%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : RNG007-1 : 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 05:23:52 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.56s
% Output : Proof 0.44s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : RNG007-1 : TPTP v9.0.0. Released v1.0.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n006.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Fri Jun 20 03:05:20 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.20/0.56 % SZS status Unsatisfiable
% 0.20/0.56 % SZS output start Proof
% 0.20/0.56 tff(sum_type, type, (
% 0.20/0.56 sum: ( $i * $i * $i ) > $o)).
% 0.20/0.56 tff(add_type, type, (
% 0.20/0.56 add: ( $i * $i ) > $i)).
% 0.20/0.56 tff(multiply_type, type, (
% 0.20/0.56 multiply: ( $i * $i ) > $i)).
% 0.20/0.56 tff(a_type, type, (
% 0.20/0.56 a: $i)).
% 0.20/0.56 tff(additive_identity_type, type, (
% 0.20/0.56 additive_identity: $i)).
% 0.20/0.56 tff(additive_inverse_type, type, (
% 0.20/0.56 additive_inverse: $i > $i)).
% 0.20/0.56 tff(product_type, type, (
% 0.20/0.56 product: ( $i * $i * $i ) > $o)).
% 0.20/0.56 tff(1,plain,
% 0.20/0.56 (^[X: $i] : refl(sum(additive_identity, X, X) <=> sum(additive_identity, X, X))),
% 0.20/0.56 inference(bind,[status(th)],[])).
% 0.20/0.56 tff(2,plain,
% 0.20/0.56 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 0.20/0.56 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.56 tff(3,plain,
% 0.20/0.56 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 0.20/0.56 inference(rewrite,[status(thm)],[])).
% 0.20/0.56 tff(4,axiom,(![X: $i] : sum(additive_identity, X, X)), file('/export/starexec/sandbox/benchmark/Axioms/RNG001-0.ax','additive_identity1')).
% 0.20/0.56 tff(5,plain,
% 0.20/0.56 (![X: $i] : sum(additive_identity, X, X)),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.20/0.56 tff(6,plain,(
% 0.20/0.56 ![X: $i] : sum(additive_identity, X, X)),
% 0.20/0.56 inference(skolemize,[status(sab)],[5])).
% 0.20/0.56 tff(7,plain,
% 0.20/0.56 (![X: $i] : sum(additive_identity, X, X)),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.20/0.56 tff(8,plain,
% 0.20/0.56 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, additive_inverse(additive_identity), additive_inverse(additive_identity))),
% 0.20/0.56 inference(quant_inst,[status(thm)],[])).
% 0.20/0.56 tff(9,plain,
% 0.20/0.56 (sum(additive_identity, additive_inverse(additive_identity), additive_inverse(additive_identity))),
% 0.20/0.56 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.20/0.56 tff(10,plain,
% 0.20/0.56 (^[X: $i] : refl(sum(X, additive_inverse(X), additive_identity) <=> sum(X, additive_inverse(X), additive_identity))),
% 0.20/0.56 inference(bind,[status(th)],[])).
% 0.20/0.56 tff(11,plain,
% 0.20/0.56 (![X: $i] : sum(X, additive_inverse(X), additive_identity) <=> ![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 0.20/0.56 inference(quant_intro,[status(thm)],[10])).
% 0.20/0.56 tff(12,plain,
% 0.20/0.56 (![X: $i] : sum(X, additive_inverse(X), additive_identity) <=> ![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 0.20/0.56 inference(rewrite,[status(thm)],[])).
% 0.20/0.56 tff(13,axiom,(![X: $i] : sum(X, additive_inverse(X), additive_identity)), file('/export/starexec/sandbox/benchmark/Axioms/RNG001-0.ax','right_inverse')).
% 0.20/0.56 tff(14,plain,
% 0.20/0.56 (![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.20/0.56 tff(15,plain,(
% 0.20/0.56 ![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 0.20/0.56 inference(skolemize,[status(sab)],[14])).
% 0.20/0.56 tff(16,plain,
% 0.20/0.56 (![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[15, 11])).
% 0.20/0.56 tff(17,plain,
% 0.20/0.56 ((~![X: $i] : sum(X, additive_inverse(X), additive_identity)) | sum(additive_identity, additive_inverse(additive_identity), additive_identity)),
% 0.20/0.56 inference(quant_inst,[status(thm)],[])).
% 0.20/0.56 tff(18,plain,
% 0.20/0.56 (sum(additive_identity, additive_inverse(additive_identity), additive_identity)),
% 0.20/0.56 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.20/0.56 tff(19,plain,
% 0.20/0.56 (^[V: $i, Y: $i, U: $i, X: $i] : refl(((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V)) <=> ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V)))),
% 0.20/0.56 inference(bind,[status(th)],[])).
% 0.20/0.56 tff(20,plain,
% 0.20/0.56 (![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] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 0.20/0.56 inference(quant_intro,[status(thm)],[19])).
% 0.20/0.56 tff(21,plain,
% 0.20/0.56 (![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] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 0.20/0.56 inference(rewrite,[status(thm)],[])).
% 0.20/0.56 tff(22,plain,
% 0.20/0.56 (^[V: $i, Y: $i, U: $i, X: $i] : rewrite((((~sum(X, Y, U)) | (~sum(X, Y, V))) | (U = V)) <=> ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V)))),
% 0.20/0.56 inference(bind,[status(th)],[])).
% 0.20/0.56 tff(23,plain,
% 0.20/0.56 (![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] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 0.20/0.56 inference(quant_intro,[status(thm)],[22])).
% 0.20/0.56 tff(24,axiom,(![V: $i, Y: $i, U: $i, X: $i] : (((~sum(X, Y, U)) | (~sum(X, Y, V))) | (U = V))), file('/export/starexec/sandbox/benchmark/Axioms/RNG001-0.ax','addition_is_well_defined')).
% 0.20/0.56 tff(25,plain,
% 0.20/0.56 (![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[24, 23])).
% 0.20/0.56 tff(26,plain,
% 0.20/0.56 (![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[25, 21])).
% 0.20/0.56 tff(27,plain,(
% 0.20/0.56 ![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 0.20/0.56 inference(skolemize,[status(sab)],[26])).
% 0.20/0.56 tff(28,plain,
% 0.20/0.56 (![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[27, 20])).
% 0.20/0.56 tff(29,plain,
% 0.20/0.56 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | (~sum(additive_identity, additive_inverse(additive_identity), additive_inverse(additive_identity))) | (additive_identity = additive_inverse(additive_identity)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | (~sum(additive_identity, additive_inverse(additive_identity), additive_inverse(additive_identity))) | (additive_identity = additive_inverse(additive_identity)))),
% 0.20/0.56 inference(rewrite,[status(thm)],[])).
% 0.20/0.56 tff(30,plain,
% 0.20/0.56 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | (~sum(additive_identity, additive_inverse(additive_identity), additive_inverse(additive_identity))) | (additive_identity = additive_inverse(additive_identity)))),
% 0.20/0.56 inference(quant_inst,[status(thm)],[])).
% 0.20/0.56 tff(31,plain,
% 0.20/0.56 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | (~sum(additive_identity, additive_inverse(additive_identity), additive_inverse(additive_identity))) | (additive_identity = additive_inverse(additive_identity))),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[30, 29])).
% 0.20/0.56 tff(32,plain,
% 0.20/0.56 (additive_identity = additive_inverse(additive_identity)),
% 0.20/0.56 inference(unit_resolution,[status(thm)],[31, 28, 18, 9])).
% 0.20/0.56 tff(33,plain,
% 0.20/0.56 (additive_inverse(additive_identity) = additive_identity),
% 0.20/0.56 inference(symmetry,[status(thm)],[32])).
% 0.20/0.56 tff(34,plain,
% 0.20/0.56 (sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity)) <=> sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_identity)),
% 0.20/0.56 inference(monotonicity,[status(thm)],[33])).
% 0.20/0.56 tff(35,plain,
% 0.20/0.56 (sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_identity) <=> sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity))),
% 0.20/0.56 inference(symmetry,[status(thm)],[34])).
% 0.20/0.56 tff(36,plain,
% 0.20/0.56 ((~![X: $i] : sum(X, additive_inverse(X), additive_identity)) | sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_identity)),
% 0.20/0.56 inference(quant_inst,[status(thm)],[])).
% 0.20/0.56 tff(37,plain,
% 0.20/0.56 (sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_identity)),
% 0.20/0.56 inference(unit_resolution,[status(thm)],[36, 16])).
