%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : RNG041-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:23:58 AM UTC 2025
% Result : Unsatisfiable 0.23s 0.56s
% Output : Proof 0.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.14 % Problem : RNG041-1 : TPTP v9.0.0. Released v1.0.0.
% 0.08/0.14 % Command : run_E %s %d THM
% 0.14/0.36 % Computer : n011.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Fri Jun 20 03:15:56 EDT 2025
% 0.14/0.36 % CPUTime :
% 0.23/0.56 % SZS status Unsatisfiable
% 0.23/0.56 % SZS output start Proof
% 0.23/0.56 tff(product_type, type, (
% 0.23/0.56 product: ( $i * $i * $i ) > $o)).
% 0.23/0.56 tff(additive_identity_type, type, (
% 0.23/0.56 additive_identity: $i)).
% 0.23/0.56 tff(h_type, type, (
% 0.23/0.56 h: $i > $i)).
% 0.23/0.56 tff(add_type, type, (
% 0.23/0.56 add: ( $i * $i ) > $i)).
% 0.23/0.56 tff(b_type, type, (
% 0.23/0.56 b: $i)).
% 0.23/0.56 tff(a_type, type, (
% 0.23/0.56 a: $i)).
% 0.23/0.56 tff(multiplicative_identity_type, type, (
% 0.23/0.56 multiplicative_identity: $i)).
% 0.23/0.56 tff(sum_type, type, (
% 0.23/0.56 sum: ( $i * $i * $i ) > $o)).
% 0.23/0.56 tff(1,assumption,(~product(additive_identity, h(add(additive_identity, b)), additive_identity)), introduced(assumption)).
% 0.23/0.56 tff(2,plain,
% 0.23/0.56 (^[A: $i] : refl(product(additive_identity, A, additive_identity) <=> product(additive_identity, A, additive_identity))),
% 0.23/0.56 inference(bind,[status(th)],[])).
% 0.23/0.56 tff(3,plain,
% 0.23/0.56 (![A: $i] : product(additive_identity, A, additive_identity) <=> ![A: $i] : product(additive_identity, A, additive_identity)),
% 0.23/0.56 inference(quant_intro,[status(thm)],[2])).
% 0.23/0.56 tff(4,plain,
% 0.23/0.56 (![A: $i] : product(additive_identity, A, additive_identity) <=> ![A: $i] : product(additive_identity, A, additive_identity)),
% 0.23/0.56 inference(rewrite,[status(thm)],[])).
% 0.23/0.56 tff(5,axiom,(![A: $i] : product(additive_identity, A, additive_identity)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','multiplicative_identity1')).
% 0.23/0.56 tff(6,plain,
% 0.23/0.56 (![A: $i] : product(additive_identity, A, additive_identity)),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.23/0.56 tff(7,plain,(
% 0.23/0.56 ![A: $i] : product(additive_identity, A, additive_identity)),
% 0.23/0.56 inference(skolemize,[status(sab)],[6])).
% 0.23/0.56 tff(8,plain,
% 0.23/0.56 (![A: $i] : product(additive_identity, A, additive_identity)),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.23/0.56 tff(9,plain,
% 0.23/0.56 ((~![A: $i] : product(additive_identity, A, additive_identity)) | product(additive_identity, h(add(additive_identity, b)), additive_identity)),
% 0.23/0.56 inference(quant_inst,[status(thm)],[])).
% 0.23/0.56 tff(10,plain,
% 0.23/0.56 ($false),
% 0.23/0.56 inference(unit_resolution,[status(thm)],[9, 8, 1])).
% 0.23/0.56 tff(11,plain,(product(additive_identity, h(add(additive_identity, b)), additive_identity)), inference(lemma,lemma(discharge,[]))).
% 0.23/0.56 tff(12,assumption,(~product(additive_identity, h(add(additive_identity, b)), a)), introduced(assumption)).
% 0.23/0.56 tff(13,assumption,(~sum(b, additive_identity, b)), introduced(assumption)).
