%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : BOO012-3 : TPTP v9.0.0. Bugfixed v1.2.1.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n014.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 05:00:08 AM UTC 2025
% Result : Unsatisfiable 2.97s 3.19s
% Output : Proof 3.04s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.02/0.11 % Problem : BOO012-3 : TPTP v9.0.0. Bugfixed v1.2.1.
% 0.02/0.11 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n014.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
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Thu Jun 19 11:33:09 EDT 2025
% 0.11/0.32 % CPUTime :
% 2.97/3.19 % SZS status Unsatisfiable
% 2.97/3.19 % SZS output start Proof
% 2.97/3.19 tff(product_type, type, (
% 2.97/3.19 product: ( $i * $i * $i ) > $o)).
% 2.97/3.19 tff(additive_identity_type, type, (
% 2.97/3.19 additive_identity: $i)).
% 2.97/3.19 tff(add_type, type, (
% 2.97/3.19 add: ( $i * $i ) > $i)).
% 2.97/3.19 tff(x_type, type, (
% 2.97/3.19 x: $i)).
% 2.97/3.19 tff(inverse_type, type, (
% 2.97/3.19 inverse: $i > $i)).
% 2.97/3.19 tff(sum_type, type, (
% 2.97/3.19 sum: ( $i * $i * $i ) > $o)).
% 2.97/3.19 tff(multiplicative_identity_type, type, (
% 2.97/3.19 multiplicative_identity: $i)).
% 2.97/3.19 tff(1,plain,
% 2.97/3.19 (^[Y: $i, X: $i] : refl(sum(X, Y, add(X, Y)) <=> sum(X, Y, add(X, Y)))),
% 2.97/3.19 inference(bind,[status(th)],[])).
% 2.97/3.19 tff(2,plain,
% 2.97/3.19 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 2.97/3.19 inference(quant_intro,[status(thm)],[1])).
% 2.97/3.19 tff(3,plain,
% 2.97/3.19 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(4,axiom,(![Y: $i, X: $i] : sum(X, Y, add(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','closure_of_addition')).
% 2.97/3.19 tff(5,plain,
% 2.97/3.19 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[4, 3])).
% 2.97/3.19 tff(6,plain,(
% 2.97/3.19 ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 2.97/3.19 inference(skolemize,[status(sab)],[5])).
% 2.97/3.19 tff(7,plain,
% 2.97/3.19 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[6, 2])).
% 2.97/3.19 tff(8,plain,
% 2.97/3.19 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))),
% 2.97/3.19 inference(quant_inst,[status(thm)],[])).
% 2.97/3.19 tff(9,plain,
% 2.97/3.19 (sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))),
% 2.97/3.19 inference(unit_resolution,[status(thm)],[8, 7])).
% 2.97/3.19 tff(10,assumption,(~product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x)))), introduced(assumption)).
% 2.97/3.19 tff(11,plain,
% 2.97/3.19 (^[Z: $i, Y: $i, X: $i] : refl(((~sum(X, Y, Z)) | sum(Y, X, Z)) <=> ((~sum(X, Y, Z)) | sum(Y, X, Z)))),
% 2.97/3.19 inference(bind,[status(th)],[])).
% 2.97/3.19 tff(12,plain,
% 2.97/3.19 (![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))),
% 2.97/3.19 inference(quant_intro,[status(thm)],[11])).
% 2.97/3.19 tff(13,plain,
% 2.97/3.19 (![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))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(14,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')).
% 2.97/3.19 tff(15,plain,
% 2.97/3.19 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[14, 13])).
% 2.97/3.19 tff(16,plain,(
% 2.97/3.19 ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 2.97/3.19 inference(skolemize,[status(sab)],[15])).
% 2.97/3.19 tff(17,plain,
% 2.97/3.19 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[16, 12])).
% 2.97/3.19 tff(18,plain,
% 2.97/3.19 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x)))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(19,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x)))),
% 2.97/3.19 inference(quant_inst,[status(thm)],[])).
% 2.97/3.19 tff(20,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[19, 18])).
% 2.97/3.19 tff(21,plain,
% 2.97/3.19 (sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))),
% 2.97/3.19 inference(unit_resolution,[status(thm)],[20, 17, 9])).
% 2.97/3.19 tff(22,plain,
% 2.97/3.19 (^[X: $i] : refl(product(inverse(X), X, additive_identity) <=> product(inverse(X), X, additive_identity))),
% 2.97/3.19 inference(bind,[status(th)],[])).
% 2.97/3.19 tff(23,plain,
% 2.97/3.19 (![X: $i] : product(inverse(X), X, additive_identity) <=> ![X: $i] : product(inverse(X), X, additive_identity)),
% 2.97/3.19 inference(quant_intro,[status(thm)],[22])).
% 2.97/3.19 tff(24,plain,
% 2.97/3.19 (![X: $i] : product(inverse(X), X, additive_identity) <=> ![X: $i] : product(inverse(X), X, additive_identity)),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(25,axiom,(![X: $i] : product(inverse(X), X, additive_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_inverse1')).
% 2.97/3.19 tff(26,plain,
% 2.97/3.19 (![X: $i] : product(inverse(X), X, additive_identity)),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[25, 24])).
% 2.97/3.19 tff(27,plain,(
% 2.97/3.19 ![X: $i] : product(inverse(X), X, additive_identity)),
% 2.97/3.19 inference(skolemize,[status(sab)],[26])).
% 2.97/3.19 tff(28,plain,
% 2.97/3.19 (![X: $i] : product(inverse(X), X, additive_identity)),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[27, 23])).
% 2.97/3.19 tff(29,plain,
% 2.97/3.19 ((~![X: $i] : product(inverse(X), X, additive_identity)) | product(inverse(inverse(x)), inverse(x), additive_identity)),
% 2.97/3.19 inference(quant_inst,[status(thm)],[])).
% 2.97/3.19 tff(30,plain,
% 2.97/3.19 (product(inverse(inverse(x)), inverse(x), additive_identity)),
% 2.97/3.19 inference(unit_resolution,[status(thm)],[29, 28])).
% 2.97/3.19 tff(31,plain,
% 2.97/3.19 (^[Z: $i, Y: $i, X: $i] : refl(((~product(X, Y, Z)) | sum(X, Z, X)) <=> ((~product(X, Y, Z)) | sum(X, Z, X)))),
% 2.97/3.19 inference(bind,[status(th)],[])).
% 2.97/3.19 tff(32,plain,
% 2.97/3.19 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X)) <=> ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))),
% 2.97/3.19 inference(quant_intro,[status(thm)],[31])).
% 2.97/3.19 tff(33,plain,
% 2.97/3.19 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X)) <=> ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(34,axiom,(![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','sum_product_dual1')).
% 2.97/3.19 tff(35,plain,
% 2.97/3.19 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[34, 33])).
% 2.97/3.19 tff(36,plain,(
% 2.97/3.19 ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))),
% 2.97/3.19 inference(skolemize,[status(sab)],[35])).