% 0.20/0.56 tff(38,plain,
% 0.20/0.56 (sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity))),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[37, 35])).
% 0.20/0.56 tff(39,plain,
% 0.20/0.56 ((~![X: $i] : sum(X, additive_inverse(X), additive_identity)) | sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)),
% 0.20/0.56 inference(quant_inst,[status(thm)],[])).
% 0.20/0.56 tff(40,plain,
% 0.20/0.56 (sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)),
% 0.20/0.56 inference(unit_resolution,[status(thm)],[39, 16])).
% 0.20/0.56 tff(41,plain,
% 0.20/0.56 (^[Y: $i, X: $i] : refl(sum(X, Y, add(X, Y)) <=> sum(X, Y, add(X, Y)))),
% 0.20/0.56 inference(bind,[status(th)],[])).
% 0.20/0.56 tff(42,plain,
% 0.20/0.56 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.20/0.56 inference(quant_intro,[status(thm)],[41])).
% 0.20/0.56 tff(43,plain,
% 0.20/0.56 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.20/0.56 inference(rewrite,[status(thm)],[])).
% 0.20/0.56 tff(44,axiom,(![Y: $i, X: $i] : sum(X, Y, add(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/RNG001-0.ax','closure_of_addition')).
% 0.20/0.56 tff(45,plain,
% 0.20/0.56 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.20/0.56 tff(46,plain,(
% 0.20/0.56 ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.20/0.56 inference(skolemize,[status(sab)],[45])).
% 0.20/0.56 tff(47,plain,
% 0.20/0.56 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[46, 42])).
% 0.20/0.56 tff(48,plain,
% 0.20/0.56 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(a, a, add(a, a))),
% 0.20/0.56 inference(quant_inst,[status(thm)],[])).
% 0.20/0.56 tff(49,plain,
% 0.20/0.56 (sum(a, a, add(a, a))),
% 0.20/0.56 inference(unit_resolution,[status(thm)],[48, 47])).
% 0.20/0.56 tff(50,plain,
% 0.20/0.56 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, a, a)),
% 0.20/0.56 inference(quant_inst,[status(thm)],[])).
% 0.20/0.56 tff(51,plain,
% 0.20/0.56 (sum(additive_identity, a, a)),
% 0.20/0.56 inference(unit_resolution,[status(thm)],[50, 7])).
% 0.20/0.56 tff(52,plain,
% 0.20/0.56 ((~sum(a, a, additive_identity)) <=> (~sum(a, a, additive_identity))),
% 0.20/0.56 inference(rewrite,[status(thm)],[])).
% 0.20/0.56 tff(53,axiom,(~sum(a, a, additive_identity)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_a_plus_a_is_id')).
% 0.20/0.56 tff(54,plain,
% 0.20/0.56 (~sum(a, a, additive_identity)),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.20/0.56 tff(55,plain,
% 0.20/0.56 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : refl((sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) <=> (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))))),
% 0.20/0.56 inference(bind,[status(th)],[])).
% 0.20/0.56 tff(56,plain,
% 0.20/0.56 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 0.20/0.56 inference(quant_intro,[status(thm)],[55])).
% 0.20/0.56 tff(57,plain,
% 0.20/0.56 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 0.20/0.56 inference(rewrite,[status(thm)],[])).
% 0.20/0.56 tff(58,plain,
% 0.20/0.56 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : trans(monotonicity(rewrite((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) <=> ((~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))), (((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) | sum(U, Z, W)) <=> (((~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) | sum(U, Z, W)))), rewrite((((~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) | sum(U, Z, W)) <=> (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))), (((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) | sum(U, Z, W)) <=> (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))))),
% 0.20/0.56 inference(bind,[status(th)],[])).
% 0.20/0.56 tff(59,plain,
% 0.20/0.56 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) | sum(U, Z, W)) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 0.20/0.56 inference(quant_intro,[status(thm)],[58])).
% 0.20/0.56 tff(60,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) | sum(U, Z, W))), file('/export/starexec/sandbox/benchmark/Axioms/RNG001-0.ax','associativity_of_addition2')).
% 0.20/0.56 tff(61,plain,
% 0.20/0.56 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[60, 59])).
% 0.20/0.56 tff(62,plain,
% 0.20/0.56 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[61, 57])).
% 0.20/0.56 tff(63,plain,(
% 0.20/0.56 ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 0.20/0.56 inference(skolemize,[status(sab)],[62])).
% 0.20/0.56 tff(64,plain,
% 0.20/0.56 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[63, 56])).
% 0.20/0.56 tff(65,plain,
% 0.20/0.56 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(a, a, additive_identity) | (~sum(additive_identity, a, a)) | (~sum(a, a, add(a, a))) | (~sum(additive_identity, add(a, a), additive_identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | sum(a, a, additive_identity) | (~sum(additive_identity, a, a)) | (~sum(a, a, add(a, a))) | (~sum(additive_identity, add(a, a), additive_identity)))),
% 0.20/0.56 inference(rewrite,[status(thm)],[])).
% 0.20/0.56 tff(66,plain,
% 0.20/0.56 ((sum(a, a, additive_identity) | (~sum(a, a, add(a, a))) | (~sum(additive_identity, a, a)) | (~sum(additive_identity, add(a, a), additive_identity))) <=> (sum(a, a, additive_identity) | (~sum(additive_identity, a, a)) | (~sum(a, a, add(a, a))) | (~sum(additive_identity, add(a, a), additive_identity)))),
% 0.20/0.56 inference(rewrite,[status(thm)],[])).
% 0.20/0.56 tff(67,plain,
% 0.20/0.56 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(a, a, additive_identity) | (~sum(a, a, add(a, a))) | (~sum(additive_identity, a, a)) | (~sum(additive_identity, add(a, a), additive_identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(a, a, additive_identity) | (~sum(additive_identity, a, a)) | (~sum(a, a, add(a, a))) | (~sum(additive_identity, add(a, a), additive_identity))))),
% 0.20/0.56 inference(monotonicity,[status(thm)],[66])).
% 0.20/0.56 tff(68,plain,
% 0.20/0.56 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(a, a, additive_identity) | (~sum(a, a, add(a, a))) | (~sum(additive_identity, a, a)) | (~sum(additive_identity, add(a, a), additive_identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | sum(a, a, additive_identity) | (~sum(additive_identity, a, a)) | (~sum(a, a, add(a, a))) | (~sum(additive_identity, add(a, a), additive_identity)))),
% 0.20/0.56 inference(transitivity,[status(thm)],[67, 65])).
% 0.20/0.56 tff(69,plain,
% 0.20/0.56 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(a, a, additive_identity) | (~sum(a, a, add(a, a))) | (~sum(additive_identity, a, a)) | (~sum(additive_identity, add(a, a), additive_identity)))),
% 0.20/0.56 inference(quant_inst,[status(thm)],[])).
% 0.20/0.56 tff(70,plain,
% 0.20/0.56 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | sum(a, a, additive_identity) | (~sum(additive_identity, a, a)) | (~sum(a, a, add(a, a))) | (~sum(additive_identity, add(a, a), additive_identity))),
% 0.20/0.56 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.20/0.56 tff(71,plain,
% 0.20/0.56 (~sum(additive_identity, add(a, a), additive_identity)),
% 0.20/0.56 inference(unit_resolution,[status(thm)],[70, 64, 54, 51, 49])).
% 0.20/0.56 tff(72,plain,
% 0.20/0.56 (^[Z: $i, Y: $i, X: $i] : refl(((~sum(X, Y, Z)) | sum(Y, X, Z)) <=> ((~sum(X, Y, Z)) | sum(Y, X, Z)))),
% 0.20/0.57 inference(bind,[status(th)],[])).
% 0.20/0.57 tff(73,plain,
% 0.20/0.57 (![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))),
% 0.20/0.57 inference(quant_intro,[status(thm)],[72])).
% 0.20/0.57 tff(74,plain,
% 0.20/0.57 (![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))),
% 0.20/0.57 inference(rewrite,[status(thm)],[])).
% 0.20/0.57 tff(75,axiom,(![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))), file('/export/starexec/sandbox/benchmark/Axioms/RNG001-0.ax','commutativity_of_addition')).
% 0.20/0.57 tff(76,plain,
% 0.20/0.57 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.20/0.57 tff(77,plain,(
% 0.20/0.57 ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 0.20/0.57 inference(skolemize,[status(sab)],[76])).
