product(inverse(X), X, additive_identity)),
% 15.50/15.69 inference(skolemize,[status(sab)],[91])).
% 15.50/15.69 tff(93,plain,
% 15.50/15.69 (![X: $i] : product(inverse(X), X, additive_identity)),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[92, 88])).
% 15.50/15.69 tff(94,plain,
% 15.50/15.69 ((~![X: $i] : product(inverse(X), X, additive_identity)) | product(inverse(x), x, additive_identity)),
% 15.50/15.69 inference(quant_inst,[status(thm)],[])).
% 15.50/15.69 tff(95,plain,
% 15.50/15.69 (product(inverse(x), x, additive_identity)),
% 15.50/15.69 inference(unit_resolution,[status(thm)],[94, 93])).
% 15.50/15.69 tff(96,plain,
% 15.50/15.69 (^[X: $i] : refl(sum(inverse(X), X, multiplicative_identity) <=> sum(inverse(X), X, multiplicative_identity))),
% 15.50/15.69 inference(bind,[status(th)],[])).
% 15.50/15.69 tff(97,plain,
% 15.50/15.69 (![X: $i] : sum(inverse(X), X, multiplicative_identity) <=> ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 15.50/15.69 inference(quant_intro,[status(thm)],[96])).
% 15.50/15.69 tff(98,plain,
% 15.50/15.69 (![X: $i] : sum(inverse(X), X, multiplicative_identity) <=> ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(99,axiom,(![X: $i] : sum(inverse(X), X, multiplicative_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_inverse1')).
% 15.50/15.69 tff(100,plain,
% 15.50/15.69 (![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[99, 98])).
% 15.50/15.69 tff(101,plain,(
% 15.50/15.69 ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 15.50/15.69 inference(skolemize,[status(sab)],[100])).
% 15.50/15.69 tff(102,plain,
% 15.50/15.69 (![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[101, 97])).
% 15.50/15.69 tff(103,plain,
% 15.50/15.69 ((~![X: $i] : sum(inverse(X), X, multiplicative_identity)) | sum(inverse(x), x, multiplicative_identity)),
% 15.50/15.69 inference(quant_inst,[status(thm)],[])).
% 15.50/15.69 tff(104,plain,
% 15.50/15.69 (sum(inverse(x), x, multiplicative_identity)),
% 15.50/15.69 inference(unit_resolution,[status(thm)],[103, 102])).
% 15.50/15.69 tff(105,plain,
% 15.50/15.69 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, x, x)),
% 15.50/15.69 inference(quant_inst,[status(thm)],[])).
% 15.50/15.69 tff(106,plain,
% 15.50/15.69 (sum(additive_identity, x, x)),
% 15.50/15.69 inference(unit_resolution,[status(thm)],[105, 7])).
% 15.50/15.69 tff(107,plain,
% 15.50/15.69 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(x, x, multiply(x, x))),
% 15.50/15.69 inference(quant_inst,[status(thm)],[])).
% 15.50/15.69 tff(108,plain,
% 15.50/15.69 (product(x, x, multiply(x, x))),
% 15.50/15.69 inference(unit_resolution,[status(thm)],[107, 62])).
% 15.50/15.69 tff(109,plain,
% 15.50/15.69 (^[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))))),
% 15.50/15.69 inference(bind,[status(th)],[])).
% 15.50/15.69 tff(110,plain,
% 15.50/15.69 (![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)))),
% 15.50/15.69 inference(quant_intro,[status(thm)],[109])).
% 15.50/15.69 tff(111,plain,
% 15.50/15.69 (![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)))),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(112,plain,
% 15.50/15.69 (^[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)))))),
% 15.50/15.69 inference(bind,[status(th)],[])).
% 15.50/15.69 tff(113,plain,
% 15.50/15.69 (![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)))),
% 15.50/15.69 inference(quant_intro,[status(thm)],[112])).
% 15.50/15.69 tff(114,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(X, V3, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity6')).
% 15.50/15.69 tff(115,plain,
% 15.50/15.69 (![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)))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[114, 113])).
% 15.50/15.69 tff(116,plain,
% 15.50/15.69 (![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)))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[115, 111])).
% 15.50/15.69 tff(117,plain,(
% 15.50/15.69 ![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)))),
% 15.50/15.69 inference(skolemize,[status(sab)],[116])).
% 15.50/15.69 tff(118,plain,
% 15.50/15.69 (![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)))),
% 15.50/15.69 inference(modus_ponens,[status(thm)],[117, 110])).
% 15.50/15.69 tff(119,plain,
% 15.50/15.69 (((~![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))))),
% 15.50/15.69 inference(rewrite,[status(thm)],[])).
% 15.50/15.69 tff(120,plain,
% 15.50/15.69 ((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))))),
% 15.50/15.70 inference(rewrite,[status(thm)],[])).
% 15.50/15.70 tff(121,plain,
% 15.50/15.70 (((~![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)))))),
% 15.50/15.70 inference(monotonicity,[status(thm)],[120])).