% 0.23/0.56 tff(14,plain,
% 0.23/0.56 (^[X: $i] : refl(sum(X, additive_identity, X) <=> sum(X, additive_identity, X))),
% 0.23/0.56 inference(bind,[status(th)],[])).
% 0.23/0.56 tff(15,plain,
% 0.23/0.56 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 0.23/0.56 inference(quant_intro,[status(thm)],[14])).
% 0.23/0.56 tff(16,plain,
% 0.23/0.56 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 0.23/0.56 inference(rewrite,[status(thm)],[])).
% 0.23/0.56 tff(17,axiom,(![X: $i] : sum(X, additive_identity, X)), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','additive_identity2')).
% 0.23/0.56 tff(18,plain,
% 0.23/0.56 (![X: $i] : sum(X, additive_identity, X)),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.23/0.56 tff(19,plain,(
% 0.23/0.56 ![X: $i] : sum(X, additive_identity, X)),
% 0.23/0.56 inference(skolemize,[status(sab)],[18])).
% 0.23/0.56 tff(20,plain,
% 0.23/0.56 (![X: $i] : sum(X, additive_identity, X)),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[19, 15])).
% 0.23/0.56 tff(21,plain,
% 0.23/0.56 ((~![X: $i] : sum(X, additive_identity, X)) | sum(b, additive_identity, b)),
% 0.23/0.56 inference(quant_inst,[status(thm)],[])).
% 0.23/0.56 tff(22,plain,
% 0.23/0.56 ($false),
% 0.23/0.56 inference(unit_resolution,[status(thm)],[21, 20, 13])).
% 0.23/0.56 tff(23,plain,(sum(b, additive_identity, b)), inference(lemma,lemma(discharge,[]))).
% 0.23/0.56 tff(24,plain,
% 0.23/0.56 (^[Y: $i, X: $i] : refl(sum(X, Y, add(X, Y)) <=> sum(X, Y, add(X, Y)))),
% 0.23/0.56 inference(bind,[status(th)],[])).
% 0.23/0.56 tff(25,plain,
% 0.23/0.56 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.23/0.56 inference(quant_intro,[status(thm)],[24])).
% 0.23/0.56 tff(26,plain,
% 0.23/0.56 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.23/0.56 inference(rewrite,[status(thm)],[])).
% 0.23/0.56 tff(27,axiom,(![Y: $i, X: $i] : sum(X, Y, add(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','closure_of_addition')).
% 0.23/0.56 tff(28,plain,
% 0.23/0.56 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.23/0.56 tff(29,plain,(
% 0.23/0.56 ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.23/0.56 inference(skolemize,[status(sab)],[28])).
% 0.23/0.56 tff(30,plain,
% 0.23/0.56 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[29, 25])).
% 0.23/0.56 tff(31,plain,
% 0.23/0.56 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(additive_identity, b, add(additive_identity, b))),
% 0.23/0.56 inference(quant_inst,[status(thm)],[])).
% 0.23/0.56 tff(32,plain,
% 0.23/0.56 (sum(additive_identity, b, add(additive_identity, b))),
% 0.23/0.56 inference(unit_resolution,[status(thm)],[31, 30])).
% 0.23/0.56 tff(33,plain,
% 0.23/0.56 (^[Z: $i, Y: $i, X: $i] : refl(((~sum(X, Y, Z)) | sum(Y, X, Z)) <=> ((~sum(X, Y, Z)) | sum(Y, X, Z)))),
% 0.23/0.56 inference(bind,[status(th)],[])).
% 0.23/0.56 tff(34,plain,
% 0.23/0.56 (![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))),
% 0.23/0.56 inference(quant_intro,[status(thm)],[33])).
% 0.23/0.56 tff(35,plain,
% 0.23/0.56 (![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))),
% 0.23/0.56 inference(rewrite,[status(thm)],[])).
% 0.23/0.56 tff(36,axiom,(![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','commutativity_of_addition')).
% 0.23/0.56 tff(37,plain,
% 0.23/0.56 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.23/0.56 tff(38,plain,(
% 0.23/0.56 ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 0.23/0.56 inference(skolemize,[status(sab)],[37])).