% 0.20/0.57 tff(78,plain,
% 0.20/0.57 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[77, 73])).
% 0.20/0.57 tff(79,plain,
% 0.20/0.57 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(add(a, a), additive_identity, additive_identity)) | sum(additive_identity, add(a, a), additive_identity))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(add(a, a), additive_identity, additive_identity)) | sum(additive_identity, add(a, a), additive_identity))),
% 0.20/0.57 inference(rewrite,[status(thm)],[])).
% 0.20/0.57 tff(80,plain,
% 0.20/0.57 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(add(a, a), additive_identity, additive_identity)) | sum(additive_identity, add(a, a), additive_identity))),
% 0.20/0.57 inference(quant_inst,[status(thm)],[])).
% 0.20/0.57 tff(81,plain,
% 0.20/0.57 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(add(a, a), additive_identity, additive_identity)) | sum(additive_identity, add(a, a), additive_identity)),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[80, 79])).
% 0.20/0.57 tff(82,plain,
% 0.20/0.57 (~sum(add(a, a), additive_identity, additive_identity)),
% 0.20/0.57 inference(unit_resolution,[status(thm)],[81, 78, 71])).
% 0.20/0.57 tff(83,plain,
% 0.20/0.57 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : refl((sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U))) <=> (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U))))),
% 0.20/0.57 inference(bind,[status(th)],[])).
% 0.20/0.57 tff(84,plain,
% 0.20/0.57 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))),
% 0.20/0.57 inference(quant_intro,[status(thm)],[83])).
% 0.20/0.57 tff(85,plain,
% 0.20/0.57 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))),
% 0.20/0.57 inference(rewrite,[status(thm)],[])).
% 0.20/0.57 tff(86,plain,
% 0.20/0.57 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : trans(monotonicity(rewrite((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(U, Z, W))) <=> ((~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))), (((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(U, Z, W))) | sum(X, V, W)) <=> (((~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U))) | sum(X, V, W)))), rewrite((((~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U))) | sum(X, V, W)) <=> (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))), (((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(U, Z, W))) | sum(X, V, W)) <=> (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))))),
% 0.20/0.57 inference(bind,[status(th)],[])).
% 0.20/0.57 tff(87,plain,
% 0.20/0.57 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(U, Z, W))) | sum(X, V, W)) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))),
% 0.20/0.57 inference(quant_intro,[status(thm)],[86])).
% 0.20/0.57 tff(88,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(U, Z, W))) | sum(X, V, W))), file('/export/starexec/sandbox/benchmark/Axioms/RNG001-0.ax','associativity_of_addition1')).
% 0.20/0.57 tff(89,plain,
% 0.20/0.57 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.20/0.57 tff(90,plain,
% 0.20/0.57 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[89, 85])).
% 0.20/0.57 tff(91,plain,(
% 0.20/0.57 ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))),
% 0.20/0.57 inference(skolemize,[status(sab)],[90])).
% 0.20/0.57 tff(92,plain,
% 0.20/0.57 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[91, 84])).
% 0.20/0.57 tff(93,plain,
% 0.20/0.57 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(a, a), additive_identity, additive_identity) | (~sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)) | (~sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | sum(add(a, a), additive_identity, additive_identity) | (~sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)) | (~sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a))))),
% 0.20/0.57 inference(rewrite,[status(thm)],[])).
% 0.20/0.57 tff(94,plain,
% 0.20/0.57 ((sum(add(a, a), additive_identity, additive_identity) | (~sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)) | (~sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)) | (~sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a)))) <=> (sum(add(a, a), additive_identity, additive_identity) | (~sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)) | (~sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a))))),
% 0.20/0.57 inference(rewrite,[status(thm)],[])).
% 0.20/0.57 tff(95,plain,
% 0.20/0.57 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(a, a), additive_identity, additive_identity) | (~sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)) | (~sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)) | (~sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(a, a), additive_identity, additive_identity) | (~sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)) | (~sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a)))))),
% 0.20/0.57 inference(monotonicity,[status(thm)],[94])).
% 0.20/0.57 tff(96,plain,
% 0.20/0.57 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(a, a), additive_identity, additive_identity) | (~sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)) | (~sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)) | (~sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | sum(add(a, a), additive_identity, additive_identity) | (~sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)) | (~sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a))))),
% 0.20/0.57 inference(transitivity,[status(thm)],[95, 93])).
% 0.20/0.57 tff(97,plain,
% 0.20/0.57 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(a, a), additive_identity, additive_identity) | (~sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)) | (~sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)) | (~sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a))))),
% 0.20/0.57 inference(quant_inst,[status(thm)],[])).
% 0.20/0.57 tff(98,plain,
% 0.20/0.57 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | sum(add(a, a), additive_identity, additive_identity) | (~sum(multiply(additive_identity, a), additive_inverse(multiply(additive_identity, a)), additive_identity)) | (~sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a)))),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[97, 96])).
% 0.20/0.57 tff(99,plain,
% 0.20/0.57 (~sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a))),
% 0.20/0.57 inference(unit_resolution,[status(thm)],[98, 92, 82, 40])).
% 0.20/0.57 tff(100,plain,
% 0.20/0.57 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a))) | sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a))) | sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a)))),
% 0.20/0.57 inference(rewrite,[status(thm)],[])).
% 0.20/0.57 tff(101,plain,
% 0.20/0.57 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a))) | sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a)))),
% 0.20/0.57 inference(quant_inst,[status(thm)],[])).
% 0.20/0.57 tff(102,plain,
% 0.20/0.57 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a))) | sum(add(a, a), multiply(additive_identity, a), multiply(additive_identity, a))),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.20/0.57 tff(103,plain,
% 0.20/0.57 (~sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a))),
% 0.20/0.57 inference(unit_resolution,[status(thm)],[102, 78, 99])).
% 0.20/0.57 tff(104,plain,
% 0.20/0.57 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(add(a, a), multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)))),
% 0.20/0.57 inference(quant_inst,[status(thm)],[])).
% 0.20/0.57 tff(105,plain,
% 0.20/0.57 (sum(add(a, a), multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)))),
% 0.20/0.57 inference(unit_resolution,[status(thm)],[104, 47])).
% 0.20/0.57 tff(106,plain,
% 0.20/0.57 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(add(a, a), multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)))) | sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(add(a, a), multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)))) | sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a))))),
% 0.20/0.57 inference(rewrite,[status(thm)],[])).
% 0.20/0.57 tff(107,plain,
% 0.20/0.57 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(add(a, a), multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)))) | sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a))))),
% 0.20/0.57 inference(quant_inst,[status(thm)],[])).
% 0.20/0.57 tff(108,plain,
% 0.20/0.57 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(add(a, a), multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)))) | sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a)))),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.20/0.57 tff(109,plain,
% 0.20/0.57 (sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a)))),
% 0.20/0.57 inference(unit_resolution,[status(thm)],[108, 78, 105])).
% 0.20/0.57 tff(110,plain,
% 0.20/0.57 (^[Y: $i, X: $i] : refl(product(X, Y, multiply(X, Y)) <=> product(X, Y, multiply(X, Y)))),
% 0.20/0.57 inference(bind,[status(th)],[])).
% 0.20/0.57 tff(111,plain,
% 0.20/0.57 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.20/0.57 inference(quant_intro,[status(thm)],[110])).
% 0.20/0.57 tff(112,plain,
% 0.20/0.57 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.20/0.57 inference(rewrite,[status(thm)],[])).
% 0.20/0.57 tff(113,axiom,(![Y: $i, X: $i] : product(X, Y, multiply(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/RNG001-0.ax','closure_of_multiplication')).
% 0.20/0.57 tff(114,plain,
% 0.20/0.57 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[113, 112])).
% 0.20/0.57 tff(115,plain,(
% 0.20/0.57 ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.20/0.57 inference(skolemize,[status(sab)],[114])).
% 0.20/0.57 tff(116,plain,
% 0.20/0.57 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[115, 111])).
% 0.20/0.57 tff(117,plain,
% 0.20/0.57 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(additive_identity, a, multiply(additive_identity, a))),
% 0.20/0.57 inference(quant_inst,[status(thm)],[])).
% 0.20/0.57 tff(118,plain,
% 0.20/0.57 (product(additive_identity, a, multiply(additive_identity, a))),
% 0.20/0.57 inference(unit_resolution,[status(thm)],[117, 116])).