% 15.50/15.70 tff(122,plain,
% 15.50/15.70 (((~![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))))),
% 15.50/15.70 inference(transitivity,[status(thm)],[121, 119])).
% 15.50/15.70 tff(123,plain,
% 15.50/15.70 ((~![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))))),
% 15.50/15.70 inference(quant_inst,[status(thm)],[])).
% 15.50/15.70 tff(124,plain,
% 15.50/15.70 ((~![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)))),
% 15.50/15.70 inference(modus_ponens,[status(thm)],[123, 122])).
% 15.50/15.70 tff(125,plain,
% 15.50/15.70 (sum(additive_identity, multiply(x, x), multiply(x, x))),
% 15.50/15.70 inference(unit_resolution,[status(thm)],[124, 118, 108, 106])).
% 15.50/15.70 tff(126,plain,
% 15.50/15.70 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~product(x, x, multiply(x, x))) | (~product(inverse(x), x, additive_identity)) | (~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | product(multiplicative_identity, x, multiply(x, x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(x, x, multiply(x, x))) | (~product(inverse(x), x, additive_identity)) | (~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | product(multiplicative_identity, x, multiply(x, x)))),
% 15.50/15.70 inference(rewrite,[status(thm)],[])).
% 15.50/15.70 tff(127,plain,
% 15.50/15.70 (((~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | product(multiplicative_identity, x, multiply(x, x)) | (~product(x, x, multiply(x, x))) | (~product(inverse(x), x, additive_identity))) <=> ((~product(x, x, multiply(x, x))) | (~product(inverse(x), x, additive_identity)) | (~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | product(multiplicative_identity, x, multiply(x, x)))),
% 15.50/15.70 inference(rewrite,[status(thm)],[])).
% 15.50/15.70 tff(128,plain,
% 15.50/15.70 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | product(multiplicative_identity, x, multiply(x, x)) | (~product(x, x, multiply(x, x))) | (~product(inverse(x), x, additive_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~product(x, x, multiply(x, x))) | (~product(inverse(x), x, additive_identity)) | (~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | product(multiplicative_identity, x, multiply(x, x))))),
% 15.50/15.70 inference(monotonicity,[status(thm)],[127])).
% 15.50/15.70 tff(129,plain,
% 15.50/15.70 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | product(multiplicative_identity, x, multiply(x, x)) | (~product(x, x, multiply(x, x))) | (~product(inverse(x), x, additive_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(x, x, multiply(x, x))) | (~product(inverse(x), x, additive_identity)) | (~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | product(multiplicative_identity, x, multiply(x, x)))),
% 15.50/15.70 inference(transitivity,[status(thm)],[128, 126])).
% 15.50/15.70 tff(130,plain,
% 15.50/15.70 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | product(multiplicative_identity, x, multiply(x, x)) | (~product(x, x, multiply(x, x))) | (~product(inverse(x), x, additive_identity)))),
% 15.50/15.70 inference(quant_inst,[status(thm)],[])).
% 15.50/15.70 tff(131,plain,
% 15.50/15.70 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(x, x, multiply(x, x))) | (~product(inverse(x), x, additive_identity)) | (~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, multiply(x, x), multiply(x, x))) | product(multiplicative_identity, x, multiply(x, x))),
% 15.50/15.70 inference(modus_ponens,[status(thm)],[130, 129])).
% 15.50/15.70 tff(132,plain,
% 15.50/15.70 ($false),
% 15.50/15.70 inference(unit_resolution,[status(thm)],[131, 47, 108, 125, 104, 95, 86])).
% 15.50/15.70 tff(133,plain,(product(multiplicative_identity, x, multiply(x, x))), inference(lemma,lemma(discharge,[]))).
% 15.50/15.70 tff(134,plain,
% 15.50/15.70 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, x, x)),
% 15.50/15.70 inference(quant_inst,[status(thm)],[])).
% 15.50/15.70 tff(135,plain,
% 15.50/15.70 (product(multiplicative_identity, x, x)),
% 15.50/15.70 inference(unit_resolution,[status(thm)],[134, 26])).
% 15.50/15.70 tff(136,plain,
% 15.50/15.70 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((~product(multiplicative_identity, x, x)) | (~product(multiplicative_identity, x, multiply(x, x))) | (multiply(x, x) = x))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | (~product(multiplicative_identity, x, x)) | (~product(multiplicative_identity, x, multiply(x, x))) | (multiply(x, x) = x))),
% 15.50/15.70 inference(rewrite,[status(thm)],[])).
% 15.50/15.70 tff(137,plain,
% 15.50/15.70 (((multiply(x, x) = x) | (~product(multiplicative_identity, x, x)) | (~product(multiplicative_identity, x, multiply(x, x)))) <=> ((~product(multiplicative_identity, x, x)) | (~product(multiplicative_identity, x, multiply(x, x))) | (multiply(x, x) = x))),
% 15.50/15.70 inference(rewrite,[status(thm)],[])).
