%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : BOO011-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:00:07 AM UTC 2025
% Result : Unsatisfiable 0.21s 0.43s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.14 % Problem : BOO011-1 : TPTP v9.0.0. Released v1.0.0.
% 0.08/0.14 % Command : run_E %s %d THM
% 0.14/0.35 % Computer : n003.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % 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 : Thu Jun 19 11:38:24 EDT 2025
% 0.14/0.36 % CPUTime :
% 0.21/0.43 % SZS status Unsatisfiable
% 0.21/0.43 % SZS output start Proof
% 0.21/0.43 tff(sum_type, type, (
% 0.21/0.43 sum: ( $i * $i * $i ) > $o)).
% 0.21/0.43 tff(inverse_type, type, (
% 0.21/0.43 inverse: $i > $i)).
% 0.21/0.43 tff(additive_identity_type, type, (
% 0.21/0.43 additive_identity: $i)).
% 0.21/0.43 tff(add_type, type, (
% 0.21/0.43 add: ( $i * $i ) > $i)).
% 0.21/0.43 tff(multiplicative_identity_type, type, (
% 0.21/0.43 multiplicative_identity: $i)).
% 0.21/0.43 tff(1,plain,
% 0.21/0.43 (^[Y: $i, X: $i] : refl(sum(X, Y, add(X, Y)) <=> sum(X, Y, add(X, Y)))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(2,plain,
% 0.21/0.43 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[1])).
% 0.21/0.43 tff(3,plain,
% 0.21/0.43 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(4,axiom,(![Y: $i, X: $i] : sum(X, Y, add(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','closure_of_addition')).
% 0.21/0.43 tff(5,plain,
% 0.21/0.43 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.21/0.43 tff(6,plain,(
% 0.21/0.43 ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.21/0.43 inference(skolemize,[status(sab)],[5])).
% 0.21/0.43 tff(7,plain,
% 0.21/0.43 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.21/0.43 tff(8,plain,
% 0.21/0.43 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(additive_identity, additive_identity, add(additive_identity, additive_identity))),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(9,plain,
% 0.21/0.43 (sum(additive_identity, additive_identity, add(additive_identity, additive_identity))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.21/0.43 tff(10,assumption,(~sum(additive_identity, additive_identity, additive_identity)), introduced(assumption)).
% 0.21/0.43 tff(11,plain,
% 0.21/0.43 (^[X: $i] : refl(sum(X, additive_identity, X) <=> sum(X, additive_identity, X))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(12,plain,
% 0.21/0.43 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 0.21/0.43 inference(quant_intro,[status(thm)],[11])).
% 0.21/0.43 tff(13,plain,
% 0.21/0.43 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(14,axiom,(![X: $i] : sum(X, additive_identity, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_identity2')).
% 0.21/0.43 tff(15,plain,
% 0.21/0.43 (![X: $i] : sum(X, additive_identity, X)),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.21/0.43 tff(16,plain,(
% 0.21/0.43 ![X: $i] : sum(X, additive_identity, X)),
% 0.21/0.43 inference(skolemize,[status(sab)],[15])).
% 0.21/0.43 tff(17,plain,
% 0.21/0.43 (![X: $i] : sum(X, additive_identity, X)),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.21/0.43 tff(18,plain,
% 0.21/0.43 ((~![X: $i] : sum(X, additive_identity, X)) | sum(additive_identity, additive_identity, additive_identity)),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(19,plain,
% 0.21/0.43 ($false),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[18, 17, 10])).
% 0.21/0.43 tff(20,plain,(sum(additive_identity, additive_identity, additive_identity)), inference(lemma,lemma(discharge,[]))).
% 0.21/0.43 tff(21,plain,
% 0.21/0.43 (^[V: $i, Y: $i, U: $i, X: $i] : refl(((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))) <=> ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(22,plain,
% 0.21/0.43 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[21])).