% 0.20/0.57 tff(119,plain,
% 0.20/0.57 (^[X: $i] : refl(sum(X, additive_identity, X) <=> sum(X, additive_identity, X))),
% 0.20/0.57 inference(bind,[status(th)],[])).
% 0.20/0.57 tff(120,plain,
% 0.20/0.57 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 0.20/0.57 inference(quant_intro,[status(thm)],[119])).
% 0.20/0.57 tff(121,plain,
% 0.20/0.57 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 0.20/0.57 inference(rewrite,[status(thm)],[])).
% 0.20/0.57 tff(122,axiom,(![X: $i] : sum(X, additive_identity, X)), file('/export/starexec/sandbox/benchmark/Axioms/RNG001-0.ax','additive_identity2')).
% 0.20/0.57 tff(123,plain,
% 0.20/0.57 (![X: $i] : sum(X, additive_identity, X)),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[122, 121])).
% 0.20/0.57 tff(124,plain,(
% 0.20/0.57 ![X: $i] : sum(X, additive_identity, X)),
% 0.20/0.57 inference(skolemize,[status(sab)],[123])).
% 0.20/0.57 tff(125,plain,
% 0.20/0.57 (![X: $i] : sum(X, additive_identity, X)),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[124, 120])).
% 0.20/0.57 tff(126,plain,
% 0.20/0.57 ((~![X: $i] : sum(X, additive_identity, X)) | sum(additive_identity, additive_identity, additive_identity)),
% 0.20/0.57 inference(quant_inst,[status(thm)],[])).
% 0.20/0.57 tff(127,plain,
% 0.20/0.57 (sum(additive_identity, additive_identity, additive_identity)),
% 0.20/0.57 inference(unit_resolution,[status(thm)],[126, 125])).
% 0.20/0.57 tff(128,plain,
% 0.20/0.57 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) <=> (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))))),
% 0.20/0.57 inference(bind,[status(th)],[])).
% 0.20/0.57 tff(129,plain,
% 0.20/0.57 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 0.20/0.57 inference(quant_intro,[status(thm)],[128])).
% 0.20/0.57 tff(130,plain,
% 0.20/0.57 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 0.20/0.57 inference(rewrite,[status(thm)],[])).
% 0.20/0.57 tff(131,plain,
% 0.20/0.57 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) <=> ((~sum(Y, Z, V3)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))), (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) <=> (((~sum(Y, Z, V3)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | (~product(V3, X, V4))))), rewrite((((~sum(Y, Z, V3)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | (~product(V3, X, V4))) <=> ((~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))), (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) <=> ((~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))))), ((((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) | sum(V1, V2, V4)) <=> (((~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | sum(V1, V2, V4)))), rewrite((((~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | sum(V1, V2, V4)) <=> (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))), ((((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) | sum(V1, V2, V4)) <=> (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))))),
% 0.20/0.57 inference(bind,[status(th)],[])).
% 0.20/0.57 tff(132,plain,
% 0.20/0.57 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) | sum(V1, V2, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 0.20/0.57 inference(quant_intro,[status(thm)],[131])).
% 0.20/0.57 tff(133,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) | sum(V1, V2, V4))), file('/export/starexec/sandbox/benchmark/Axioms/RNG001-0.ax','distributivity3')).
% 0.20/0.57 tff(134,plain,
% 0.20/0.57 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[133, 132])).
% 0.20/0.57 tff(135,plain,
% 0.20/0.57 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[134, 130])).
% 0.20/0.57 tff(136,plain,(
% 0.20/0.57 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 0.20/0.57 inference(skolemize,[status(sab)],[135])).
% 0.20/0.57 tff(137,plain,
% 0.20/0.57 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 0.20/0.57 inference(modus_ponens,[status(thm)],[136, 129])).
% 0.20/0.57 tff(138,plain,
% 0.20/0.57 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a)))),
% 0.20/0.57 inference(rewrite,[status(thm)],[])).
% 0.20/0.57 tff(139,plain,
% 0.20/0.57 ((sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, a, multiply(additive_identity, a)))) <=> ((~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a)))),
% 0.42/0.57 inference(rewrite,[status(thm)],[])).
% 0.42/0.57 tff(140,plain,
% 0.42/0.57 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, a, multiply(additive_identity, a))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))))),
% 0.42/0.57 inference(monotonicity,[status(thm)],[139])).
% 0.42/0.57 tff(141,plain,
% 0.42/0.57 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, a, multiply(additive_identity, a))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a)))),
% 0.42/0.57 inference(transitivity,[status(thm)],[140, 138])).
% 0.42/0.57 tff(142,plain,
% 0.42/0.57 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, a, multiply(additive_identity, a))))),
% 0.42/0.57 inference(quant_inst,[status(thm)],[])).
% 0.42/0.57 tff(143,plain,
% 0.42/0.57 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))),
% 0.42/0.57 inference(modus_ponens,[status(thm)],[142, 141])).
% 0.42/0.57 tff(144,plain,
% 0.42/0.57 (sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))),
% 0.42/0.57 inference(unit_resolution,[status(thm)],[143, 137, 127, 118])).
% 0.42/0.57 tff(145,plain,
% 0.42/0.57 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | ((~sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a)))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a)) | (~sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a)))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a)) | (~sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a))))),
% 0.42/0.58 inference(rewrite,[status(thm)],[])).
% 0.42/0.58 tff(146,plain,
% 0.42/0.58 ((sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a)) | (~sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a)))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a)))) <=> ((~sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a)))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a)) | (~sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a))))),
% 0.42/0.58 inference(rewrite,[status(thm)],[])).
% 0.42/0.58 tff(147,plain,
% 0.42/0.58 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a)) | (~sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a)))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | ((~sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a)))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a)) | (~sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a)))))),
% 0.42/0.58 inference(monotonicity,[status(thm)],[146])).
% 0.42/0.58 tff(148,plain,
% 0.42/0.58 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a)) | (~sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a)))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a)))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a)) | (~sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a))))),
% 0.42/0.58 inference(transitivity,[status(thm)],[147, 145])).
% 0.42/0.58 tff(149,plain,
% 0.42/0.58 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a)) | (~sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a)))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a))))),
% 0.42/0.58 inference(quant_inst,[status(thm)],[])).
% 0.42/0.58 tff(150,plain,
% 0.42/0.58 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(multiply(additive_identity, a), add(a, a), add(add(a, a), multiply(additive_identity, a)))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(multiply(additive_identity, a), add(a, a), multiply(additive_identity, a)) | (~sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a)))),
% 0.42/0.58 inference(modus_ponens,[status(thm)],[149, 148])).
% 0.42/0.58 tff(151,plain,
% 0.42/0.58 (~sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a))),
% 0.42/0.58 inference(unit_resolution,[status(thm)],[150, 64, 144, 109, 103])).
% 0.42/0.58 tff(152,plain,
% 0.42/0.58 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a)))),
% 0.42/0.58 inference(rewrite,[status(thm)],[])).
% 0.42/0.58 tff(153,plain,
% 0.42/0.58 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a)))),
% 0.42/0.58 inference(quant_inst,[status(thm)],[])).
% 0.42/0.58 tff(154,plain,
% 0.42/0.58 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a))),
% 0.42/0.58 inference(modus_ponens,[status(thm)],[153, 152])).
% 0.42/0.58 tff(155,plain,
% 0.42/0.58 (~sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a))),
% 0.42/0.58 inference(unit_resolution,[status(thm)],[154, 78, 151])).
% 0.42/0.58 tff(156,plain,
% 0.42/0.58 (sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a)) <=> sum(additive_identity, multiply(additive_identity, a), multiply(additive_identity, a))),
% 0.42/0.58 inference(monotonicity,[status(thm)],[33])).
% 0.42/0.58 tff(157,plain,
% 0.42/0.58 (sum(additive_identity, multiply(additive_identity, a), multiply(additive_identity, a)) <=> sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))),
% 0.42/0.58 inference(symmetry,[status(thm)],[156])).
% 0.42/0.58 tff(158,plain,
% 0.42/0.58 (^[X: $i] : refl(product(X, X, X) <=> product(X, X, X))),
% 0.42/0.58 inference(bind,[status(th)],[])).
% 0.42/0.58 tff(159,plain,
% 0.42/0.58 (![X: $i] : product(X, X, X) <=> ![X: $i] : product(X, X, X)),
% 0.42/0.58 inference(quant_intro,[status(thm)],[158])).