% 2.97/3.19 tff(37,plain,
% 2.97/3.19 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[36, 32])).
% 2.97/3.19 tff(38,plain,
% 2.97/3.19 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | ((~product(inverse(inverse(x)), inverse(x), additive_identity)) | sum(inverse(inverse(x)), additive_identity, inverse(inverse(x))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | (~product(inverse(inverse(x)), inverse(x), additive_identity)) | sum(inverse(inverse(x)), additive_identity, inverse(inverse(x))))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(39,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | ((~product(inverse(inverse(x)), inverse(x), additive_identity)) | sum(inverse(inverse(x)), additive_identity, inverse(inverse(x))))),
% 2.97/3.19 inference(quant_inst,[status(thm)],[])).
% 2.97/3.19 tff(40,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | (~product(inverse(inverse(x)), inverse(x), additive_identity)) | sum(inverse(inverse(x)), additive_identity, inverse(inverse(x)))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[39, 38])).
% 2.97/3.19 tff(41,plain,
% 2.97/3.19 (sum(inverse(inverse(x)), additive_identity, inverse(inverse(x)))),
% 2.97/3.19 inference(unit_resolution,[status(thm)],[40, 37, 30])).
% 2.97/3.19 tff(42,plain,
% 2.97/3.19 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(inverse(inverse(x)), additive_identity, inverse(inverse(x)))) | sum(additive_identity, inverse(inverse(x)), inverse(inverse(x))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(inverse(inverse(x)), additive_identity, inverse(inverse(x)))) | sum(additive_identity, inverse(inverse(x)), inverse(inverse(x))))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(43,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(inverse(inverse(x)), additive_identity, inverse(inverse(x)))) | sum(additive_identity, inverse(inverse(x)), inverse(inverse(x))))),
% 2.97/3.19 inference(quant_inst,[status(thm)],[])).
% 2.97/3.19 tff(44,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(inverse(inverse(x)), additive_identity, inverse(inverse(x)))) | sum(additive_identity, inverse(inverse(x)), inverse(inverse(x)))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[43, 42])).
% 2.97/3.19 tff(45,plain,
% 2.97/3.19 (sum(additive_identity, inverse(inverse(x)), inverse(inverse(x)))),
% 2.97/3.19 inference(unit_resolution,[status(thm)],[44, 17, 41])).
% 2.97/3.19 tff(46,plain,
% 2.97/3.19 (^[X: $i] : refl(sum(X, inverse(X), multiplicative_identity) <=> sum(X, inverse(X), multiplicative_identity))),
% 2.97/3.19 inference(bind,[status(th)],[])).
% 2.97/3.19 tff(47,plain,
% 2.97/3.19 (![X: $i] : sum(X, inverse(X), multiplicative_identity) <=> ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 2.97/3.19 inference(quant_intro,[status(thm)],[46])).
% 2.97/3.19 tff(48,plain,
% 2.97/3.19 (![X: $i] : sum(X, inverse(X), multiplicative_identity) <=> ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(49,axiom,(![X: $i] : sum(X, inverse(X), multiplicative_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_inverse2')).
% 2.97/3.19 tff(50,plain,
% 2.97/3.19 (![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[49, 48])).
% 2.97/3.19 tff(51,plain,(
% 2.97/3.19 ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 2.97/3.19 inference(skolemize,[status(sab)],[50])).
% 2.97/3.19 tff(52,plain,
% 2.97/3.19 (![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[51, 47])).
% 2.97/3.19 tff(53,plain,
% 2.97/3.19 ((~![X: $i] : sum(X, inverse(X), multiplicative_identity)) | sum(inverse(x), inverse(inverse(x)), multiplicative_identity)),
% 2.97/3.19 inference(quant_inst,[status(thm)],[])).
% 2.97/3.19 tff(54,plain,
% 2.97/3.19 (sum(inverse(x), inverse(inverse(x)), multiplicative_identity)),
% 2.97/3.19 inference(unit_resolution,[status(thm)],[53, 52])).
% 2.97/3.19 tff(55,plain,
% 2.97/3.19 (^[X: $i] : refl(product(X, inverse(X), additive_identity) <=> product(X, inverse(X), additive_identity))),
% 2.97/3.19 inference(bind,[status(th)],[])).
% 2.97/3.19 tff(56,plain,
% 2.97/3.19 (![X: $i] : product(X, inverse(X), additive_identity) <=> ![X: $i] : product(X, inverse(X), additive_identity)),
% 2.97/3.19 inference(quant_intro,[status(thm)],[55])).
% 2.97/3.19 tff(57,plain,
% 2.97/3.19 (![X: $i] : product(X, inverse(X), additive_identity) <=> ![X: $i] : product(X, inverse(X), additive_identity)),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(58,axiom,(![X: $i] : product(X, inverse(X), additive_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_inverse2')).
% 2.97/3.19 tff(59,plain,
% 2.97/3.19 (![X: $i] : product(X, inverse(X), additive_identity)),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[58, 57])).
% 2.97/3.19 tff(60,plain,(
% 2.97/3.19 ![X: $i] : product(X, inverse(X), additive_identity)),
% 2.97/3.19 inference(skolemize,[status(sab)],[59])).
% 2.97/3.19 tff(61,plain,
% 2.97/3.19 (![X: $i] : product(X, inverse(X), additive_identity)),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[60, 56])).
% 2.97/3.19 tff(62,plain,
% 2.97/3.19 ((~![X: $i] : product(X, inverse(X), additive_identity)) | product(x, inverse(x), additive_identity)),
% 2.97/3.19 inference(quant_inst,[status(thm)],[])).
% 2.97/3.19 tff(63,plain,
% 2.97/3.19 (product(x, inverse(x), additive_identity)),
% 2.97/3.19 inference(unit_resolution,[status(thm)],[62, 61])).
% 2.97/3.19 tff(64,plain,
% 2.97/3.19 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))))),
% 2.97/3.19 inference(bind,[status(th)],[])).
% 2.97/3.19 tff(65,plain,
% 2.97/3.19 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 2.97/3.19 inference(quant_intro,[status(thm)],[64])).
% 2.97/3.19 tff(66,plain,
% 2.97/3.19 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(67,plain,
% 2.97/3.19 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) <=> ((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) <=> (((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | (~sum(V3, X, V4))))), rewrite((((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | (~sum(V3, X, V4))) <=> ((~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) <=> ((~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))))), ((((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) | product(V1, V2, V4)) <=> (((~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | product(V1, V2, V4)))), rewrite((((~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | product(V1, V2, V4)) <=> (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), ((((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) | product(V1, V2, V4)) <=> (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))))),
% 2.97/3.19 inference(bind,[status(th)],[])).
% 2.97/3.19 tff(68,plain,
% 2.97/3.19 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) | product(V1, V2, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 2.97/3.19 inference(quant_intro,[status(thm)],[67])).
% 2.97/3.19 tff(69,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) | product(V1, V2, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity7')).
% 2.97/3.19 tff(70,plain,
% 2.97/3.19 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[69, 68])).