% 0.23/0.56 tff(39,plain,
% 0.23/0.56 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[38, 34])).
% 0.23/0.56 tff(40,plain,
% 0.23/0.56 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(additive_identity, b, add(additive_identity, b))) | sum(b, additive_identity, add(additive_identity, b)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(additive_identity, b, add(additive_identity, b))) | sum(b, additive_identity, add(additive_identity, b)))),
% 0.23/0.56 inference(rewrite,[status(thm)],[])).
% 0.23/0.56 tff(41,plain,
% 0.23/0.56 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(additive_identity, b, add(additive_identity, b))) | sum(b, additive_identity, add(additive_identity, b)))),
% 0.23/0.56 inference(quant_inst,[status(thm)],[])).
% 0.23/0.56 tff(42,plain,
% 0.23/0.56 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(additive_identity, b, add(additive_identity, b))) | sum(b, additive_identity, add(additive_identity, b))),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.23/0.56 tff(43,plain,
% 0.23/0.56 (sum(b, additive_identity, add(additive_identity, b))),
% 0.23/0.56 inference(unit_resolution,[status(thm)],[42, 39, 32])).
% 0.23/0.56 tff(44,plain,
% 0.23/0.56 (^[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)))),
% 0.23/0.56 inference(bind,[status(th)],[])).
% 0.23/0.56 tff(45,plain,
% 0.23/0.56 (![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))),
% 0.23/0.56 inference(quant_intro,[status(thm)],[44])).
% 0.23/0.56 tff(46,plain,
% 0.23/0.56 (![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))),
% 0.23/0.56 inference(rewrite,[status(thm)],[])).
% 0.23/0.56 tff(47,plain,
% 0.23/0.56 (^[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)))),
% 0.23/0.56 inference(bind,[status(th)],[])).
% 0.23/0.56 tff(48,plain,
% 0.23/0.56 (![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))),
% 0.23/0.56 inference(quant_intro,[status(thm)],[47])).
% 0.23/0.56 tff(49,axiom,(![V: $i, Y: $i, U: $i, X: $i] : (((~sum(X, Y, U)) | (~sum(X, Y, V))) | (U = V))), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','addition_is_well_defined')).
% 0.23/0.56 tff(50,plain,
% 0.23/0.56 (![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.23/0.56 tff(51,plain,
% 0.23/0.56 (![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[50, 46])).
% 0.23/0.56 tff(52,plain,(
% 0.23/0.56 ![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 0.23/0.56 inference(skolemize,[status(sab)],[51])).
% 0.23/0.56 tff(53,plain,
% 0.23/0.56 (![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[52, 45])).
% 0.23/0.56 tff(54,plain,
% 0.23/0.56 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(b, additive_identity, add(additive_identity, b))) | (add(additive_identity, b) = b) | (~sum(b, additive_identity, b)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(b, additive_identity, add(additive_identity, b))) | (add(additive_identity, b) = b) | (~sum(b, additive_identity, b)))),
% 0.23/0.56 inference(rewrite,[status(thm)],[])).
% 0.23/0.56 tff(55,plain,
% 0.23/0.56 (((~sum(b, additive_identity, add(additive_identity, b))) | (~sum(b, additive_identity, b)) | (add(additive_identity, b) = b)) <=> ((~sum(b, additive_identity, add(additive_identity, b))) | (add(additive_identity, b) = b) | (~sum(b, additive_identity, b)))),
% 0.23/0.56 inference(rewrite,[status(thm)],[])).
% 0.23/0.56 tff(56,plain,
% 0.23/0.56 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(b, additive_identity, add(additive_identity, b))) | (~sum(b, additive_identity, b)) | (add(additive_identity, b) = b))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(b, additive_identity, add(additive_identity, b))) | (add(additive_identity, b) = b) | (~sum(b, additive_identity, b))))),
% 0.23/0.56 inference(monotonicity,[status(thm)],[55])).