% 0.42/0.58 tff(160,plain,
% 0.42/0.58 (![X: $i] : product(X, X, X) <=> ![X: $i] : product(X, X, X)),
% 0.42/0.58 inference(rewrite,[status(thm)],[])).
% 0.42/0.58 tff(161,axiom,(![X: $i] : product(X, X, X)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','x_squared_is_x')).
% 0.42/0.58 tff(162,plain,
% 0.42/0.58 (![X: $i] : product(X, X, X)),
% 0.42/0.58 inference(modus_ponens,[status(thm)],[161, 160])).
% 0.42/0.58 tff(163,plain,(
% 0.42/0.58 ![X: $i] : product(X, X, X)),
% 0.42/0.58 inference(skolemize,[status(sab)],[162])).
% 0.42/0.58 tff(164,plain,
% 0.42/0.58 (![X: $i] : product(X, X, X)),
% 0.42/0.58 inference(modus_ponens,[status(thm)],[163, 159])).
% 0.42/0.58 tff(165,plain,
% 0.42/0.58 ((~![X: $i] : product(X, X, X)) | product(additive_identity, additive_identity, additive_identity)),
% 0.42/0.58 inference(quant_inst,[status(thm)],[])).
% 0.42/0.58 tff(166,plain,
% 0.42/0.58 (product(additive_identity, additive_identity, additive_identity)),
% 0.42/0.58 inference(unit_resolution,[status(thm)],[165, 164])).
% 0.42/0.58 tff(167,plain,
% 0.42/0.58 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) <=> (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))))),
% 0.42/0.58 inference(bind,[status(th)],[])).
% 0.42/0.58 tff(168,plain,
% 0.42/0.58 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 0.42/0.58 inference(quant_intro,[status(thm)],[167])).
% 0.42/0.58 tff(169,plain,
% 0.42/0.58 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 0.42/0.58 inference(rewrite,[status(thm)],[])).
% 0.42/0.58 tff(170,plain,
% 0.42/0.58 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) <=> (((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~product(X, V3, V4))))), rewrite((((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~product(X, V3, V4))) <=> ((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) <=> ((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4)) <=> (((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | sum(V1, V2, V4)))), rewrite((((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | sum(V1, V2, V4)) <=> (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4)) <=> (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))))),
% 0.42/0.58 inference(bind,[status(th)],[])).
% 0.42/0.58 tff(171,plain,
% 0.42/0.58 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 0.42/0.58 inference(quant_intro,[status(thm)],[170])).
% 0.42/0.58 tff(172,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4))), file('/export/starexec/sandbox/benchmark/Axioms/RNG001-0.ax','distributivity1')).
% 0.42/0.58 tff(173,plain,
% 0.42/0.58 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 0.42/0.58 inference(modus_ponens,[status(thm)],[172, 171])).
% 0.42/0.58 tff(174,plain,
% 0.42/0.58 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 0.42/0.58 inference(modus_ponens,[status(thm)],[173, 169])).
% 0.42/0.58 tff(175,plain,(
% 0.42/0.58 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 0.42/0.58 inference(skolemize,[status(sab)],[174])).
% 0.42/0.58 tff(176,plain,
% 0.42/0.58 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 0.42/0.58 inference(modus_ponens,[status(thm)],[175, 168])).
% 0.42/0.58 tff(177,plain,
% 0.42/0.58 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~sum(additive_identity, a, a)) | (~product(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | sum(additive_identity, multiply(additive_identity, a), multiply(additive_identity, a)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(additive_identity, a, a)) | (~product(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | sum(additive_identity, multiply(additive_identity, a), multiply(additive_identity, a)))),
% 0.42/0.58 inference(rewrite,[status(thm)],[])).
% 0.42/0.58 tff(178,plain,
% 0.42/0.58 ((sum(additive_identity, multiply(additive_identity, a), multiply(additive_identity, a)) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~sum(additive_identity, a, a)) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, additive_identity, additive_identity))) <=> ((~sum(additive_identity, a, a)) | (~product(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | sum(additive_identity, multiply(additive_identity, a), multiply(additive_identity, a)))),
% 0.42/0.58 inference(rewrite,[status(thm)],[])).
% 0.42/0.58 tff(179,plain,
% 0.42/0.58 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(additive_identity, multiply(additive_identity, a), multiply(additive_identity, a)) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~sum(additive_identity, a, a)) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, additive_identity, additive_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~sum(additive_identity, a, a)) | (~product(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | sum(additive_identity, multiply(additive_identity, a), multiply(additive_identity, a))))),
% 0.42/0.58 inference(monotonicity,[status(thm)],[178])).
% 0.42/0.58 tff(180,plain,
% 0.42/0.58 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(additive_identity, multiply(additive_identity, a), multiply(additive_identity, a)) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~sum(additive_identity, a, a)) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, additive_identity, additive_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(additive_identity, a, a)) | (~product(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | sum(additive_identity, multiply(additive_identity, a), multiply(additive_identity, a)))),
% 0.42/0.58 inference(transitivity,[status(thm)],[179, 177])).
% 0.42/0.58 tff(181,plain,
% 0.42/0.58 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(additive_identity, multiply(additive_identity, a), multiply(additive_identity, a)) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~sum(additive_identity, a, a)) | (~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, additive_identity, additive_identity)))),
% 0.42/0.58 inference(quant_inst,[status(thm)],[])).
% 0.42/0.58 tff(182,plain,
% 0.42/0.58 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(additive_identity, a, a)) | (~product(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | sum(additive_identity, multiply(additive_identity, a), multiply(additive_identity, a))),
% 0.42/0.58 inference(modus_ponens,[status(thm)],[181, 180])).
% 0.42/0.58 tff(183,plain,
% 0.42/0.58 (sum(additive_identity, multiply(additive_identity, a), multiply(additive_identity, a))),
% 0.42/0.58 inference(unit_resolution,[status(thm)],[182, 176, 51, 118, 166])).
% 0.42/0.58 tff(184,plain,
% 0.42/0.58 (sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))),
% 0.42/0.58 inference(modus_ponens,[status(thm)],[183, 157])).
% 0.42/0.58 tff(185,plain,
% 0.42/0.58 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | ((~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a)) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a)) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity))))),
% 0.42/0.58 inference(rewrite,[status(thm)],[])).
% 0.42/0.58 tff(186,plain,
% 0.42/0.58 ((sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a)) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity)))) <=> ((~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a)) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity))))),
% 0.42/0.58 inference(rewrite,[status(thm)],[])).
% 0.42/0.58 tff(187,plain,
% 0.42/0.58 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a)) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | ((~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a)) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity)))))),
% 0.42/0.59 inference(monotonicity,[status(thm)],[186])).
% 0.42/0.59 tff(188,plain,
% 0.42/0.59 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a)) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a)) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity))))),
% 0.42/0.59 inference(transitivity,[status(thm)],[187, 185])).
% 0.42/0.59 tff(189,plain,
% 0.42/0.59 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a)) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity))))),
% 0.42/0.59 inference(quant_inst,[status(thm)],[])).
% 0.42/0.59 tff(190,plain,
% 0.42/0.59 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(add(add(a, a), multiply(additive_identity, a)), multiply(additive_identity, a), multiply(additive_identity, a)) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity)))),
% 0.42/0.59 inference(modus_ponens,[status(thm)],[189, 188])).
% 0.42/0.59 tff(191,plain,
% 0.42/0.59 (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity))),
% 0.42/0.59 inference(unit_resolution,[status(thm)],[190, 92, 184, 155])).
% 0.42/0.59 tff(192,plain,
% 0.42/0.59 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity)) | (~sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity)) | (~sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity))))),
% 0.42/0.59 inference(rewrite,[status(thm)],[])).
% 0.42/0.59 tff(193,plain,
% 0.42/0.59 ((sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity)) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity))) | (~sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))))) <=> (sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity)) | (~sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity))))),
% 0.42/0.59 inference(rewrite,[status(thm)],[])).
% 0.42/0.59 tff(194,plain,
% 0.42/0.59 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity)) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity))) | (~sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity)) | (~sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity)))))),
% 0.42/0.59 inference(monotonicity,[status(thm)],[193])).
% 0.42/0.59 tff(195,plain,
% 0.42/0.59 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity)) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity))) | (~sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity)) | (~sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity))))),
% 0.42/0.59 inference(transitivity,[status(thm)],[194, 192])).