% 0.21/0.43 tff(23,plain,
% 0.21/0.43 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(24,plain,
% 0.21/0.43 (^[V: $i, Y: $i, U: $i, X: $i] : rewrite((((~sum(X, Y, U)) | (~sum(X, Y, V))) | (U = V)) <=> ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(25,plain,
% 0.21/0.43 (![V: $i, Y: $i, U: $i, X: $i] : (((~sum(X, Y, U)) | (~sum(X, Y, V))) | (U = V)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[24])).
% 0.21/0.43 tff(26,axiom,(![V: $i, Y: $i, U: $i, X: $i] : (((~sum(X, Y, U)) | (~sum(X, Y, V))) | (U = V))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','addition_is_well_defined')).
% 0.21/0.43 tff(27,plain,
% 0.21/0.43 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.21/0.43 tff(28,plain,
% 0.21/0.43 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[27, 23])).
% 0.21/0.43 tff(29,plain,(
% 0.21/0.43 ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 0.21/0.43 inference(skolemize,[status(sab)],[28])).
% 0.21/0.43 tff(30,plain,
% 0.21/0.43 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[29, 22])).
% 0.21/0.43 tff(31,plain,
% 0.21/0.43 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((~sum(additive_identity, additive_identity, additive_identity)) | (additive_identity = add(additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, add(additive_identity, additive_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(additive_identity, additive_identity, additive_identity)) | (additive_identity = add(additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, add(additive_identity, additive_identity))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(32,plain,
% 0.21/0.43 (((additive_identity = add(additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, add(additive_identity, additive_identity))) | (~sum(additive_identity, additive_identity, additive_identity))) <=> ((~sum(additive_identity, additive_identity, additive_identity)) | (additive_identity = add(additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, add(additive_identity, additive_identity))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(33,plain,
% 0.21/0.43 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((additive_identity = add(additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, add(additive_identity, additive_identity))) | (~sum(additive_identity, additive_identity, additive_identity)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((~sum(additive_identity, additive_identity, additive_identity)) | (additive_identity = add(additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, add(additive_identity, additive_identity)))))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[32])).
% 0.21/0.43 tff(34,plain,
% 0.21/0.43 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((additive_identity = add(additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, add(additive_identity, additive_identity))) | (~sum(additive_identity, additive_identity, additive_identity)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(additive_identity, additive_identity, additive_identity)) | (additive_identity = add(additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, add(additive_identity, additive_identity))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[33, 31])).
% 0.21/0.43 tff(35,plain,
% 0.21/0.43 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((additive_identity = add(additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, add(additive_identity, additive_identity))) | (~sum(additive_identity, additive_identity, additive_identity)))),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(36,plain,
% 0.21/0.43 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(additive_identity, additive_identity, additive_identity)) | (additive_identity = add(additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, add(additive_identity, additive_identity)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.21/0.43 tff(37,plain,
% 0.21/0.43 (additive_identity = add(additive_identity, additive_identity)),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[36, 30, 20, 9])).
% 0.21/0.44 tff(38,plain,
% 0.21/0.44 (add(additive_identity, additive_identity) = additive_identity),
% 0.21/0.44 inference(symmetry,[status(thm)],[37])).
% 0.21/0.44 tff(39,plain,
% 0.21/0.44 (sum(add(additive_identity, additive_identity), inverse(additive_identity), inverse(additive_identity)) <=> sum(additive_identity, inverse(additive_identity), inverse(additive_identity))),
% 0.21/0.44 inference(monotonicity,[status(thm)],[38])).
% 0.21/0.44 tff(40,plain,
% 0.21/0.44 (sum(additive_identity, inverse(additive_identity), inverse(additive_identity)) <=> sum(add(additive_identity, additive_identity), inverse(additive_identity), inverse(additive_identity))),
% 0.21/0.44 inference(symmetry,[status(thm)],[39])).
% 0.21/0.44 tff(41,assumption,(~sum(additive_identity, inverse(additive_identity), inverse(additive_identity))), introduced(assumption)).