% 15.50/15.70 tff(138,plain,
% 15.50/15.70 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((multiply(x, x) = x) | (~product(multiplicative_identity, x, x)) | (~product(multiplicative_identity, x, multiply(x, x))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((~product(multiplicative_identity, x, x)) | (~product(multiplicative_identity, x, multiply(x, x))) | (multiply(x, x) = x)))),
% 15.50/15.70 inference(monotonicity,[status(thm)],[137])).
% 15.50/15.70 tff(139,plain,
% 15.50/15.70 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((multiply(x, x) = x) | (~product(multiplicative_identity, x, x)) | (~product(multiplicative_identity, x, multiply(x, x))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | (~product(multiplicative_identity, x, x)) | (~product(multiplicative_identity, x, multiply(x, x))) | (multiply(x, x) = x))),
% 15.50/15.70 inference(transitivity,[status(thm)],[138, 136])).
% 15.50/15.70 tff(140,plain,
% 15.50/15.70 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | ((multiply(x, x) = x) | (~product(multiplicative_identity, x, x)) | (~product(multiplicative_identity, x, multiply(x, x))))),
% 15.50/15.70 inference(quant_inst,[status(thm)],[])).
% 15.50/15.70 tff(141,plain,
% 15.50/15.70 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~product(X, Y, V)) | (~product(X, Y, U)))) | (~product(multiplicative_identity, x, x)) | (~product(multiplicative_identity, x, multiply(x, x))) | (multiply(x, x) = x)),
% 15.50/15.70 inference(modus_ponens,[status(thm)],[140, 139])).
% 15.50/15.70 tff(142,plain,
% 15.50/15.70 (multiply(x, x) = x),
% 15.50/15.70 inference(unit_resolution,[status(thm)],[141, 74, 135, 133])).
% 15.50/15.70 tff(143,plain,
% 15.50/15.70 (inverse(multiply(x, x)) = inverse(x)),
% 15.50/15.70 inference(monotonicity,[status(thm)],[142])).
% 15.50/15.70 tff(144,plain,
% 15.50/15.70 (inverse(x) = inverse(multiply(x, x))),
% 15.50/15.70 inference(symmetry,[status(thm)],[143])).
% 15.50/15.70 tff(145,plain,
% 15.50/15.70 (sum(inverse(x), x, add(x, multiplicative_identity)) <=> sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity))),
% 15.50/15.70 inference(monotonicity,[status(thm)],[144])).
% 15.50/15.70 tff(146,plain,
% 15.50/15.70 (sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity)) <=> sum(inverse(x), x, add(x, multiplicative_identity))),
% 15.50/15.70 inference(symmetry,[status(thm)],[145])).
% 15.50/15.70 tff(147,plain,
% 15.50/15.70 ((~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity))) <=> (~sum(inverse(x), x, add(x, multiplicative_identity)))),
% 15.50/15.70 inference(monotonicity,[status(thm)],[146])).
% 15.50/15.70 tff(148,plain,
% 15.50/15.70 ((~![X: $i] : sum(inverse(X), X, multiplicative_identity)) | sum(inverse(multiply(x, x)), multiply(x, x), multiplicative_identity)),
% 15.50/15.70 inference(quant_inst,[status(thm)],[])).
% 15.50/15.70 tff(149,plain,
% 15.50/15.70 (sum(inverse(multiply(x, x)), multiply(x, x), multiplicative_identity)),
% 15.50/15.70 inference(unit_resolution,[status(thm)],[148, 102])).
% 15.50/15.70 tff(150,plain,
% 15.50/15.70 ((~sum(x, multiplicative_identity, multiplicative_identity)) <=> (~sum(x, multiplicative_identity, multiplicative_identity))),
% 15.50/15.70 inference(rewrite,[status(thm)],[])).
% 15.50/15.70 tff(151,axiom,(~sum(x, multiplicative_identity, multiplicative_identity)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_equations')).
% 15.50/15.70 tff(152,plain,
% 15.50/15.70 (~sum(x, multiplicative_identity, multiplicative_identity)),
% 15.50/15.70 inference(modus_ponens,[status(thm)],[151, 150])).
% 15.50/15.70 tff(153,plain,
% 15.50/15.70 (((~![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(x, multiplicative_identity, multiplicative_identity) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_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)))) | sum(x, multiplicative_identity, multiplicative_identity) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity)))),
% 15.50/15.70 inference(rewrite,[status(thm)],[])).
% 15.50/15.70 tff(154,plain,
% 15.50/15.70 ((sum(x, multiplicative_identity, multiplicative_identity) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity))) <=> (sum(x, multiplicative_identity, multiplicative_identity) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity)))),
% 15.50/15.70 inference(rewrite,[status(thm)],[])).