% 0.23/0.56 tff(57,plain,
% 0.23/0.56 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(b, additive_identity, add(additive_identity, b))) | (~sum(b, additive_identity, b)) | (add(additive_identity, b) = b))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(b, additive_identity, add(additive_identity, b))) | (add(additive_identity, b) = b) | (~sum(b, additive_identity, b)))),
% 0.23/0.56 inference(transitivity,[status(thm)],[56, 54])).
% 0.23/0.56 tff(58,plain,
% 0.23/0.56 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(b, additive_identity, add(additive_identity, b))) | (~sum(b, additive_identity, b)) | (add(additive_identity, b) = b))),
% 0.23/0.56 inference(quant_inst,[status(thm)],[])).
% 0.23/0.56 tff(59,plain,
% 0.23/0.56 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(b, additive_identity, add(additive_identity, b))) | (add(additive_identity, b) = b) | (~sum(b, additive_identity, b))),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.23/0.56 tff(60,plain,
% 0.23/0.56 (add(additive_identity, b) = b),
% 0.23/0.56 inference(unit_resolution,[status(thm)],[59, 53, 43, 23])).
% 0.23/0.56 tff(61,plain,
% 0.23/0.56 (b = add(additive_identity, b)),
% 0.23/0.56 inference(symmetry,[status(thm)],[60])).
% 0.23/0.56 tff(62,plain,
% 0.23/0.56 (h(b) = h(add(additive_identity, b))),
% 0.23/0.56 inference(monotonicity,[status(thm)],[61])).
% 0.23/0.56 tff(63,plain,
% 0.23/0.56 (h(add(additive_identity, b)) = h(b)),
% 0.23/0.56 inference(symmetry,[status(thm)],[62])).
% 0.23/0.56 tff(64,plain,
% 0.23/0.56 (product(b, h(add(additive_identity, b)), multiplicative_identity) <=> product(b, h(b), multiplicative_identity)),
% 0.23/0.56 inference(monotonicity,[status(thm)],[63])).
% 0.23/0.56 tff(65,plain,
% 0.23/0.56 (product(b, h(b), multiplicative_identity) <=> product(b, h(add(additive_identity, b)), multiplicative_identity)),
% 0.23/0.56 inference(symmetry,[status(thm)],[64])).
% 0.23/0.56 tff(66,plain,
% 0.23/0.56 ((~(b = additive_identity)) <=> (~(b = additive_identity))),
% 0.23/0.56 inference(rewrite,[status(thm)],[])).
% 0.23/0.56 tff(67,axiom,(~(b = additive_identity)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_b_is_additive_identity')).
% 0.23/0.56 tff(68,plain,
% 0.23/0.56 (~(b = additive_identity)),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[67, 66])).
% 0.23/0.56 tff(69,plain,
% 0.23/0.56 (^[A: $i] : refl(((A = additive_identity) | product(A, h(A), multiplicative_identity)) <=> ((A = additive_identity) | product(A, h(A), multiplicative_identity)))),
% 0.23/0.56 inference(bind,[status(th)],[])).
% 0.23/0.56 tff(70,plain,
% 0.23/0.56 (![A: $i] : ((A = additive_identity) | product(A, h(A), multiplicative_identity)) <=> ![A: $i] : ((A = additive_identity) | product(A, h(A), multiplicative_identity))),
% 0.23/0.56 inference(quant_intro,[status(thm)],[69])).
% 0.23/0.56 tff(71,plain,
% 0.23/0.56 (![A: $i] : ((A = additive_identity) | product(A, h(A), multiplicative_identity)) <=> ![A: $i] : ((A = additive_identity) | product(A, h(A), multiplicative_identity))),
% 0.23/0.56 inference(rewrite,[status(thm)],[])).
% 0.23/0.56 tff(72,plain,
% 0.23/0.56 (^[A: $i] : rewrite((product(A, h(A), multiplicative_identity) | (A = additive_identity)) <=> ((A = additive_identity) | product(A, h(A), multiplicative_identity)))),
% 0.23/0.56 inference(bind,[status(th)],[])).