% 2.97/3.19 tff(71,plain,
% 2.97/3.19 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[70, 66])).
% 2.97/3.19 tff(72,plain,(
% 2.97/3.19 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 2.97/3.19 inference(skolemize,[status(sab)],[71])).
% 2.97/3.19 tff(73,plain,
% 2.97/3.19 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[72, 65])).
% 2.97/3.19 tff(74,plain,
% 2.97/3.19 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(x, inverse(x), additive_identity)) | (~sum(additive_identity, inverse(inverse(x)), inverse(inverse(x)))) | (~sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))) | (~sum(inverse(x), inverse(inverse(x)), multiplicative_identity)) | product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(x, inverse(x), additive_identity)) | (~sum(additive_identity, inverse(inverse(x)), inverse(inverse(x)))) | (~sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))) | (~sum(inverse(x), inverse(inverse(x)), multiplicative_identity)) | product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x))))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(75,plain,
% 2.97/3.19 ((product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x))) | (~product(x, inverse(x), additive_identity)) | (~sum(additive_identity, inverse(inverse(x)), inverse(inverse(x)))) | (~sum(inverse(x), inverse(inverse(x)), multiplicative_identity)) | (~sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x)))) <=> ((~product(x, inverse(x), additive_identity)) | (~sum(additive_identity, inverse(inverse(x)), inverse(inverse(x)))) | (~sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))) | (~sum(inverse(x), inverse(inverse(x)), multiplicative_identity)) | product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x))))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(76,plain,
% 2.97/3.19 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x))) | (~product(x, inverse(x), additive_identity)) | (~sum(additive_identity, inverse(inverse(x)), inverse(inverse(x)))) | (~sum(inverse(x), inverse(inverse(x)), multiplicative_identity)) | (~sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(x, inverse(x), additive_identity)) | (~sum(additive_identity, inverse(inverse(x)), inverse(inverse(x)))) | (~sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))) | (~sum(inverse(x), inverse(inverse(x)), multiplicative_identity)) | product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x)))))),
% 2.97/3.19 inference(monotonicity,[status(thm)],[75])).
% 2.97/3.19 tff(77,plain,
% 2.97/3.19 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x))) | (~product(x, inverse(x), additive_identity)) | (~sum(additive_identity, inverse(inverse(x)), inverse(inverse(x)))) | (~sum(inverse(x), inverse(inverse(x)), multiplicative_identity)) | (~sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(x, inverse(x), additive_identity)) | (~sum(additive_identity, inverse(inverse(x)), inverse(inverse(x)))) | (~sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))) | (~sum(inverse(x), inverse(inverse(x)), multiplicative_identity)) | product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x))))),
% 2.97/3.19 inference(transitivity,[status(thm)],[76, 74])).
% 2.97/3.19 tff(78,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x))) | (~product(x, inverse(x), additive_identity)) | (~sum(additive_identity, inverse(inverse(x)), inverse(inverse(x)))) | (~sum(inverse(x), inverse(inverse(x)), multiplicative_identity)) | (~sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))))),
% 2.97/3.19 inference(quant_inst,[status(thm)],[])).
% 2.97/3.19 tff(79,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(x, inverse(x), additive_identity)) | (~sum(additive_identity, inverse(inverse(x)), inverse(inverse(x)))) | (~sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))) | (~sum(inverse(x), inverse(inverse(x)), multiplicative_identity)) | product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x)))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[78, 77])).
% 2.97/3.19 tff(80,plain,
% 2.97/3.19 ($false),
% 2.97/3.19 inference(unit_resolution,[status(thm)],[79, 73, 63, 54, 45, 21, 10])).
% 2.97/3.19 tff(81,plain,(product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x)))), inference(lemma,lemma(discharge,[]))).
% 2.97/3.19 tff(82,assumption,(~sum(inverse(inverse(x)), x, inverse(inverse(x)))), introduced(assumption)).
% 2.97/3.19 tff(83,plain,
% 2.97/3.19 ((~![X: $i] : sum(X, inverse(X), multiplicative_identity)) | sum(x, inverse(x), multiplicative_identity)),
% 2.97/3.19 inference(quant_inst,[status(thm)],[])).
% 2.97/3.19 tff(84,plain,
% 2.97/3.19 (sum(x, inverse(x), multiplicative_identity)),
% 2.97/3.19 inference(unit_resolution,[status(thm)],[83, 52])).
% 2.97/3.19 tff(85,plain,
% 2.97/3.19 (^[Z: $i, Y: $i, X: $i] : refl(((~sum(X, Y, Z)) | product(X, Z, X)) <=> ((~sum(X, Y, Z)) | product(X, Z, X)))),
% 2.97/3.19 inference(bind,[status(th)],[])).
% 2.97/3.19 tff(86,plain,
% 2.97/3.19 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X)) <=> ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))),
% 2.97/3.19 inference(quant_intro,[status(thm)],[85])).
% 2.97/3.19 tff(87,plain,
% 2.97/3.19 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X)) <=> ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(88,axiom,(![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','sum_product_dual2')).
% 2.97/3.19 tff(89,plain,
% 2.97/3.19 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[88, 87])).
% 2.97/3.19 tff(90,plain,(
% 2.97/3.19 ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))),
% 2.97/3.19 inference(skolemize,[status(sab)],[89])).
% 2.97/3.19 tff(91,plain,
% 2.97/3.19 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[90, 86])).
% 2.97/3.19 tff(92,plain,
% 2.97/3.19 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))) | ((~sum(x, inverse(x), multiplicative_identity)) | product(x, multiplicative_identity, x))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))) | (~sum(x, inverse(x), multiplicative_identity)) | product(x, multiplicative_identity, x))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(93,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))) | ((~sum(x, inverse(x), multiplicative_identity)) | product(x, multiplicative_identity, x))),
% 2.97/3.19 inference(quant_inst,[status(thm)],[])).
% 2.97/3.19 tff(94,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))) | (~sum(x, inverse(x), multiplicative_identity)) | product(x, multiplicative_identity, x)),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[93, 92])).
% 2.97/3.19 tff(95,plain,
% 2.97/3.19 (product(x, multiplicative_identity, x)),
% 2.97/3.19 inference(unit_resolution,[status(thm)],[94, 91, 84])).
% 2.97/3.19 tff(96,plain,
% 2.97/3.19 (^[Z: $i, Y: $i, X: $i] : refl(((~product(X, Y, Z)) | product(Y, X, Z)) <=> ((~product(X, Y, Z)) | product(Y, X, Z)))),
% 2.97/3.19 inference(bind,[status(th)],[])).
% 2.97/3.19 tff(97,plain,
% 2.97/3.19 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z)) <=> ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 2.97/3.19 inference(quant_intro,[status(thm)],[96])).
% 2.97/3.19 tff(98,plain,
% 2.97/3.19 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z)) <=> ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(99,axiom,(![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','commutativity_of_multiplication')).
% 2.97/3.19 tff(100,plain,
% 2.97/3.19 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[99, 98])).