% 15.50/15.70 tff(155,plain,
% 15.50/15.70 (((~![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(x, multiplicative_identity, multiplicative_identity) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_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)))) | (sum(x, multiplicative_identity, multiplicative_identity) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity))))),
% 15.50/15.70 inference(monotonicity,[status(thm)],[154])).
% 15.50/15.70 tff(156,plain,
% 15.50/15.70 (((~![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(x, multiplicative_identity, multiplicative_identity) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_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)))) | sum(x, multiplicative_identity, multiplicative_identity) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity)))),
% 15.50/15.70 inference(transitivity,[status(thm)],[155, 153])).
% 15.50/15.70 tff(157,plain,
% 15.50/15.70 ((~![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(x, multiplicative_identity, multiplicative_identity) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity)))),
% 15.50/15.70 inference(quant_inst,[status(thm)],[])).
% 15.50/15.70 tff(158,plain,
% 15.50/15.70 ((~![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(x, multiplicative_identity, multiplicative_identity) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity))),
% 15.50/15.70 inference(modus_ponens,[status(thm)],[157, 156])).
% 15.50/15.70 tff(159,plain,
% 15.50/15.70 (~product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity)),
% 15.50/15.70 inference(unit_resolution,[status(thm)],[158, 118, 152, 37, 28])).
% 15.50/15.70 tff(160,plain,
% 15.50/15.70 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((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)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))))),
% 15.50/15.70 inference(bind,[status(th)],[])).
% 15.50/15.70 tff(161,plain,
% 15.50/15.70 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))),
% 15.50/15.70 inference(quant_intro,[status(thm)],[160])).
% 15.50/15.70 tff(162,plain,
% 15.50/15.70 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))),
% 15.50/15.70 inference(rewrite,[status(thm)],[])).
% 15.50/15.70 tff(163,plain,
% 15.50/15.70 (^[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))) | (~sum(X, V3, V4))) <=> (((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) | (~sum(X, V3, V4))))), rewrite((((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) | (~sum(X, V3, V4))) <=> ((~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))), (((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~sum(X, V3, V4))) <=> ((~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))))), ((((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~sum(X, V3, V4))) | product(V1, V2, V4)) <=> (((~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) | product(V1, V2, V4)))), rewrite((((~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) | product(V1, V2, V4)) <=> (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))), ((((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~sum(X, V3, V4))) | product(V1, V2, V4)) <=> (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))))),
% 15.50/15.70 inference(bind,[status(th)],[])).
% 15.50/15.70 tff(164,plain,
% 15.50/15.70 (![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))) | (~sum(X, V3, V4))) | product(V1, V2, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))),
% 15.50/15.70 inference(quant_intro,[status(thm)],[163])).
% 15.50/15.70 tff(165,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))) | (~sum(X, V3, V4))) | product(V1, V2, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity5')).
% 15.50/15.70 tff(166,plain,
% 15.50/15.70 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))),
% 15.50/15.70 inference(modus_ponens,[status(thm)],[165, 164])).
% 15.50/15.70 tff(167,plain,
% 15.50/15.70 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))),
% 15.50/15.70 inference(modus_ponens,[status(thm)],[166, 162])).
% 15.50/15.70 tff(168,plain,(
% 15.50/15.70 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))),
% 15.50/15.70 inference(skolemize,[status(sab)],[167])).
% 15.50/15.70 tff(169,plain,
% 15.50/15.70 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))),
% 15.50/15.70 inference(modus_ponens,[status(thm)],[168, 161])).
% 15.50/15.70 tff(170,plain,
% 15.50/15.70 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity) | (~product(x, x, multiply(x, x))) | (~sum(inverse(multiply(x, x)), multiply(x, x), multiplicative_identity)) | (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity) | (~product(x, x, multiply(x, x))) | (~sum(inverse(multiply(x, x)), multiply(x, x), multiplicative_identity)) | (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity))))),
% 15.50/15.70 inference(rewrite,[status(thm)],[])).
% 15.50/15.70 tff(171,plain,
% 15.50/15.70 ((product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity) | (~sum(inverse(multiply(x, x)), multiply(x, x), multiplicative_identity)) | (~product(x, x, multiply(x, x))) | (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity))) | (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity)))) <=> (product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity) | (~product(x, x, multiply(x, x))) | (~sum(inverse(multiply(x, x)), multiply(x, x), multiplicative_identity)) | (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity))))),
% 15.50/15.70 inference(rewrite,[status(thm)],[])).
% 15.50/15.70 tff(172,plain,
% 15.50/15.70 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity) | (~sum(inverse(multiply(x, x)), multiply(x, x), multiplicative_identity)) | (~product(x, x, multiply(x, x))) | (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity))) | (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity) | (~product(x, x, multiply(x, x))) | (~sum(inverse(multiply(x, x)), multiply(x, x), multiplicative_identity)) | (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity)))))),
% 15.50/15.70 inference(monotonicity,[status(thm)],[171])).