% 0.23/0.56 tff(73,plain,
% 0.23/0.56 (![A: $i] : (product(A, h(A), multiplicative_identity) | (A = additive_identity)) <=> ![A: $i] : ((A = additive_identity) | product(A, h(A), multiplicative_identity))),
% 0.23/0.56 inference(quant_intro,[status(thm)],[72])).
% 0.23/0.56 tff(74,axiom,(![A: $i] : (product(A, h(A), multiplicative_identity) | (A = additive_identity))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','clause41')).
% 0.23/0.56 tff(75,plain,
% 0.23/0.56 (![A: $i] : ((A = additive_identity) | product(A, h(A), multiplicative_identity))),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[74, 73])).
% 0.23/0.56 tff(76,plain,
% 0.23/0.56 (![A: $i] : ((A = additive_identity) | product(A, h(A), multiplicative_identity))),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[75, 71])).
% 0.23/0.56 tff(77,plain,(
% 0.23/0.56 ![A: $i] : ((A = additive_identity) | product(A, h(A), multiplicative_identity))),
% 0.23/0.56 inference(skolemize,[status(sab)],[76])).
% 0.23/0.56 tff(78,plain,
% 0.23/0.56 (![A: $i] : ((A = additive_identity) | product(A, h(A), multiplicative_identity))),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[77, 70])).
% 0.23/0.56 tff(79,plain,
% 0.23/0.56 (((~![A: $i] : ((A = additive_identity) | product(A, h(A), multiplicative_identity))) | ((b = additive_identity) | product(b, h(b), multiplicative_identity))) <=> ((~![A: $i] : ((A = additive_identity) | product(A, h(A), multiplicative_identity))) | (b = additive_identity) | product(b, h(b), multiplicative_identity))),
% 0.23/0.56 inference(rewrite,[status(thm)],[])).
% 0.23/0.56 tff(80,plain,
% 0.23/0.56 ((~![A: $i] : ((A = additive_identity) | product(A, h(A), multiplicative_identity))) | ((b = additive_identity) | product(b, h(b), multiplicative_identity))),
% 0.23/0.56 inference(quant_inst,[status(thm)],[])).
% 0.23/0.56 tff(81,plain,
% 0.23/0.56 ((~![A: $i] : ((A = additive_identity) | product(A, h(A), multiplicative_identity))) | (b = additive_identity) | product(b, h(b), multiplicative_identity)),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[80, 79])).
% 0.23/0.56 tff(82,plain,
% 0.23/0.56 (product(b, h(b), multiplicative_identity)),
% 0.23/0.56 inference(unit_resolution,[status(thm)],[81, 78, 68])).
% 0.23/0.56 tff(83,plain,
% 0.23/0.56 (product(b, h(add(additive_identity, b)), multiplicative_identity)),
% 0.23/0.56 inference(modus_ponens,[status(thm)],[82, 65])).
% 0.23/0.56 tff(84,plain,
% 0.23/0.56 (^[A: $i] : refl(product(A, multiplicative_identity, A) <=> product(A, multiplicative_identity, A))),
% 0.23/0.56 inference(bind,[status(th)],[])).
% 0.23/0.57 tff(85,plain,
% 0.23/0.57 (![A: $i] : product(A, multiplicative_identity, A) <=> ![A: $i] : product(A, multiplicative_identity, A)),
% 0.23/0.57 inference(quant_intro,[status(thm)],[84])).
% 0.23/0.57 tff(86,plain,
% 0.23/0.57 (![A: $i] : product(A, multiplicative_identity, A) <=> ![A: $i] : product(A, multiplicative_identity, A)),
% 0.23/0.57 inference(rewrite,[status(thm)],[])).
% 0.23/0.57 tff(87,axiom,(![A: $i] : product(A, multiplicative_identity, A)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','right_multiplicative_identity')).
% 0.23/0.57 tff(88,plain,
% 0.23/0.57 (![A: $i] : product(A, multiplicative_identity, A)),
% 0.23/0.57 inference(modus_ponens,[status(thm)],[87, 86])).