% 2.97/3.19 tff(101,plain,(
% 2.97/3.19 ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 2.97/3.19 inference(skolemize,[status(sab)],[100])).
% 2.97/3.19 tff(102,plain,
% 2.97/3.19 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[101, 97])).
% 2.97/3.19 tff(103,plain,
% 2.97/3.19 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | ((~product(x, multiplicative_identity, x)) | product(multiplicative_identity, x, x))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | (~product(x, multiplicative_identity, x)) | product(multiplicative_identity, x, x))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(104,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | ((~product(x, multiplicative_identity, x)) | product(multiplicative_identity, x, x))),
% 2.97/3.19 inference(quant_inst,[status(thm)],[])).
% 2.97/3.19 tff(105,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | (~product(x, multiplicative_identity, x)) | product(multiplicative_identity, x, x)),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[104, 103])).
% 2.97/3.19 tff(106,plain,
% 2.97/3.19 (product(multiplicative_identity, x, x)),
% 2.97/3.19 inference(unit_resolution,[status(thm)],[105, 102, 95])).
% 2.97/3.19 tff(107,plain,
% 2.97/3.19 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | ((~product(multiplicative_identity, x, x)) | sum(multiplicative_identity, x, multiplicative_identity))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | (~product(multiplicative_identity, x, x)) | sum(multiplicative_identity, x, multiplicative_identity))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(108,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | ((~product(multiplicative_identity, x, x)) | sum(multiplicative_identity, x, multiplicative_identity))),
% 2.97/3.19 inference(quant_inst,[status(thm)],[])).
% 2.97/3.19 tff(109,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | (~product(multiplicative_identity, x, x)) | sum(multiplicative_identity, x, multiplicative_identity)),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[108, 107])).
% 2.97/3.19 tff(110,plain,
% 2.97/3.19 (sum(multiplicative_identity, x, multiplicative_identity)),
% 2.97/3.19 inference(unit_resolution,[status(thm)],[109, 37, 106])).
% 2.97/3.19 tff(111,plain,
% 2.97/3.19 (^[X: $i] : refl(sum(inverse(X), X, multiplicative_identity) <=> sum(inverse(X), X, multiplicative_identity))),
% 2.97/3.19 inference(bind,[status(th)],[])).
% 2.97/3.19 tff(112,plain,
% 2.97/3.19 (![X: $i] : sum(inverse(X), X, multiplicative_identity) <=> ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 2.97/3.19 inference(quant_intro,[status(thm)],[111])).
% 2.97/3.19 tff(113,plain,
% 2.97/3.19 (![X: $i] : sum(inverse(X), X, multiplicative_identity) <=> ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(114,axiom,(![X: $i] : sum(inverse(X), X, multiplicative_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_inverse1')).
% 2.97/3.19 tff(115,plain,
% 2.97/3.19 (![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[114, 113])).
% 2.97/3.19 tff(116,plain,(
% 2.97/3.19 ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 2.97/3.19 inference(skolemize,[status(sab)],[115])).
% 2.97/3.19 tff(117,plain,
% 2.97/3.19 (![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 2.97/3.19 inference(modus_ponens,[status(thm)],[116, 112])).
% 2.97/3.19 tff(118,plain,
% 2.97/3.19 ((~![X: $i] : sum(inverse(X), X, multiplicative_identity)) | sum(inverse(inverse(x)), inverse(x), multiplicative_identity)),
% 2.97/3.19 inference(quant_inst,[status(thm)],[])).
% 2.97/3.19 tff(119,plain,
% 2.97/3.19 (sum(inverse(inverse(x)), inverse(x), multiplicative_identity)),
% 2.97/3.19 inference(unit_resolution,[status(thm)],[118, 117])).
% 2.97/3.19 tff(120,plain,
% 2.97/3.19 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))) | ((~sum(inverse(inverse(x)), inverse(x), multiplicative_identity)) | product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))) | (~sum(inverse(inverse(x)), inverse(x), multiplicative_identity)) | product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x))))),
% 2.97/3.19 inference(rewrite,[status(thm)],[])).
% 2.97/3.19 tff(121,plain,
% 2.97/3.19 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))) | ((~sum(inverse(inverse(x)), inverse(x), multiplicative_identity)) | product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x))))),
% 3.04/3.20 inference(quant_inst,[status(thm)],[])).
% 3.04/3.20 tff(122,plain,
% 3.04/3.20 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))) | (~sum(inverse(inverse(x)), inverse(x), multiplicative_identity)) | product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[121, 120])).
% 3.04/3.20 tff(123,plain,
% 3.04/3.20 (product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))),
% 3.04/3.20 inference(unit_resolution,[status(thm)],[122, 91, 119])).
% 3.04/3.20 tff(124,plain,
% 3.04/3.20 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl(((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))))),
% 3.04/3.20 inference(bind,[status(th)],[])).
% 3.04/3.20 tff(125,plain,
% 3.04/3.20 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 3.04/3.20 inference(quant_intro,[status(thm)],[124])).
% 3.04/3.20 tff(126,plain,
% 3.04/3.20 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 3.04/3.20 inference(rewrite,[status(thm)],[])).
% 3.04/3.20 tff(127,plain,
% 3.04/3.20 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) <=> ((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) <=> (((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | (~product(V1, V2, V4))))), rewrite((((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | (~product(V1, V2, V4))) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))))), ((((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(V3, X, V4)) <=> (((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | sum(V3, X, V4)))), rewrite((((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | sum(V3, X, V4)) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), ((((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(V3, X, V4)) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))))),
% 3.04/3.20 inference(bind,[status(th)],[])).
% 3.04/3.20 tff(128,plain,
% 3.04/3.20 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(V3, X, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 3.04/3.20 inference(quant_intro,[status(thm)],[127])).
% 3.04/3.20 tff(129,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(V3, X, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity8')).
% 3.04/3.20 tff(130,plain,
% 3.04/3.20 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[129, 128])).
% 3.04/3.20 tff(131,plain,
% 3.04/3.20 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[130, 126])).
% 3.04/3.20 tff(132,plain,(
% 3.04/3.20 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 3.04/3.20 inference(skolemize,[status(sab)],[131])).
% 3.04/3.20 tff(133,plain,
% 3.04/3.20 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[132, 125])).
% 3.04/3.20 tff(134,plain,
% 3.04/3.20 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))) | (~sum(multiplicative_identity, x, multiplicative_identity)) | sum(inverse(inverse(x)), x, inverse(inverse(x))) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))) | (~sum(multiplicative_identity, x, multiplicative_identity)) | sum(inverse(inverse(x)), x, inverse(inverse(x))) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x)))))),
% 3.04/3.20 inference(rewrite,[status(thm)],[])).
% 3.04/3.20 tff(135,plain,
% 3.04/3.20 (((~product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))) | (~product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x)))) | sum(inverse(inverse(x)), x, inverse(inverse(x))) | (~sum(multiplicative_identity, x, multiplicative_identity)) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x)))) <=> ((~product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))) | (~sum(multiplicative_identity, x, multiplicative_identity)) | sum(inverse(inverse(x)), x, inverse(inverse(x))) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x)))))),
% 3.04/3.20 inference(rewrite,[status(thm)],[])).