% 15.50/15.70 tff(173,plain,
% 15.50/15.70 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity) | (~sum(inverse(multiply(x, x)), multiply(x, x), multiplicative_identity)) | (~product(x, x, multiply(x, x))) | (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity))) | (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity) | (~product(x, x, multiply(x, x))) | (~sum(inverse(multiply(x, x)), multiply(x, x), multiplicative_identity)) | (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity))))),
% 15.50/15.71 inference(transitivity,[status(thm)],[172, 170])).
% 15.50/15.71 tff(174,plain,
% 15.50/15.71 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity) | (~sum(inverse(multiply(x, x)), multiply(x, x), multiplicative_identity)) | (~product(x, x, multiply(x, x))) | (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity))) | (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity))))),
% 15.50/15.71 inference(quant_inst,[status(thm)],[])).
% 15.50/15.71 tff(175,plain,
% 15.50/15.71 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | product(add(x, multiplicative_identity), add(x, multiplicative_identity), multiplicative_identity) | (~product(x, x, multiply(x, x))) | (~sum(inverse(multiply(x, x)), multiply(x, x), multiplicative_identity)) | (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity)))),
% 15.50/15.71 inference(modus_ponens,[status(thm)],[174, 173])).
% 15.50/15.71 tff(176,plain,
% 15.50/15.71 (~sum(inverse(multiply(x, x)), x, add(x, multiplicative_identity))),
% 15.50/15.71 inference(unit_resolution,[status(thm)],[175, 169, 108, 159, 149])).
% 15.50/15.71 tff(177,plain,
% 15.50/15.71 (~sum(inverse(x), x, add(x, multiplicative_identity))),
% 15.50/15.71 inference(modus_ponens,[status(thm)],[176, 147])).
% 15.50/15.71 tff(178,assumption,(~product(multiplicative_identity, additive_identity, additive_identity)), introduced(assumption)).
% 15.50/15.71 tff(179,plain,
% 15.50/15.71 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, additive_identity, additive_identity)),
% 15.50/15.71 inference(quant_inst,[status(thm)],[])).
% 15.50/15.71 tff(180,plain,
% 15.50/15.71 ($false),
% 15.50/15.71 inference(unit_resolution,[status(thm)],[179, 26, 178])).
% 15.50/15.71 tff(181,plain,(product(multiplicative_identity, additive_identity, additive_identity)), inference(lemma,lemma(discharge,[]))).
% 15.50/15.71 tff(182,plain,
% 15.50/15.71 (^[Z: $i, Y: $i, X: $i] : refl(((~sum(X, Y, Z)) | sum(Y, X, Z)) <=> ((~sum(X, Y, Z)) | sum(Y, X, Z)))),
% 15.50/15.71 inference(bind,[status(th)],[])).
% 15.50/15.71 tff(183,plain,
% 15.50/15.71 (![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))),
% 15.50/15.71 inference(quant_intro,[status(thm)],[182])).
% 15.50/15.71 tff(184,plain,
% 15.50/15.71 (![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))),
% 15.50/15.71 inference(rewrite,[status(thm)],[])).
% 15.50/15.71 tff(185,axiom,(![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','commutativity_of_addition')).
% 15.50/15.71 tff(186,plain,
% 15.50/15.71 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 15.50/15.71 inference(modus_ponens,[status(thm)],[185, 184])).
% 15.50/15.71 tff(187,plain,(
% 15.50/15.71 ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 15.50/15.71 inference(skolemize,[status(sab)],[186])).
% 15.50/15.71 tff(188,plain,
% 15.50/15.71 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 15.50/15.71 inference(modus_ponens,[status(thm)],[187, 183])).
% 15.50/15.71 tff(189,plain,
% 15.50/15.71 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(additive_identity, x, x)) | sum(x, additive_identity, x))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(additive_identity, x, x)) | sum(x, additive_identity, x))),
% 15.50/15.71 inference(rewrite,[status(thm)],[])).
% 15.50/15.71 tff(190,plain,
% 15.50/15.71 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(additive_identity, x, x)) | sum(x, additive_identity, x))),
% 15.50/15.71 inference(quant_inst,[status(thm)],[])).
% 15.50/15.71 tff(191,plain,
% 15.50/15.71 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(additive_identity, x, x)) | sum(x, additive_identity, x)),
% 15.50/15.71 inference(modus_ponens,[status(thm)],[190, 189])).
% 15.50/15.71 tff(192,plain,
% 15.50/15.71 (sum(x, additive_identity, x)),
% 15.50/15.71 inference(unit_resolution,[status(thm)],[191, 188, 106])).
% 15.50/15.71 tff(193,plain,
% 15.50/15.71 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | ((~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~sum(x, additive_identity, x)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | product(add(x, multiplicative_identity), x, x))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~sum(x, additive_identity, x)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | product(add(x, multiplicative_identity), x, x))),
% 15.50/15.71 inference(rewrite,[status(thm)],[])).