% 0.42/0.59 tff(196,plain,
% 0.42/0.59 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity)) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity))) | (~sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))),
% 0.44/0.59 inference(quant_inst,[status(thm)],[])).
% 0.44/0.59 tff(197,plain,
% 0.44/0.59 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(additive_identity), additive_inverse(additive_identity)) | (~sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity)))),
% 0.44/0.59 inference(modus_ponens,[status(thm)],[196, 195])).
% 0.44/0.59 tff(198,plain,
% 0.44/0.59 ((~sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~sum(add(add(a, a), multiply(additive_identity, a)), additive_inverse(add(add(a, a), multiply(additive_identity, a))), additive_inverse(additive_identity)))),
% 0.44/0.59 inference(unit_resolution,[status(thm)],[197, 92, 191])).
% 0.44/0.59 tff(199,plain,
% 0.44/0.59 (~sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))),
% 0.44/0.59 inference(unit_resolution,[status(thm)],[198, 38])).
% 0.44/0.59 tff(200,plain,
% 0.44/0.59 ((~![X: $i] : product(X, X, X)) | product(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))),
% 0.44/0.59 inference(quant_inst,[status(thm)],[])).
% 0.44/0.59 tff(201,plain,
% 0.44/0.59 (product(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))),
% 0.44/0.59 inference(unit_resolution,[status(thm)],[200, 164])).
% 0.44/0.59 tff(202,assumption,(~sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity))), introduced(assumption)).
% 0.44/0.59 tff(203,plain,
% 0.44/0.59 (sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity)) <=> sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_identity)),
% 0.44/0.59 inference(monotonicity,[status(thm)],[33])).
% 0.44/0.59 tff(204,plain,
% 0.44/0.59 (sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_identity) <=> sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))),
% 0.44/0.59 inference(symmetry,[status(thm)],[203])).
% 0.44/0.59 tff(205,plain,
% 0.44/0.59 (^[X: $i] : refl(sum(additive_inverse(X), X, additive_identity) <=> sum(additive_inverse(X), X, additive_identity))),
% 0.44/0.59 inference(bind,[status(th)],[])).
% 0.44/0.59 tff(206,plain,
% 0.44/0.59 (![X: $i] : sum(additive_inverse(X), X, additive_identity) <=> ![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 0.44/0.59 inference(quant_intro,[status(thm)],[205])).
% 0.44/0.59 tff(207,plain,
% 0.44/0.59 (![X: $i] : sum(additive_inverse(X), X, additive_identity) <=> ![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 0.44/0.59 inference(rewrite,[status(thm)],[])).
% 0.44/0.59 tff(208,axiom,(![X: $i] : sum(additive_inverse(X), X, additive_identity)), file('/export/starexec/sandbox/benchmark/Axioms/RNG001-0.ax','left_inverse')).
% 0.44/0.59 tff(209,plain,
% 0.44/0.59 (![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 0.44/0.59 inference(modus_ponens,[status(thm)],[208, 207])).
% 0.44/0.59 tff(210,plain,(
% 0.44/0.59 ![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 0.44/0.59 inference(skolemize,[status(sab)],[209])).
% 0.44/0.59 tff(211,plain,
% 0.44/0.59 (![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 0.44/0.59 inference(modus_ponens,[status(thm)],[210, 206])).
% 0.44/0.59 tff(212,plain,
% 0.44/0.59 ((~![X: $i] : sum(additive_inverse(X), X, additive_identity)) | sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_identity)),
% 0.44/0.59 inference(quant_inst,[status(thm)],[])).
% 0.44/0.59 tff(213,plain,
% 0.44/0.59 (sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_identity)),
% 0.44/0.59 inference(unit_resolution,[status(thm)],[212, 211])).
% 0.44/0.59 tff(214,plain,
% 0.44/0.59 (sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))),
% 0.44/0.59 inference(modus_ponens,[status(thm)],[213, 204])).
% 0.44/0.59 tff(215,plain,
% 0.44/0.59 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | ((~sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity)))),
% 0.44/0.59 inference(rewrite,[status(thm)],[])).
% 0.44/0.59 tff(216,plain,
% 0.44/0.59 ((sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity)) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))) | (~sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity)))) <=> ((~sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity)))),
% 0.44/0.59 inference(rewrite,[status(thm)],[])).
% 0.44/0.59 tff(217,plain,
% 0.44/0.59 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity)) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))) | (~sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | ((~sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity))))),
% 0.44/0.59 inference(monotonicity,[status(thm)],[216])).
% 0.44/0.59 tff(218,plain,
% 0.44/0.59 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity)) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))) | (~sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity)))),
% 0.44/0.59 inference(transitivity,[status(thm)],[217, 215])).
% 0.44/0.59 tff(219,plain,
% 0.44/0.59 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity)) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))) | (~sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))))),
% 0.44/0.59 inference(quant_inst,[status(thm)],[])).
% 0.44/0.59 tff(220,plain,
% 0.44/0.59 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(additive_inverse(multiply(additive_identity, a)), multiply(additive_identity, a), additive_inverse(additive_identity))) | (~sum(multiply(additive_identity, a), multiply(additive_identity, a), multiply(additive_identity, a))) | sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity))),
% 0.44/0.59 inference(modus_ponens,[status(thm)],[219, 218])).
% 0.44/0.59 tff(221,plain,
% 0.44/0.59 ($false),
% 0.44/0.59 inference(unit_resolution,[status(thm)],[220, 64, 144, 214, 202])).
% 0.44/0.59 tff(222,plain,(sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity))), inference(lemma,lemma(discharge,[]))).
% 0.44/0.59 tff(223,plain,
% 0.44/0.59 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (multiply(additive_identity, a) = additive_inverse(additive_identity)) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (multiply(additive_identity, a) = additive_inverse(additive_identity)) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity))))),
% 0.44/0.59 inference(rewrite,[status(thm)],[])).
% 0.44/0.59 tff(224,plain,
% 0.44/0.59 (((~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity))) | (multiply(additive_identity, a) = additive_inverse(additive_identity))) <=> ((~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (multiply(additive_identity, a) = additive_inverse(additive_identity)) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity))))),
% 0.44/0.59 inference(rewrite,[status(thm)],[])).
% 0.44/0.59 tff(225,plain,
% 0.44/0.59 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity))) | (multiply(additive_identity, a) = additive_inverse(additive_identity)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (multiply(additive_identity, a) = additive_inverse(additive_identity)) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity)))))),
% 0.44/0.60 inference(monotonicity,[status(thm)],[224])).
% 0.44/0.60 tff(226,plain,
% 0.44/0.60 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity))) | (multiply(additive_identity, a) = additive_inverse(additive_identity)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (multiply(additive_identity, a) = additive_inverse(additive_identity)) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity))))),
% 0.44/0.60 inference(transitivity,[status(thm)],[225, 223])).
% 0.44/0.60 tff(227,plain,
% 0.44/0.60 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity))) | (multiply(additive_identity, a) = additive_inverse(additive_identity)))),
% 0.44/0.60 inference(quant_inst,[status(thm)],[])).
% 0.44/0.60 tff(228,plain,
% 0.44/0.60 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), multiply(additive_identity, a))) | (multiply(additive_identity, a) = additive_inverse(additive_identity)) | (~sum(additive_inverse(additive_identity), multiply(additive_identity, a), additive_inverse(additive_identity)))),
% 0.44/0.60 inference(modus_ponens,[status(thm)],[227, 226])).
% 0.44/0.60 tff(229,plain,
% 0.44/0.60 (multiply(additive_identity, a) = additive_inverse(additive_identity)),
% 0.44/0.60 inference(unit_resolution,[status(thm)],[228, 28, 184, 222])).
% 0.44/0.60 tff(230,plain,
% 0.44/0.60 (additive_inverse(additive_identity) = multiply(additive_identity, a)),
% 0.44/0.60 inference(symmetry,[status(thm)],[229])).
% 0.44/0.60 tff(231,plain,
% 0.44/0.60 (sum(a, additive_inverse(additive_identity), add(a, multiply(additive_identity, a))) <=> sum(a, multiply(additive_identity, a), add(a, multiply(additive_identity, a)))),
% 0.44/0.60 inference(monotonicity,[status(thm)],[230])).