% 3.04/3.20 tff(136,plain,
% 3.04/3.20 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))) | (~product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x)))) | sum(inverse(inverse(x)), x, inverse(inverse(x))) | (~sum(multiplicative_identity, x, multiplicative_identity)) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))) | (~sum(multiplicative_identity, x, multiplicative_identity)) | sum(inverse(inverse(x)), x, inverse(inverse(x))) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x))))))),
% 3.04/3.20 inference(monotonicity,[status(thm)],[135])).
% 3.04/3.20 tff(137,plain,
% 3.04/3.20 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))) | (~product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x)))) | sum(inverse(inverse(x)), x, inverse(inverse(x))) | (~sum(multiplicative_identity, x, multiplicative_identity)) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))) | (~sum(multiplicative_identity, x, multiplicative_identity)) | sum(inverse(inverse(x)), x, inverse(inverse(x))) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x)))))),
% 3.04/3.20 inference(transitivity,[status(thm)],[136, 134])).
% 3.04/3.20 tff(138,plain,
% 3.04/3.20 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))) | (~product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x)))) | sum(inverse(inverse(x)), x, inverse(inverse(x))) | (~sum(multiplicative_identity, x, multiplicative_identity)) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))))),
% 3.04/3.20 inference(quant_inst,[status(thm)],[])).
% 3.04/3.20 tff(139,plain,
% 3.04/3.20 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))) | (~sum(multiplicative_identity, x, multiplicative_identity)) | sum(inverse(inverse(x)), x, inverse(inverse(x))) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~product(add(inverse(inverse(x)), x), multiplicative_identity, inverse(inverse(x))))),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[138, 137])).
% 3.04/3.20 tff(140,plain,
% 3.04/3.20 ($false),
% 3.04/3.20 inference(unit_resolution,[status(thm)],[139, 133, 123, 110, 82, 9, 81])).
% 3.04/3.20 tff(141,plain,(sum(inverse(inverse(x)), x, inverse(inverse(x)))), inference(lemma,lemma(discharge,[]))).
% 3.04/3.20 tff(142,plain,
% 3.04/3.20 (^[V: $i, Y: $i, U: $i, X: $i] : refl(((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))) <=> ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))))),
% 3.04/3.20 inference(bind,[status(th)],[])).
% 3.04/3.20 tff(143,plain,
% 3.04/3.20 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 3.04/3.20 inference(quant_intro,[status(thm)],[142])).
% 3.04/3.20 tff(144,plain,
% 3.04/3.20 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 3.04/3.20 inference(rewrite,[status(thm)],[])).
% 3.04/3.20 tff(145,plain,
% 3.04/3.20 (^[V: $i, Y: $i, U: $i, X: $i] : rewrite((((~sum(X, Y, U)) | (~sum(X, Y, V))) | (U = V)) <=> ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))))),
% 3.04/3.20 inference(bind,[status(th)],[])).
% 3.04/3.20 tff(146,plain,
% 3.04/3.20 (![V: $i, Y: $i, U: $i, X: $i] : (((~sum(X, Y, U)) | (~sum(X, Y, V))) | (U = V)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 3.04/3.20 inference(quant_intro,[status(thm)],[145])).
% 3.04/3.20 tff(147,axiom,(![V: $i, Y: $i, U: $i, X: $i] : (((~sum(X, Y, U)) | (~sum(X, Y, V))) | (U = V))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','addition_is_well_defined')).
% 3.04/3.20 tff(148,plain,
% 3.04/3.20 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[147, 146])).
% 3.04/3.20 tff(149,plain,
% 3.04/3.20 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[148, 144])).
% 3.04/3.20 tff(150,plain,(
% 3.04/3.20 ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 3.04/3.20 inference(skolemize,[status(sab)],[149])).
% 3.04/3.20 tff(151,plain,
% 3.04/3.20 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[150, 143])).
% 3.04/3.20 tff(152,plain,
% 3.04/3.20 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((~sum(inverse(inverse(x)), x, inverse(inverse(x)))) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (inverse(inverse(x)) = add(inverse(inverse(x)), x)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(inverse(inverse(x)), x, inverse(inverse(x)))) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (inverse(inverse(x)) = add(inverse(inverse(x)), x)))),
% 3.04/3.20 inference(rewrite,[status(thm)],[])).
% 3.04/3.20 tff(153,plain,
% 3.04/3.20 (((inverse(inverse(x)) = add(inverse(inverse(x)), x)) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~sum(inverse(inverse(x)), x, inverse(inverse(x))))) <=> ((~sum(inverse(inverse(x)), x, inverse(inverse(x)))) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (inverse(inverse(x)) = add(inverse(inverse(x)), x)))),
% 3.04/3.20 inference(rewrite,[status(thm)],[])).
% 3.04/3.20 tff(154,plain,
% 3.04/3.20 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((inverse(inverse(x)) = add(inverse(inverse(x)), x)) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~sum(inverse(inverse(x)), x, inverse(inverse(x)))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((~sum(inverse(inverse(x)), x, inverse(inverse(x)))) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (inverse(inverse(x)) = add(inverse(inverse(x)), x))))),
% 3.04/3.20 inference(monotonicity,[status(thm)],[153])).
% 3.04/3.20 tff(155,plain,
% 3.04/3.20 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((inverse(inverse(x)) = add(inverse(inverse(x)), x)) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~sum(inverse(inverse(x)), x, inverse(inverse(x)))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(inverse(inverse(x)), x, inverse(inverse(x)))) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (inverse(inverse(x)) = add(inverse(inverse(x)), x)))),
% 3.04/3.20 inference(transitivity,[status(thm)],[154, 152])).
% 3.04/3.20 tff(156,plain,
% 3.04/3.20 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((inverse(inverse(x)) = add(inverse(inverse(x)), x)) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~sum(inverse(inverse(x)), x, inverse(inverse(x)))))),
% 3.04/3.20 inference(quant_inst,[status(thm)],[])).
% 3.04/3.20 tff(157,plain,
% 3.04/3.20 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(inverse(inverse(x)), x, inverse(inverse(x)))) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (inverse(inverse(x)) = add(inverse(inverse(x)), x))),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[156, 155])).
% 3.04/3.20 tff(158,plain,
% 3.04/3.20 (inverse(inverse(x)) = add(inverse(inverse(x)), x)),
% 3.04/3.20 inference(unit_resolution,[status(thm)],[157, 151, 141, 9])).
% 3.04/3.20 tff(159,plain,
% 3.04/3.20 (add(inverse(inverse(x)), x) = inverse(inverse(x))),
% 3.04/3.20 inference(symmetry,[status(thm)],[158])).
% 3.04/3.20 tff(160,plain,
% 3.04/3.20 (product(inverse(x), add(inverse(inverse(x)), x), additive_identity) <=> product(inverse(x), inverse(inverse(x)), additive_identity)),
% 3.04/3.20 inference(monotonicity,[status(thm)],[159])).