% 15.50/15.71 tff(194,plain,
% 15.50/15.71 ((product(add(x, multiplicative_identity), x, x) | (~sum(x, additive_identity, x)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(x, additive_identity, x)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity)))) <=> ((~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~sum(x, additive_identity, x)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | product(add(x, multiplicative_identity), x, x))),
% 15.50/15.71 inference(rewrite,[status(thm)],[])).
% 15.50/15.71 tff(195,plain,
% 15.50/15.71 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (product(add(x, multiplicative_identity), x, x) | (~sum(x, additive_identity, x)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(x, additive_identity, x)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | ((~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~sum(x, additive_identity, x)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | product(add(x, multiplicative_identity), x, x)))),
% 15.50/15.71 inference(monotonicity,[status(thm)],[194])).
% 15.50/15.71 tff(196,plain,
% 15.50/15.71 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (product(add(x, multiplicative_identity), x, x) | (~sum(x, additive_identity, x)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(x, additive_identity, x)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~sum(x, additive_identity, x)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | product(add(x, multiplicative_identity), x, x))),
% 15.50/15.71 inference(transitivity,[status(thm)],[195, 193])).
% 15.50/15.71 tff(197,plain,
% 15.50/15.71 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (product(add(x, multiplicative_identity), x, x) | (~sum(x, additive_identity, x)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(x, additive_identity, x)) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))))),
% 15.50/15.71 inference(quant_inst,[status(thm)],[])).
% 15.50/15.71 tff(198,plain,
% 15.50/15.71 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (~sum(x, multiplicative_identity, add(x, multiplicative_identity))) | (~sum(x, additive_identity, x)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | product(add(x, multiplicative_identity), x, x)),
% 15.50/15.71 inference(modus_ponens,[status(thm)],[197, 196])).
% 15.50/15.71 tff(199,plain,
% 15.50/15.71 (product(add(x, multiplicative_identity), x, x)),
% 15.50/15.71 inference(unit_resolution,[status(thm)],[198, 169, 37, 192, 181])).
% 15.50/15.71 tff(200,plain,
% 15.50/15.71 (^[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))))),
% 15.50/15.71 inference(bind,[status(th)],[])).
% 15.50/15.71 tff(201,plain,
% 15.50/15.71 (![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)))),
% 15.50/15.71 inference(quant_intro,[status(thm)],[200])).
% 15.50/15.71 tff(202,plain,
% 15.50/15.71 (![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)))),
% 15.50/15.71 inference(rewrite,[status(thm)],[])).
% 15.50/15.71 tff(203,plain,
% 15.50/15.71 (^[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)))))),
% 15.50/15.71 inference(bind,[status(th)],[])).
% 15.50/15.71 tff(204,plain,
% 15.50/15.71 (![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)))),
% 15.50/15.71 inference(quant_intro,[status(thm)],[203])).
% 15.50/15.71 tff(205,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity1')).
% 15.50/15.71 tff(206,plain,
% 15.50/15.71 (![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)))),
% 15.50/15.71 inference(modus_ponens,[status(thm)],[205, 204])).
% 15.50/15.71 tff(207,plain,
% 15.50/15.71 (![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)))),
% 15.50/15.71 inference(modus_ponens,[status(thm)],[206, 202])).
% 15.50/15.71 tff(208,plain,(
% 15.50/15.71 ![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)))),
% 15.50/15.71 inference(skolemize,[status(sab)],[207])).
% 15.50/15.71 tff(209,plain,
% 15.50/15.71 (![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)))),
% 15.50/15.71 inference(modus_ponens,[status(thm)],[208, 201])).
% 15.50/15.71 tff(210,plain,
% 15.50/15.71 (((~![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(inverse(x), x, multiplicative_identity)) | sum(inverse(x), x, add(x, multiplicative_identity)) | (~product(add(x, multiplicative_identity), x, x)) | (~product(add(x, multiplicative_identity), inverse(x), inverse(x))) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_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(inverse(x), x, multiplicative_identity)) | sum(inverse(x), x, add(x, multiplicative_identity)) | (~product(add(x, multiplicative_identity), x, x)) | (~product(add(x, multiplicative_identity), inverse(x), inverse(x))) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))))),
% 15.50/15.71 inference(rewrite,[status(thm)],[])).
% 15.50/15.71 tff(211,plain,
% 15.50/15.71 ((sum(inverse(x), x, add(x, multiplicative_identity)) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(add(x, multiplicative_identity), x, x)) | (~product(add(x, multiplicative_identity), inverse(x), inverse(x)))) <=> ((~sum(inverse(x), x, multiplicative_identity)) | sum(inverse(x), x, add(x, multiplicative_identity)) | (~product(add(x, multiplicative_identity), x, x)) | (~product(add(x, multiplicative_identity), inverse(x), inverse(x))) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))))),
% 15.50/15.71 inference(rewrite,[status(thm)],[])).