% 0.44/0.60 tff(232,plain,
% 0.44/0.60 (sum(a, multiply(additive_identity, a), add(a, multiply(additive_identity, a))) <=> sum(a, additive_inverse(additive_identity), add(a, multiply(additive_identity, a)))),
% 0.44/0.60 inference(symmetry,[status(thm)],[231])).
% 0.44/0.60 tff(233,plain,
% 0.44/0.60 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(a, multiply(additive_identity, a), add(a, multiply(additive_identity, a)))),
% 0.44/0.60 inference(quant_inst,[status(thm)],[])).
% 0.44/0.60 tff(234,plain,
% 0.44/0.60 (sum(a, multiply(additive_identity, a), add(a, multiply(additive_identity, a)))),
% 0.44/0.60 inference(unit_resolution,[status(thm)],[233, 47])).
% 0.44/0.60 tff(235,plain,
% 0.44/0.60 (sum(a, additive_inverse(additive_identity), add(a, multiply(additive_identity, a)))),
% 0.44/0.60 inference(modus_ponens,[status(thm)],[234, 232])).
% 0.44/0.60 tff(236,plain,
% 0.44/0.60 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(additive_identity, a, a)) | sum(a, additive_identity, a))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(additive_identity, a, a)) | sum(a, additive_identity, a))),
% 0.44/0.60 inference(rewrite,[status(thm)],[])).
% 0.44/0.60 tff(237,plain,
% 0.44/0.60 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(additive_identity, a, a)) | sum(a, additive_identity, a))),
% 0.44/0.60 inference(quant_inst,[status(thm)],[])).
% 0.44/0.60 tff(238,plain,
% 0.44/0.60 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(additive_identity, a, a)) | sum(a, additive_identity, a)),
% 0.44/0.60 inference(modus_ponens,[status(thm)],[237, 236])).
% 0.44/0.60 tff(239,plain,
% 0.44/0.60 (sum(a, additive_identity, a)),
% 0.44/0.60 inference(unit_resolution,[status(thm)],[238, 78, 51])).
% 0.44/0.60 tff(240,plain,
% 0.44/0.60 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | ((~sum(a, additive_identity, a)) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | sum(a, additive_inverse(additive_identity), a))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(a, additive_identity, a)) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | sum(a, additive_inverse(additive_identity), a))),
% 0.44/0.60 inference(rewrite,[status(thm)],[])).
% 0.44/0.60 tff(241,plain,
% 0.44/0.60 ((sum(a, additive_inverse(additive_identity), a) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | (~sum(a, additive_identity, a)) | (~sum(a, additive_identity, a))) <=> ((~sum(a, additive_identity, a)) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | sum(a, additive_inverse(additive_identity), a))),
% 0.44/0.60 inference(rewrite,[status(thm)],[])).
% 0.44/0.60 tff(242,plain,
% 0.44/0.60 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(a, additive_inverse(additive_identity), a) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | (~sum(a, additive_identity, a)) | (~sum(a, additive_identity, a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | ((~sum(a, additive_identity, a)) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | sum(a, additive_inverse(additive_identity), a)))),
% 0.44/0.60 inference(monotonicity,[status(thm)],[241])).
% 0.44/0.60 tff(243,plain,
% 0.44/0.60 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(a, additive_inverse(additive_identity), a) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | (~sum(a, additive_identity, a)) | (~sum(a, additive_identity, a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(a, additive_identity, a)) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | sum(a, additive_inverse(additive_identity), a))),
% 0.44/0.60 inference(transitivity,[status(thm)],[242, 240])).
% 0.44/0.60 tff(244,plain,
% 0.44/0.60 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(a, additive_inverse(additive_identity), a) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | (~sum(a, additive_identity, a)) | (~sum(a, additive_identity, a)))),
% 0.44/0.60 inference(quant_inst,[status(thm)],[])).
% 0.44/0.60 tff(245,plain,
% 0.44/0.60 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(a, additive_identity, a)) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | sum(a, additive_inverse(additive_identity), a)),
% 0.44/0.60 inference(modus_ponens,[status(thm)],[244, 243])).
% 0.44/0.60 tff(246,plain,
% 0.44/0.60 (sum(a, additive_inverse(additive_identity), a)),
% 0.44/0.60 inference(unit_resolution,[status(thm)],[245, 64, 18, 239])).
% 0.44/0.60 tff(247,plain,
% 0.44/0.60 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(a, additive_inverse(additive_identity), a)) | (~sum(a, additive_inverse(additive_identity), add(a, multiply(additive_identity, a)))) | (a = add(a, multiply(additive_identity, a))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(a, additive_inverse(additive_identity), a)) | (~sum(a, additive_inverse(additive_identity), add(a, multiply(additive_identity, a)))) | (a = add(a, multiply(additive_identity, a))))),
% 0.44/0.60 inference(rewrite,[status(thm)],[])).
% 0.44/0.60 tff(248,plain,
% 0.44/0.60 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(a, additive_inverse(additive_identity), a)) | (~sum(a, additive_inverse(additive_identity), add(a, multiply(additive_identity, a)))) | (a = add(a, multiply(additive_identity, a))))),
% 0.44/0.60 inference(quant_inst,[status(thm)],[])).
% 0.44/0.60 tff(249,plain,
% 0.44/0.60 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(a, additive_inverse(additive_identity), a)) | (~sum(a, additive_inverse(additive_identity), add(a, multiply(additive_identity, a)))) | (a = add(a, multiply(additive_identity, a)))),
% 0.44/0.60 inference(modus_ponens,[status(thm)],[248, 247])).
% 0.44/0.60 tff(250,plain,
% 0.44/0.60 ((~sum(a, additive_inverse(additive_identity), add(a, multiply(additive_identity, a)))) | (a = add(a, multiply(additive_identity, a)))),
% 0.44/0.60 inference(unit_resolution,[status(thm)],[249, 28, 246])).
% 0.44/0.60 tff(251,plain,
% 0.44/0.60 (a = add(a, multiply(additive_identity, a))),
% 0.44/0.60 inference(unit_resolution,[status(thm)],[250, 235])).
% 0.44/0.60 tff(252,plain,
% 0.44/0.60 (add(a, multiply(additive_identity, a)) = a),
% 0.44/0.60 inference(symmetry,[status(thm)],[251])).
% 0.44/0.60 tff(253,plain,
% 0.44/0.60 (product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a))) <=> product(a, a, a)),
% 0.44/0.60 inference(monotonicity,[status(thm)],[252, 252, 252])).
% 0.44/0.60 tff(254,plain,
% 0.44/0.60 (product(a, a, a) <=> product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))),
% 0.44/0.60 inference(symmetry,[status(thm)],[253])).
% 0.44/0.60 tff(255,plain,
% 0.44/0.60 ((~![X: $i] : product(X, X, X)) | product(a, a, a)),
% 0.44/0.60 inference(quant_inst,[status(thm)],[])).
% 0.44/0.60 tff(256,plain,
% 0.44/0.60 (product(a, a, a)),
% 0.44/0.60 inference(unit_resolution,[status(thm)],[255, 164])).
% 0.44/0.60 tff(257,plain,
% 0.44/0.60 (product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))),
% 0.44/0.60 inference(modus_ponens,[status(thm)],[256, 254])).
% 0.44/0.60 tff(258,plain,
% 0.44/0.60 (sum(add(a, a), additive_inverse(additive_identity), add(add(a, a), multiply(additive_identity, a))) <=> sum(add(a, a), multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a)))),
% 0.44/0.60 inference(monotonicity,[status(thm)],[230])).
% 0.44/0.60 tff(259,plain,
% 0.44/0.60 (sum(add(a, a), multiply(additive_identity, a), add(add(a, a), multiply(additive_identity, a))) <=> sum(add(a, a), additive_inverse(additive_identity), add(add(a, a), multiply(additive_identity, a)))),
% 0.44/0.60 inference(symmetry,[status(thm)],[258])).
% 0.44/0.60 tff(260,plain,
% 0.44/0.60 (sum(add(a, a), additive_inverse(additive_identity), add(add(a, a), multiply(additive_identity, a)))),
% 0.44/0.60 inference(modus_ponens,[status(thm)],[105, 259])).
% 0.44/0.60 tff(261,plain,
% 0.44/0.60 (sum(add(a, a), additive_inverse(additive_identity), add(a, a)) <=> sum(add(a, a), additive_identity, add(a, a))),
% 0.44/0.60 inference(monotonicity,[status(thm)],[33])).