% 3.04/3.20 tff(161,plain,
% 3.04/3.20 (product(inverse(x), inverse(inverse(x)), additive_identity) <=> product(inverse(x), add(inverse(inverse(x)), x), additive_identity)),
% 3.04/3.20 inference(symmetry,[status(thm)],[160])).
% 3.04/3.20 tff(162,plain,
% 3.04/3.20 ((~![X: $i] : product(X, inverse(X), additive_identity)) | product(inverse(x), inverse(inverse(x)), additive_identity)),
% 3.04/3.20 inference(quant_inst,[status(thm)],[])).
% 3.04/3.20 tff(163,plain,
% 3.04/3.20 (product(inverse(x), inverse(inverse(x)), additive_identity)),
% 3.04/3.20 inference(unit_resolution,[status(thm)],[162, 61])).
% 3.04/3.20 tff(164,plain,
% 3.04/3.20 (product(inverse(x), add(inverse(inverse(x)), x), additive_identity)),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[163, 161])).
% 3.04/3.20 tff(165,plain,
% 3.04/3.20 ((~(inverse(inverse(x)) = x)) <=> (~(inverse(inverse(x)) = x))),
% 3.04/3.20 inference(rewrite,[status(thm)],[])).
% 3.04/3.20 tff(166,axiom,(~(inverse(inverse(x)) = x)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_inverse_is_an_involution')).
% 3.04/3.20 tff(167,plain,
% 3.04/3.20 (~(inverse(inverse(x)) = x)),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[166, 165])).
% 3.04/3.20 tff(168,plain,
% 3.04/3.20 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((inverse(inverse(x)) = x) | (~sum(inverse(inverse(x)), x, x)) | (~sum(inverse(inverse(x)), x, inverse(inverse(x)))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (inverse(inverse(x)) = x) | (~sum(inverse(inverse(x)), x, x)) | (~sum(inverse(inverse(x)), x, inverse(inverse(x)))))),
% 3.04/3.20 inference(rewrite,[status(thm)],[])).
% 3.04/3.20 tff(169,plain,
% 3.04/3.20 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((inverse(inverse(x)) = x) | (~sum(inverse(inverse(x)), x, x)) | (~sum(inverse(inverse(x)), x, inverse(inverse(x)))))),
% 3.04/3.20 inference(quant_inst,[status(thm)],[])).
% 3.04/3.20 tff(170,plain,
% 3.04/3.20 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (inverse(inverse(x)) = x) | (~sum(inverse(inverse(x)), x, x)) | (~sum(inverse(inverse(x)), x, inverse(inverse(x))))),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[169, 168])).
% 3.04/3.20 tff(171,plain,
% 3.04/3.20 ((~sum(inverse(inverse(x)), x, x)) | (~sum(inverse(inverse(x)), x, inverse(inverse(x))))),
% 3.04/3.20 inference(unit_resolution,[status(thm)],[170, 151, 167])).
% 3.04/3.20 tff(172,plain,
% 3.04/3.20 (~sum(inverse(inverse(x)), x, x)),
% 3.04/3.20 inference(unit_resolution,[status(thm)],[171, 141])).
% 3.04/3.20 tff(173,plain,
% 3.04/3.20 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | ((~product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))) | product(multiplicative_identity, inverse(inverse(x)), inverse(inverse(x))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | (~product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))) | product(multiplicative_identity, inverse(inverse(x)), inverse(inverse(x))))),
% 3.04/3.20 inference(rewrite,[status(thm)],[])).
% 3.04/3.20 tff(174,plain,
% 3.04/3.20 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | ((~product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))) | product(multiplicative_identity, inverse(inverse(x)), inverse(inverse(x))))),
% 3.04/3.20 inference(quant_inst,[status(thm)],[])).
% 3.04/3.20 tff(175,plain,
% 3.04/3.20 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | (~product(inverse(inverse(x)), multiplicative_identity, inverse(inverse(x)))) | product(multiplicative_identity, inverse(inverse(x)), inverse(inverse(x)))),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[174, 173])).
% 3.04/3.20 tff(176,plain,
% 3.04/3.20 (product(multiplicative_identity, inverse(inverse(x)), inverse(inverse(x)))),
% 3.04/3.20 inference(unit_resolution,[status(thm)],[175, 102, 123])).
% 3.04/3.20 tff(177,plain,
% 3.04/3.20 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | ((~product(multiplicative_identity, inverse(inverse(x)), inverse(inverse(x)))) | sum(multiplicative_identity, inverse(inverse(x)), multiplicative_identity))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | (~product(multiplicative_identity, inverse(inverse(x)), inverse(inverse(x)))) | sum(multiplicative_identity, inverse(inverse(x)), multiplicative_identity))),
% 3.04/3.20 inference(rewrite,[status(thm)],[])).
% 3.04/3.20 tff(178,plain,
% 3.04/3.20 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | ((~product(multiplicative_identity, inverse(inverse(x)), inverse(inverse(x)))) | sum(multiplicative_identity, inverse(inverse(x)), multiplicative_identity))),
% 3.04/3.20 inference(quant_inst,[status(thm)],[])).
% 3.04/3.20 tff(179,plain,
% 3.04/3.20 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | (~product(multiplicative_identity, inverse(inverse(x)), inverse(inverse(x)))) | sum(multiplicative_identity, inverse(inverse(x)), multiplicative_identity)),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[178, 177])).
% 3.04/3.20 tff(180,plain,
% 3.04/3.20 (sum(multiplicative_identity, inverse(inverse(x)), multiplicative_identity)),
% 3.04/3.20 inference(unit_resolution,[status(thm)],[179, 37, 176])).
% 3.04/3.20 tff(181,plain,
% 3.04/3.20 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(multiplicative_identity, inverse(inverse(x)), multiplicative_identity)) | sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(multiplicative_identity, inverse(inverse(x)), multiplicative_identity)) | sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity))),
% 3.04/3.20 inference(rewrite,[status(thm)],[])).
% 3.04/3.20 tff(182,plain,
% 3.04/3.20 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(multiplicative_identity, inverse(inverse(x)), multiplicative_identity)) | sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity))),
% 3.04/3.20 inference(quant_inst,[status(thm)],[])).
% 3.04/3.20 tff(183,plain,
% 3.04/3.20 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(multiplicative_identity, inverse(inverse(x)), multiplicative_identity)) | sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity)),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[182, 181])).
% 3.04/3.20 tff(184,plain,
% 3.04/3.20 (sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity)),
% 3.04/3.20 inference(unit_resolution,[status(thm)],[183, 17, 180])).
% 3.04/3.20 tff(185,plain,
% 3.04/3.20 (^[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))))),
% 3.04/3.20 inference(bind,[status(th)],[])).
% 3.04/3.20 tff(186,plain,
% 3.04/3.20 (![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)))),
% 3.04/3.20 inference(quant_intro,[status(thm)],[185])).