% 0.23/0.57 tff(89,plain,(
% 0.23/0.57 ![A: $i] : product(A, multiplicative_identity, A)),
% 0.23/0.57 inference(skolemize,[status(sab)],[88])).
% 0.23/0.57 tff(90,plain,
% 0.23/0.57 (![A: $i] : product(A, multiplicative_identity, A)),
% 0.23/0.57 inference(modus_ponens,[status(thm)],[89, 85])).
% 0.23/0.57 tff(91,plain,
% 0.23/0.57 ((~![A: $i] : product(A, multiplicative_identity, A)) | product(a, multiplicative_identity, a)),
% 0.23/0.57 inference(quant_inst,[status(thm)],[])).
% 0.23/0.57 tff(92,plain,
% 0.23/0.57 (product(a, multiplicative_identity, a)),
% 0.23/0.57 inference(unit_resolution,[status(thm)],[91, 90])).
% 0.23/0.57 tff(93,plain,
% 0.23/0.57 (product(a, b, additive_identity) <=> product(a, b, additive_identity)),
% 0.23/0.57 inference(rewrite,[status(thm)],[])).
% 0.23/0.57 tff(94,axiom,(product(a, b, additive_identity)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','a_times_b')).
% 0.23/0.57 tff(95,plain,
% 0.23/0.57 (product(a, b, additive_identity)),
% 0.23/0.57 inference(modus_ponens,[status(thm)],[94, 93])).
% 0.23/0.57 tff(96,plain,
% 0.23/0.57 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : refl((product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) <=> (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))))),
% 0.23/0.57 inference(bind,[status(th)],[])).
% 0.23/0.57 tff(97,plain,
% 0.23/0.57 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 0.23/0.57 inference(quant_intro,[status(thm)],[96])).
% 0.23/0.57 tff(98,plain,
% 0.23/0.57 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 0.23/0.57 inference(rewrite,[status(thm)],[])).
% 0.23/0.57 tff(99,plain,
% 0.23/0.57 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : trans(monotonicity(rewrite((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) <=> ((~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))), (((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W)) <=> (((~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) | product(U, Z, W)))), rewrite((((~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) | product(U, Z, W)) <=> (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))), (((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W)) <=> (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))))),
% 0.23/0.57 inference(bind,[status(th)],[])).
% 0.23/0.57 tff(100,plain,
% 0.23/0.57 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W)) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 0.23/0.57 inference(quant_intro,[status(thm)],[99])).
% 0.23/0.57 tff(101,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W))), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','associativity_of_multiplication2')).
% 0.23/0.57 tff(102,plain,
% 0.23/0.57 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 0.23/0.57 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.23/0.57 tff(103,plain,
% 0.23/0.57 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 0.23/0.57 inference(modus_ponens,[status(thm)],[102, 98])).
% 0.23/0.57 tff(104,plain,(
% 0.23/0.57 ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 0.23/0.57 inference(skolemize,[status(sab)],[103])).
% 0.23/0.57 tff(105,plain,
% 0.23/0.57 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 0.23/0.57 inference(modus_ponens,[status(thm)],[104, 97])).
% 0.23/0.57 tff(106,plain,
% 0.23/0.57 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(a, b, additive_identity)) | (~product(b, h(add(additive_identity, b)), multiplicative_identity)) | (~product(a, multiplicative_identity, a)) | product(additive_identity, h(add(additive_identity, b)), a))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(a, b, additive_identity)) | (~product(b, h(add(additive_identity, b)), multiplicative_identity)) | (~product(a, multiplicative_identity, a)) | product(additive_identity, h(add(additive_identity, b)), a))),
% 0.23/0.57 inference(rewrite,[status(thm)],[])).
% 0.23/0.57 tff(107,plain,
% 0.23/0.57 ((product(additive_identity, h(add(additive_identity, b)), a) | (~product(b, h(add(additive_identity, b)), multiplicative_identity)) | (~product(a, b, additive_identity)) | (~product(a, multiplicative_identity, a))) <=> ((~product(a, b, additive_identity)) | (~product(b, h(add(additive_identity, b)), multiplicative_identity)) | (~product(a, multiplicative_identity, a)) | product(additive_identity, h(add(additive_identity, b)), a))),
% 0.23/0.57 inference(rewrite,[status(thm)],[])).