% 15.50/15.71 tff(212,plain,
% 15.50/15.71 (((~![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(inverse(x), x, add(x, multiplicative_identity)) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(add(x, multiplicative_identity), x, x)) | (~product(add(x, multiplicative_identity), inverse(x), inverse(x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~sum(inverse(x), x, multiplicative_identity)) | sum(inverse(x), x, add(x, multiplicative_identity)) | (~product(add(x, multiplicative_identity), x, x)) | (~product(add(x, multiplicative_identity), inverse(x), inverse(x))) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity)))))),
% 15.50/15.71 inference(monotonicity,[status(thm)],[211])).
% 15.50/15.71 tff(213,plain,
% 15.50/15.71 (((~![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(inverse(x), x, add(x, multiplicative_identity)) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(add(x, multiplicative_identity), x, x)) | (~product(add(x, multiplicative_identity), inverse(x), inverse(x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(inverse(x), x, multiplicative_identity)) | sum(inverse(x), x, add(x, multiplicative_identity)) | (~product(add(x, multiplicative_identity), x, x)) | (~product(add(x, multiplicative_identity), inverse(x), inverse(x))) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))))),
% 15.50/15.71 inference(transitivity,[status(thm)],[212, 210])).
% 15.50/15.71 tff(214,plain,
% 15.50/15.71 ((~![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(inverse(x), x, add(x, multiplicative_identity)) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity))) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(add(x, multiplicative_identity), x, x)) | (~product(add(x, multiplicative_identity), inverse(x), inverse(x))))),
% 15.50/15.71 inference(quant_inst,[status(thm)],[])).
% 15.50/15.71 tff(215,plain,
% 15.50/15.71 ((~![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(inverse(x), x, multiplicative_identity)) | sum(inverse(x), x, add(x, multiplicative_identity)) | (~product(add(x, multiplicative_identity), x, x)) | (~product(add(x, multiplicative_identity), inverse(x), inverse(x))) | (~product(add(x, multiplicative_identity), multiplicative_identity, add(x, multiplicative_identity)))),
% 15.50/15.71 inference(modus_ponens,[status(thm)],[214, 213])).
% 15.50/15.71 tff(216,plain,
% 15.50/15.71 (~product(add(x, multiplicative_identity), inverse(x), inverse(x))),
% 15.50/15.71 inference(unit_resolution,[status(thm)],[215, 209, 199, 104, 177, 55])).
% 15.50/15.71 tff(217,plain,
% 15.50/15.71 (~product(multiply(add(x, multiplicative_identity), multiplicative_identity), inverse(x), inverse(x))),
% 15.50/15.71 inference(modus_ponens,[status(thm)],[216, 85])).
% 15.50/15.71 tff(218,plain,
% 15.50/15.71 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, inverse(x), inverse(x))),
% 15.50/15.71 inference(quant_inst,[status(thm)],[])).
% 15.50/15.71 tff(219,plain,
% 15.50/15.71 (product(multiplicative_identity, inverse(x), inverse(x))),
% 15.50/15.71 inference(unit_resolution,[status(thm)],[218, 26])).
% 15.50/15.71 tff(220,plain,
% 15.50/15.71 (sum(x, multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity)) <=> sum(x, multiplicative_identity, add(x, multiplicative_identity))),
% 15.50/15.71 inference(monotonicity,[status(thm)],[82])).
% 15.50/15.71 tff(221,plain,
% 15.50/15.71 (sum(x, multiplicative_identity, add(x, multiplicative_identity)) <=> sum(x, multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))),
% 15.50/15.71 inference(symmetry,[status(thm)],[220])).
% 15.50/15.71 tff(222,plain,
% 15.50/15.71 (sum(x, multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))),
% 15.50/15.71 inference(modus_ponens,[status(thm)],[37, 221])).
% 15.50/15.71 tff(223,plain,
% 15.50/15.71 (^[X: $i] : refl(product(X, inverse(X), additive_identity) <=> product(X, inverse(X), additive_identity))),
% 15.50/15.71 inference(bind,[status(th)],[])).
% 15.50/15.71 tff(224,plain,
% 15.50/15.71 (![X: $i] : product(X, inverse(X), additive_identity) <=> ![X: $i] : product(X, inverse(X), additive_identity)),
% 15.50/15.71 inference(quant_intro,[status(thm)],[223])).
% 15.50/15.71 tff(225,plain,
% 15.50/15.71 (![X: $i] : product(X, inverse(X), additive_identity) <=> ![X: $i] : product(X, inverse(X), additive_identity)),
% 15.50/15.71 inference(rewrite,[status(thm)],[])).
% 15.50/15.71 tff(226,axiom,(![X: $i] : product(X, inverse(X), additive_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_inverse2')).
% 15.50/15.71 tff(227,plain,
% 15.50/15.71 (![X: $i] : product(X, inverse(X), additive_identity)),
% 15.50/15.71 inference(modus_ponens,[status(thm)],[226, 225])).