% 3.04/3.20 tff(187,plain,
% 3.04/3.20 (![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)))),
% 3.04/3.20 inference(rewrite,[status(thm)],[])).
% 3.04/3.20 tff(188,plain,
% 3.04/3.20 (^[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)))))),
% 3.04/3.20 inference(bind,[status(th)],[])).
% 3.04/3.20 tff(189,plain,
% 3.04/3.20 (![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)))),
% 3.04/3.20 inference(quant_intro,[status(thm)],[188])).
% 3.04/3.20 tff(190,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')).
% 3.04/3.20 tff(191,plain,
% 3.04/3.20 (![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)))),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[190, 189])).
% 3.04/3.20 tff(192,plain,
% 3.04/3.20 (![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)))),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[191, 187])).
% 3.04/3.20 tff(193,plain,(
% 3.04/3.20 ![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)))),
% 3.04/3.20 inference(skolemize,[status(sab)],[192])).
% 3.04/3.20 tff(194,plain,
% 3.04/3.20 (![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)))),
% 3.04/3.20 inference(modus_ponens,[status(thm)],[193, 186])).
% 3.04/3.20 tff(195,plain,
% 3.04/3.20 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | ((~product(multiplicative_identity, x, x)) | sum(inverse(inverse(x)), x, x) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity)) | (~product(multiplicative_identity, add(inverse(inverse(x)), x), x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (~product(multiplicative_identity, x, x)) | sum(inverse(inverse(x)), x, x) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity)) | (~product(multiplicative_identity, add(inverse(inverse(x)), x), x)))),
% 3.04/3.20 inference(rewrite,[status(thm)],[])).
% 3.04/3.20 tff(196,plain,
% 3.04/3.20 ((sum(inverse(inverse(x)), x, x) | (~product(multiplicative_identity, x, x)) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity)) | (~product(multiplicative_identity, add(inverse(inverse(x)), x), x))) <=> ((~product(multiplicative_identity, x, x)) | sum(inverse(inverse(x)), x, x) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity)) | (~product(multiplicative_identity, add(inverse(inverse(x)), x), x)))),
% 3.04/3.20 inference(rewrite,[status(thm)],[])).
% 3.04/3.20 tff(197,plain,
% 3.04/3.20 (((~![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(inverse(inverse(x)), x, x) | (~product(multiplicative_identity, x, x)) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity)) | (~product(multiplicative_identity, add(inverse(inverse(x)), x), x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | ((~product(multiplicative_identity, x, x)) | sum(inverse(inverse(x)), x, x) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity)) | (~product(multiplicative_identity, add(inverse(inverse(x)), x), x))))),
% 3.04/3.20 inference(monotonicity,[status(thm)],[196])).
% 3.04/3.20 tff(198,plain,
% 3.04/3.20 (((~![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(inverse(inverse(x)), x, x) | (~product(multiplicative_identity, x, x)) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity)) | (~product(multiplicative_identity, add(inverse(inverse(x)), x), x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (~product(multiplicative_identity, x, x)) | sum(inverse(inverse(x)), x, x) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity)) | (~product(multiplicative_identity, add(inverse(inverse(x)), x), x)))),
% 3.04/3.21 inference(transitivity,[status(thm)],[197, 195])).
% 3.04/3.21 tff(199,plain,
% 3.04/3.21 ((~![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(inverse(inverse(x)), x, x) | (~product(multiplicative_identity, x, x)) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity)) | (~product(multiplicative_identity, add(inverse(inverse(x)), x), x)))),
% 3.04/3.21 inference(quant_inst,[status(thm)],[])).
% 3.04/3.21 tff(200,plain,
% 3.04/3.21 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (~product(multiplicative_identity, x, x)) | sum(inverse(inverse(x)), x, x) | (~sum(inverse(inverse(x)), x, add(inverse(inverse(x)), x))) | (~sum(inverse(inverse(x)), multiplicative_identity, multiplicative_identity)) | (~product(multiplicative_identity, add(inverse(inverse(x)), x), x))),
% 3.04/3.21 inference(modus_ponens,[status(thm)],[199, 198])).
% 3.04/3.21 tff(201,plain,
% 3.04/3.21 (~product(multiplicative_identity, add(inverse(inverse(x)), x), x)),
% 3.04/3.21 inference(unit_resolution,[status(thm)],[200, 194, 106, 184, 172, 9])).
% 3.04/3.21 tff(202,plain,
% 3.04/3.21 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))) | ((~sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))) | product(x, add(inverse(inverse(x)), x), x))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))) | (~sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))) | product(x, add(inverse(inverse(x)), x), x))),
% 3.04/3.21 inference(rewrite,[status(thm)],[])).
% 3.04/3.21 tff(203,plain,
% 3.04/3.21 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))) | ((~sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))) | product(x, add(inverse(inverse(x)), x), x))),
% 3.04/3.21 inference(quant_inst,[status(thm)],[])).
% 3.04/3.21 tff(204,plain,
% 3.04/3.21 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | product(X, Z, X))) | (~sum(x, inverse(inverse(x)), add(inverse(inverse(x)), x))) | product(x, add(inverse(inverse(x)), x), x)),
% 3.04/3.21 inference(modus_ponens,[status(thm)],[203, 202])).
% 3.04/3.21 tff(205,plain,
% 3.04/3.21 (product(x, add(inverse(inverse(x)), x), x)),
% 3.04/3.21 inference(unit_resolution,[status(thm)],[204, 91, 21])).
% 3.04/3.21 tff(206,plain,
% 3.04/3.21 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | ((~product(x, add(inverse(inverse(x)), x), x)) | product(add(inverse(inverse(x)), x), x, x))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | (~product(x, add(inverse(inverse(x)), x), x)) | product(add(inverse(inverse(x)), x), x, x))),
% 3.04/3.21 inference(rewrite,[status(thm)],[])).
% 3.04/3.21 tff(207,plain,
% 3.04/3.21 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | ((~product(x, add(inverse(inverse(x)), x), x)) | product(add(inverse(inverse(x)), x), x, x))),
% 3.04/3.21 inference(quant_inst,[status(thm)],[])).
% 3.04/3.21 tff(208,plain,
% 3.04/3.21 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | (~product(x, add(inverse(inverse(x)), x), x)) | product(add(inverse(inverse(x)), x), x, x)),
% 3.04/3.21 inference(modus_ponens,[status(thm)],[207, 206])).
% 3.04/3.21 tff(209,plain,
% 3.04/3.21 (product(add(inverse(inverse(x)), x), x, x)),
% 3.04/3.21 inference(unit_resolution,[status(thm)],[208, 102, 205])).
% 3.04/3.21 tff(210,plain,
% 3.04/3.21 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | ((~product(add(inverse(inverse(x)), x), x, x)) | sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | (~product(add(inverse(inverse(x)), x), x, x)) | sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x)))),
% 3.04/3.21 inference(rewrite,[status(thm)],[])).