% 0.23/0.57 tff(108,plain,
% 0.23/0.57 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(additive_identity, h(add(additive_identity, b)), a) | (~product(b, h(add(additive_identity, b)), multiplicative_identity)) | (~product(a, b, additive_identity)) | (~product(a, multiplicative_identity, a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(a, b, additive_identity)) | (~product(b, h(add(additive_identity, b)), multiplicative_identity)) | (~product(a, multiplicative_identity, a)) | product(additive_identity, h(add(additive_identity, b)), a)))),
% 0.23/0.57 inference(monotonicity,[status(thm)],[107])).
% 0.23/0.57 tff(109,plain,
% 0.23/0.57 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(additive_identity, h(add(additive_identity, b)), a) | (~product(b, h(add(additive_identity, b)), multiplicative_identity)) | (~product(a, b, additive_identity)) | (~product(a, multiplicative_identity, a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(a, b, additive_identity)) | (~product(b, h(add(additive_identity, b)), multiplicative_identity)) | (~product(a, multiplicative_identity, a)) | product(additive_identity, h(add(additive_identity, b)), a))),
% 0.23/0.57 inference(transitivity,[status(thm)],[108, 106])).
% 0.23/0.57 tff(110,plain,
% 0.23/0.57 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(additive_identity, h(add(additive_identity, b)), a) | (~product(b, h(add(additive_identity, b)), multiplicative_identity)) | (~product(a, b, additive_identity)) | (~product(a, multiplicative_identity, a)))),
% 0.23/0.57 inference(quant_inst,[status(thm)],[])).
% 0.23/0.57 tff(111,plain,
% 0.23/0.57 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(a, b, additive_identity)) | (~product(b, h(add(additive_identity, b)), multiplicative_identity)) | (~product(a, multiplicative_identity, a)) | product(additive_identity, h(add(additive_identity, b)), a)),
% 0.23/0.57 inference(modus_ponens,[status(thm)],[110, 109])).
% 0.23/0.57 tff(112,plain,
% 0.23/0.57 ($false),
% 0.23/0.57 inference(unit_resolution,[status(thm)],[111, 105, 95, 92, 83, 12])).
% 0.23/0.57 tff(113,plain,(product(additive_identity, h(add(additive_identity, b)), a)), inference(lemma,lemma(discharge,[]))).
% 0.23/0.57 tff(114,plain,
% 0.23/0.57 ((~(a = additive_identity)) <=> (~(a = additive_identity))),
% 0.23/0.57 inference(rewrite,[status(thm)],[])).
% 0.23/0.57 tff(115,axiom,(~(a = additive_identity)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','a_not_additive_identity')).
% 0.23/0.57 tff(116,plain,
% 0.23/0.57 (~(a = additive_identity)),
% 0.23/0.57 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.23/0.57 tff(117,plain,
% 0.23/0.57 (^[V: $i, Y: $i, U: $i, X: $i] : refl(((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V)) <=> ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V)))),
% 0.23/0.57 inference(bind,[status(th)],[])).
% 0.23/0.57 tff(118,plain,
% 0.23/0.57 (![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))),
% 0.23/0.57 inference(quant_intro,[status(thm)],[117])).
% 0.23/0.57 tff(119,plain,
% 0.23/0.57 (![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))),
% 0.23/0.57 inference(rewrite,[status(thm)],[])).
% 0.23/0.57 tff(120,plain,
% 0.23/0.57 (^[V: $i, Y: $i, U: $i, X: $i] : rewrite((((~product(X, Y, U)) | (~product(X, Y, V))) | (U = V)) <=> ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V)))),
% 0.23/0.57 inference(bind,[status(th)],[])).
% 0.23/0.57 tff(121,plain,
% 0.23/0.57 (![V: $i, Y: $i, U: $i, X: $i] : (((~product(X, Y, U)) | (~product(X, Y, V))) | (U = V)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))),
% 0.23/0.57 inference(quant_intro,[status(thm)],[120])).