% 15.50/15.71 tff(228,plain,(
% 15.50/15.71 ![X: $i] : product(X, inverse(X), additive_identity)),
% 15.50/15.71 inference(skolemize,[status(sab)],[227])).
% 15.50/15.71 tff(229,plain,
% 15.50/15.71 (![X: $i] : product(X, inverse(X), additive_identity)),
% 15.50/15.71 inference(modus_ponens,[status(thm)],[228, 224])).
% 15.50/15.71 tff(230,plain,
% 15.50/15.71 ((~![X: $i] : product(X, inverse(X), additive_identity)) | product(x, inverse(x), additive_identity)),
% 15.50/15.71 inference(quant_inst,[status(thm)],[])).
% 15.50/15.71 tff(231,plain,
% 15.50/15.71 (product(x, inverse(x), additive_identity)),
% 15.50/15.71 inference(unit_resolution,[status(thm)],[230, 229])).
% 15.50/15.71 tff(232,plain,
% 15.50/15.71 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~product(x, inverse(x), additive_identity)) | (~sum(x, multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (~sum(additive_identity, inverse(x), inverse(x))) | product(multiply(add(x, multiplicative_identity), multiplicative_identity), inverse(x), inverse(x)) | (~product(multiplicative_identity, inverse(x), inverse(x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(x, inverse(x), additive_identity)) | (~sum(x, multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (~sum(additive_identity, inverse(x), inverse(x))) | product(multiply(add(x, multiplicative_identity), multiplicative_identity), inverse(x), inverse(x)) | (~product(multiplicative_identity, inverse(x), inverse(x))))),
% 15.50/15.71 inference(rewrite,[status(thm)],[])).
% 15.50/15.71 tff(233,plain,
% 15.50/15.71 (((~sum(x, multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (~sum(additive_identity, inverse(x), inverse(x))) | product(multiply(add(x, multiplicative_identity), multiplicative_identity), inverse(x), inverse(x)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~product(x, inverse(x), additive_identity))) <=> ((~product(x, inverse(x), additive_identity)) | (~sum(x, multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (~sum(additive_identity, inverse(x), inverse(x))) | product(multiply(add(x, multiplicative_identity), multiplicative_identity), inverse(x), inverse(x)) | (~product(multiplicative_identity, inverse(x), inverse(x))))),
% 15.50/15.71 inference(rewrite,[status(thm)],[])).
% 15.50/15.71 tff(234,plain,
% 15.50/15.71 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(x, multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (~sum(additive_identity, inverse(x), inverse(x))) | product(multiply(add(x, multiplicative_identity), multiplicative_identity), inverse(x), inverse(x)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~product(x, inverse(x), additive_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~product(x, inverse(x), additive_identity)) | (~sum(x, multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (~sum(additive_identity, inverse(x), inverse(x))) | product(multiply(add(x, multiplicative_identity), multiplicative_identity), inverse(x), inverse(x)) | (~product(multiplicative_identity, inverse(x), inverse(x)))))),
% 15.50/15.71 inference(monotonicity,[status(thm)],[233])).
% 15.50/15.71 tff(235,plain,
% 15.50/15.71 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(x, multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (~sum(additive_identity, inverse(x), inverse(x))) | product(multiply(add(x, multiplicative_identity), multiplicative_identity), inverse(x), inverse(x)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~product(x, inverse(x), additive_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(x, inverse(x), additive_identity)) | (~sum(x, multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (~sum(additive_identity, inverse(x), inverse(x))) | product(multiply(add(x, multiplicative_identity), multiplicative_identity), inverse(x), inverse(x)) | (~product(multiplicative_identity, inverse(x), inverse(x))))),
% 15.50/15.73 inference(transitivity,[status(thm)],[234, 232])).
% 15.50/15.73 tff(236,plain,
% 15.50/15.73 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(x, multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (~sum(additive_identity, inverse(x), inverse(x))) | product(multiply(add(x, multiplicative_identity), multiplicative_identity), inverse(x), inverse(x)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~product(x, inverse(x), additive_identity)))),
% 15.50/15.73 inference(quant_inst,[status(thm)],[])).
% 15.50/15.73 tff(237,plain,
% 15.50/15.73 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(x, inverse(x), additive_identity)) | (~sum(x, multiplicative_identity, multiply(add(x, multiplicative_identity), multiplicative_identity))) | (~sum(additive_identity, inverse(x), inverse(x))) | product(multiply(add(x, multiplicative_identity), multiplicative_identity), inverse(x), inverse(x)) | (~product(multiplicative_identity, inverse(x), inverse(x)))),
% 15.50/15.73 inference(modus_ponens,[status(thm)],[236, 235])).
% 15.50/15.73 tff(238,plain,
% 15.50/15.73 ($false),
% 15.50/15.73 inference(unit_resolution,[status(thm)],[237, 47, 231, 222, 219, 217, 9])).
% 15.50/15.73 % SZS output end Proof
% 15.50/15.73 % E exiting
%------------------------------------------------------------------------------