% 3.04/3.21 tff(211,plain,
% 3.04/3.21 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | ((~product(add(inverse(inverse(x)), x), x, x)) | sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x)))),
% 3.04/3.21 inference(quant_inst,[status(thm)],[])).
% 3.04/3.21 tff(212,plain,
% 3.04/3.21 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | (~product(add(inverse(inverse(x)), x), x, x)) | sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x))),
% 3.04/3.21 inference(modus_ponens,[status(thm)],[211, 210])).
% 3.04/3.21 tff(213,plain,
% 3.04/3.21 (sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x))),
% 3.04/3.21 inference(unit_resolution,[status(thm)],[212, 37, 209])).
% 3.04/3.21 tff(214,plain,
% 3.04/3.21 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | ((~product(x, inverse(x), additive_identity)) | sum(x, additive_identity, x))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | (~product(x, inverse(x), additive_identity)) | sum(x, additive_identity, x))),
% 3.04/3.21 inference(rewrite,[status(thm)],[])).
% 3.04/3.21 tff(215,plain,
% 3.04/3.21 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | ((~product(x, inverse(x), additive_identity)) | sum(x, additive_identity, x))),
% 3.04/3.21 inference(quant_inst,[status(thm)],[])).
% 3.04/3.21 tff(216,plain,
% 3.04/3.21 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | sum(X, Z, X))) | (~product(x, inverse(x), additive_identity)) | sum(x, additive_identity, x)),
% 3.04/3.21 inference(modus_ponens,[status(thm)],[215, 214])).
% 3.04/3.21 tff(217,plain,
% 3.04/3.21 (sum(x, additive_identity, x)),
% 3.04/3.21 inference(unit_resolution,[status(thm)],[216, 37, 63])).
% 3.04/3.21 tff(218,plain,
% 3.04/3.21 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(x, additive_identity, x)) | sum(additive_identity, x, x))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(x, additive_identity, x)) | sum(additive_identity, x, x))),
% 3.04/3.21 inference(rewrite,[status(thm)],[])).
% 3.04/3.21 tff(219,plain,
% 3.04/3.21 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(x, additive_identity, x)) | sum(additive_identity, x, x))),
% 3.04/3.21 inference(quant_inst,[status(thm)],[])).
% 3.04/3.21 tff(220,plain,
% 3.04/3.21 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(x, additive_identity, x)) | sum(additive_identity, x, x)),
% 3.04/3.21 inference(modus_ponens,[status(thm)],[219, 218])).
% 3.04/3.21 tff(221,plain,
% 3.04/3.21 (sum(additive_identity, x, x)),
% 3.04/3.21 inference(unit_resolution,[status(thm)],[220, 17, 217])).
% 3.04/3.21 tff(222,plain,
% 3.04/3.21 ((~![X: $i] : sum(inverse(X), X, multiplicative_identity)) | sum(inverse(x), x, multiplicative_identity)),
% 3.04/3.21 inference(quant_inst,[status(thm)],[])).
% 3.04/3.21 tff(223,plain,
% 3.04/3.21 (sum(inverse(x), x, multiplicative_identity)),
% 3.04/3.21 inference(unit_resolution,[status(thm)],[222, 117])).
% 3.04/3.21 tff(224,plain,
% 3.04/3.21 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, x, x)) | (~sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x))) | (~product(inverse(x), add(inverse(inverse(x)), x), additive_identity)) | product(multiplicative_identity, add(inverse(inverse(x)), x), x))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, x, x)) | (~sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x))) | (~product(inverse(x), add(inverse(inverse(x)), x), additive_identity)) | product(multiplicative_identity, add(inverse(inverse(x)), x), x))),
% 3.04/3.21 inference(rewrite,[status(thm)],[])).
% 3.04/3.21 tff(225,plain,
% 3.04/3.21 ((product(multiplicative_identity, add(inverse(inverse(x)), x), x) | (~product(inverse(x), add(inverse(inverse(x)), x), additive_identity)) | (~sum(additive_identity, x, x)) | (~sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x))) | (~sum(inverse(x), x, multiplicative_identity))) <=> ((~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, x, x)) | (~sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x))) | (~product(inverse(x), add(inverse(inverse(x)), x), additive_identity)) | product(multiplicative_identity, add(inverse(inverse(x)), x), x))),
% 3.04/3.21 inference(rewrite,[status(thm)],[])).
% 3.04/3.21 tff(226,plain,
% 3.04/3.21 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (product(multiplicative_identity, add(inverse(inverse(x)), x), x) | (~product(inverse(x), add(inverse(inverse(x)), x), additive_identity)) | (~sum(additive_identity, x, x)) | (~sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x))) | (~sum(inverse(x), x, multiplicative_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, x, x)) | (~sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x))) | (~product(inverse(x), add(inverse(inverse(x)), x), additive_identity)) | product(multiplicative_identity, add(inverse(inverse(x)), x), x)))),
% 3.04/3.21 inference(monotonicity,[status(thm)],[225])).
% 3.04/3.21 tff(227,plain,
% 3.04/3.21 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (product(multiplicative_identity, add(inverse(inverse(x)), x), x) | (~product(inverse(x), add(inverse(inverse(x)), x), additive_identity)) | (~sum(additive_identity, x, x)) | (~sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x))) | (~sum(inverse(x), x, multiplicative_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, x, x)) | (~sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x))) | (~product(inverse(x), add(inverse(inverse(x)), x), additive_identity)) | product(multiplicative_identity, add(inverse(inverse(x)), x), x))),
% 3.04/3.21 inference(transitivity,[status(thm)],[226, 224])).
% 3.04/3.21 tff(228,plain,
% 3.04/3.21 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (product(multiplicative_identity, add(inverse(inverse(x)), x), x) | (~product(inverse(x), add(inverse(inverse(x)), x), additive_identity)) | (~sum(additive_identity, x, x)) | (~sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x))) | (~sum(inverse(x), x, multiplicative_identity)))),
% 3.04/3.21 inference(quant_inst,[status(thm)],[])).
% 3.04/3.21 tff(229,plain,
% 3.04/3.21 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~sum(inverse(x), x, multiplicative_identity)) | (~sum(additive_identity, x, x)) | (~sum(add(inverse(inverse(x)), x), x, add(inverse(inverse(x)), x))) | (~product(inverse(x), add(inverse(inverse(x)), x), additive_identity)) | product(multiplicative_identity, add(inverse(inverse(x)), x), x)),
% 3.04/3.21 inference(modus_ponens,[status(thm)],[228, 227])).
% 3.04/3.21 tff(230,plain,
% 3.04/3.21 (~product(inverse(x), add(inverse(inverse(x)), x), additive_identity)),
% 3.04/3.21 inference(unit_resolution,[status(thm)],[229, 73, 223, 221, 213, 201])).
% 3.04/3.21 tff(231,plain,
% 3.04/3.21 ($false),
% 3.04/3.21 inference(unit_resolution,[status(thm)],[230, 164])).
% 3.04/3.21 % SZS output end Proof
% 3.04/3.22 % E exiting
%------------------------------------------------------------------------------