% 0.23/0.57 tff(122,axiom,(![V: $i, Y: $i, U: $i, X: $i] : (((~product(X, Y, U)) | (~product(X, Y, V))) | (U = V))), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','multiplication_is_well_defined')).
% 0.23/0.57 tff(123,plain,
% 0.23/0.57 (![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))),
% 0.23/0.57 inference(modus_ponens,[status(thm)],[122, 121])).
% 0.23/0.57 tff(124,plain,
% 0.23/0.57 (![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))),
% 0.23/0.57 inference(modus_ponens,[status(thm)],[123, 119])).
% 0.23/0.57 tff(125,plain,(
% 0.23/0.57 ![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))),
% 0.23/0.57 inference(skolemize,[status(sab)],[124])).
% 0.23/0.57 tff(126,plain,
% 0.23/0.57 (![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))),
% 0.23/0.57 inference(modus_ponens,[status(thm)],[125, 118])).
% 0.23/0.57 tff(127,plain,
% 0.23/0.57 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))) | ((a = additive_identity) | (~product(additive_identity, h(add(additive_identity, b)), a)) | (~product(additive_identity, h(add(additive_identity, b)), additive_identity)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))) | (a = additive_identity) | (~product(additive_identity, h(add(additive_identity, b)), a)) | (~product(additive_identity, h(add(additive_identity, b)), additive_identity)))),
% 0.23/0.57 inference(rewrite,[status(thm)],[])).
% 0.23/0.57 tff(128,plain,
% 0.23/0.57 (((~product(additive_identity, h(add(additive_identity, b)), additive_identity)) | (~product(additive_identity, h(add(additive_identity, b)), a)) | (a = additive_identity)) <=> ((a = additive_identity) | (~product(additive_identity, h(add(additive_identity, b)), a)) | (~product(additive_identity, h(add(additive_identity, b)), additive_identity)))),
% 0.23/0.57 inference(rewrite,[status(thm)],[])).
% 0.23/0.57 tff(129,plain,
% 0.23/0.57 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))) | ((~product(additive_identity, h(add(additive_identity, b)), additive_identity)) | (~product(additive_identity, h(add(additive_identity, b)), a)) | (a = additive_identity))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))) | ((a = additive_identity) | (~product(additive_identity, h(add(additive_identity, b)), a)) | (~product(additive_identity, h(add(additive_identity, b)), additive_identity))))),
% 0.23/0.57 inference(monotonicity,[status(thm)],[128])).
% 0.23/0.57 tff(130,plain,
% 0.23/0.57 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))) | ((~product(additive_identity, h(add(additive_identity, b)), additive_identity)) | (~product(additive_identity, h(add(additive_identity, b)), a)) | (a = additive_identity))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))) | (a = additive_identity) | (~product(additive_identity, h(add(additive_identity, b)), a)) | (~product(additive_identity, h(add(additive_identity, b)), additive_identity)))),
% 0.23/0.57 inference(transitivity,[status(thm)],[129, 127])).
% 0.23/0.57 tff(131,plain,
% 0.23/0.57 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))) | ((~product(additive_identity, h(add(additive_identity, b)), additive_identity)) | (~product(additive_identity, h(add(additive_identity, b)), a)) | (a = additive_identity))),
% 0.23/0.57 inference(quant_inst,[status(thm)],[])).
% 0.23/0.57 tff(132,plain,
% 0.23/0.57 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))) | (a = additive_identity) | (~product(additive_identity, h(add(additive_identity, b)), a)) | (~product(additive_identity, h(add(additive_identity, b)), additive_identity))),
% 0.23/0.57 inference(modus_ponens,[status(thm)],[131, 130])).
% 0.23/0.57 tff(133,plain,
% 0.23/0.57 ($false),
% 0.23/0.57 inference(unit_resolution,[status(thm)],[132, 126, 116, 113, 11])).
% 0.23/0.57 % SZS output end Proof
% 0.23/0.57 % E exiting
%------------------------------------------------------------------------------