% 0.44/0.60 tff(262,plain,
% 0.44/0.60 (sum(add(a, a), additive_identity, add(a, a)) <=> sum(add(a, a), additive_inverse(additive_identity), add(a, a))),
% 0.44/0.60 inference(symmetry,[status(thm)],[261])).
% 0.44/0.60 tff(263,plain,
% 0.44/0.60 ((~![X: $i] : sum(X, additive_identity, X)) | sum(add(a, a), additive_identity, add(a, a))),
% 0.44/0.60 inference(quant_inst,[status(thm)],[])).
% 0.44/0.60 tff(264,plain,
% 0.44/0.60 (sum(add(a, a), additive_identity, add(a, a))),
% 0.44/0.60 inference(unit_resolution,[status(thm)],[263, 125])).
% 0.44/0.60 tff(265,plain,
% 0.44/0.60 (sum(add(a, a), additive_inverse(additive_identity), add(a, a))),
% 0.44/0.60 inference(modus_ponens,[status(thm)],[264, 262])).
% 0.44/0.60 tff(266,plain,
% 0.44/0.60 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(add(a, a), additive_inverse(additive_identity), add(a, a))) | (~sum(add(a, a), additive_inverse(additive_identity), add(add(a, a), multiply(additive_identity, a)))) | (add(a, a) = add(add(a, a), multiply(additive_identity, a))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(add(a, a), additive_inverse(additive_identity), add(a, a))) | (~sum(add(a, a), additive_inverse(additive_identity), add(add(a, a), multiply(additive_identity, a)))) | (add(a, a) = add(add(a, a), multiply(additive_identity, a))))),
% 0.44/0.60 inference(rewrite,[status(thm)],[])).
% 0.44/0.60 tff(267,plain,
% 0.44/0.60 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(add(a, a), additive_inverse(additive_identity), add(a, a))) | (~sum(add(a, a), additive_inverse(additive_identity), add(add(a, a), multiply(additive_identity, a)))) | (add(a, a) = add(add(a, a), multiply(additive_identity, a))))),
% 0.44/0.60 inference(quant_inst,[status(thm)],[])).
% 0.44/0.60 tff(268,plain,
% 0.44/0.60 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(add(a, a), additive_inverse(additive_identity), add(a, a))) | (~sum(add(a, a), additive_inverse(additive_identity), add(add(a, a), multiply(additive_identity, a)))) | (add(a, a) = add(add(a, a), multiply(additive_identity, a)))),
% 0.44/0.60 inference(modus_ponens,[status(thm)],[267, 266])).
% 0.44/0.60 tff(269,plain,
% 0.44/0.60 ((~sum(add(a, a), additive_inverse(additive_identity), add(add(a, a), multiply(additive_identity, a)))) | (add(a, a) = add(add(a, a), multiply(additive_identity, a)))),
% 0.44/0.60 inference(unit_resolution,[status(thm)],[268, 28, 265])).
% 0.44/0.60 tff(270,plain,
% 0.44/0.60 (add(a, a) = add(add(a, a), multiply(additive_identity, a))),
% 0.44/0.60 inference(unit_resolution,[status(thm)],[269, 260])).
% 0.44/0.60 tff(271,plain,
% 0.44/0.60 (add(add(a, a), multiply(additive_identity, a)) = add(a, a)),
% 0.44/0.60 inference(symmetry,[status(thm)],[270])).
% 0.44/0.60 tff(272,plain,
% 0.44/0.60 (sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) <=> sum(a, a, add(a, a))),
% 0.44/0.60 inference(monotonicity,[status(thm)],[252, 252, 271])).
% 0.44/0.60 tff(273,plain,
% 0.44/0.60 (sum(a, a, add(a, a)) <=> sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))),
% 0.44/0.60 inference(symmetry,[status(thm)],[272])).
% 0.44/0.60 tff(274,plain,
% 0.44/0.60 (sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))),
% 0.44/0.60 inference(modus_ponens,[status(thm)],[49, 273])).
% 0.44/0.60 tff(275,plain,
% 0.44/0.60 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) <=> (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))))),
% 0.44/0.60 inference(bind,[status(th)],[])).
% 0.44/0.60 tff(276,plain,
% 0.44/0.60 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.44/0.60 inference(quant_intro,[status(thm)],[275])).
% 0.44/0.60 tff(277,plain,
% 0.44/0.60 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.44/0.60 inference(rewrite,[status(thm)],[])).
% 0.44/0.60 tff(278,plain,
% 0.44/0.60 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) <=> (((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~sum(V1, V2, V4))))), rewrite((((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~sum(V1, V2, V4))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> (((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) | product(X, V3, V4)))), rewrite((((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))))),
% 0.44/0.60 inference(bind,[status(th)],[])).
% 0.44/0.60 tff(279,plain,
% 0.44/0.60 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.44/0.60 inference(quant_intro,[status(thm)],[278])).
% 0.44/0.60 tff(280,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4))), file('/export/starexec/sandbox/benchmark/Axioms/RNG001-0.ax','distributivity2')).
% 0.44/0.60 tff(281,plain,
% 0.44/0.60 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.44/0.60 inference(modus_ponens,[status(thm)],[280, 279])).
% 0.44/0.60 tff(282,plain,
% 0.44/0.60 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.44/0.60 inference(modus_ponens,[status(thm)],[281, 277])).
% 0.44/0.60 tff(283,plain,(
% 0.44/0.60 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.44/0.60 inference(skolemize,[status(sab)],[282])).
% 0.44/0.60 tff(284,plain,
% 0.44/0.60 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 0.44/0.60 inference(modus_ponens,[status(thm)],[283, 276])).
% 0.44/0.60 tff(285,plain,
% 0.44/0.60 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))),
% 0.44/0.60 inference(rewrite,[status(thm)],[])).
% 0.44/0.60 tff(286,plain,
% 0.44/0.60 ((product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))))) <=> (product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))),
% 0.44/0.61 inference(rewrite,[status(thm)],[])).
% 0.44/0.61 tff(287,plain,
% 0.44/0.61 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))))))),
% 0.44/0.61 inference(monotonicity,[status(thm)],[286])).
% 0.44/0.61 tff(288,plain,
% 0.44/0.61 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))),
% 0.44/0.61 inference(transitivity,[status(thm)],[287, 285])).
% 0.44/0.61 tff(289,plain,
% 0.44/0.61 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))),
% 0.44/0.61 inference(quant_inst,[status(thm)],[])).
% 0.44/0.61 tff(290,plain,
% 0.44/0.61 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))))),
% 0.44/0.61 inference(modus_ponens,[status(thm)],[289, 288])).
% 0.44/0.61 tff(291,plain,
% 0.44/0.61 (product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~product(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))))),
% 0.44/0.61 inference(unit_resolution,[status(thm)],[290, 284])).
% 0.44/0.61 tff(292,plain,
% 0.44/0.61 (product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))),
% 0.44/0.61 inference(unit_resolution,[status(thm)],[291, 274, 257])).
% 0.44/0.61 tff(293,plain,
% 0.44/0.61 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))),
% 0.44/0.61 inference(rewrite,[status(thm)],[])).
% 0.44/0.61 tff(294,plain,
% 0.44/0.61 ((sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))))) <=> ((~product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))),
% 0.44/0.61 inference(rewrite,[status(thm)],[])).
% 0.44/0.61 tff(295,plain,
% 0.44/0.61 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))))))),
% 0.44/0.61 inference(monotonicity,[status(thm)],[294])).
% 0.44/0.61 tff(296,plain,
% 0.44/0.61 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))),
% 0.44/0.61 inference(transitivity,[status(thm)],[295, 293])).
% 0.44/0.61 tff(297,plain,
% 0.44/0.61 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))))),
% 0.44/0.61 inference(quant_inst,[status(thm)],[])).
% 0.44/0.61 tff(298,plain,
% 0.44/0.61 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | (~product(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)))) | sum(add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))) | (~sum(add(a, multiply(additive_identity, a)), add(a, multiply(additive_identity, a)), add(add(a, a), multiply(additive_identity, a))))),
% 0.44/0.61 inference(modus_ponens,[status(thm)],[297, 296])).
% 0.44/0.61 tff(299,plain,
% 0.44/0.61 ($false),
% 0.44/0.61 inference(unit_resolution,[status(thm)],[298, 137, 274, 292, 201, 199])).
% 0.44/0.61 % SZS output end Proof
% 0.44/0.62 % E exiting
%------------------------------------------------------------------------------