% 0.21/0.44 tff(42,plain,
% 0.21/0.44 (^[X: $i] : refl(sum(additive_identity, X, X) <=> sum(additive_identity, X, X))),
% 0.21/0.44 inference(bind,[status(th)],[])).
% 0.21/0.44 tff(43,plain,
% 0.21/0.44 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 0.21/0.44 inference(quant_intro,[status(thm)],[42])).
% 0.21/0.44 tff(44,plain,
% 0.21/0.44 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(45,axiom,(![X: $i] : sum(additive_identity, X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_identity1')).
% 0.21/0.44 tff(46,plain,
% 0.21/0.44 (![X: $i] : sum(additive_identity, X, X)),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.21/0.44 tff(47,plain,(
% 0.21/0.44 ![X: $i] : sum(additive_identity, X, X)),
% 0.21/0.44 inference(skolemize,[status(sab)],[46])).
% 0.21/0.44 tff(48,plain,
% 0.21/0.44 (![X: $i] : sum(additive_identity, X, X)),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[47, 43])).
% 0.21/0.44 tff(49,plain,
% 0.21/0.44 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, inverse(additive_identity), inverse(additive_identity))),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(50,plain,
% 0.21/0.44 ($false),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[49, 48, 41])).
% 0.21/0.44 tff(51,plain,(sum(additive_identity, inverse(additive_identity), inverse(additive_identity))), inference(lemma,lemma(discharge,[]))).
% 0.21/0.44 tff(52,plain,
% 0.21/0.44 (sum(add(additive_identity, additive_identity), inverse(additive_identity), inverse(additive_identity))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[51, 40])).
% 0.21/0.44 tff(53,plain,
% 0.21/0.44 (sum(add(additive_identity, additive_identity), inverse(additive_identity), multiplicative_identity) <=> sum(additive_identity, inverse(additive_identity), multiplicative_identity)),
% 0.21/0.44 inference(monotonicity,[status(thm)],[38])).
% 0.21/0.44 tff(54,plain,
% 0.21/0.44 (sum(additive_identity, inverse(additive_identity), multiplicative_identity) <=> sum(add(additive_identity, additive_identity), inverse(additive_identity), multiplicative_identity)),
% 0.21/0.44 inference(symmetry,[status(thm)],[53])).
% 0.21/0.44 tff(55,plain,
% 0.21/0.44 (^[X: $i] : refl(sum(X, inverse(X), multiplicative_identity) <=> sum(X, inverse(X), multiplicative_identity))),
% 0.21/0.44 inference(bind,[status(th)],[])).
% 0.21/0.44 tff(56,plain,
% 0.21/0.44 (![X: $i] : sum(X, inverse(X), multiplicative_identity) <=> ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 0.21/0.44 inference(quant_intro,[status(thm)],[55])).
% 0.21/0.44 tff(57,plain,
% 0.21/0.44 (![X: $i] : sum(X, inverse(X), multiplicative_identity) <=> ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(58,axiom,(![X: $i] : sum(X, inverse(X), multiplicative_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_inverse2')).
% 0.21/0.44 tff(59,plain,
% 0.21/0.44 (![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.21/0.44 tff(60,plain,(
% 0.21/0.44 ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 0.21/0.44 inference(skolemize,[status(sab)],[59])).
% 0.21/0.44 tff(61,plain,
% 0.21/0.44 (![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[60, 56])).
% 0.21/0.44 tff(62,plain,
% 0.21/0.44 ((~![X: $i] : sum(X, inverse(X), multiplicative_identity)) | sum(additive_identity, inverse(additive_identity), multiplicative_identity)),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(63,plain,
% 0.21/0.44 (sum(additive_identity, inverse(additive_identity), multiplicative_identity)),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[62, 61])).
% 0.21/0.44 tff(64,plain,
% 0.21/0.44 (sum(add(additive_identity, additive_identity), inverse(additive_identity), multiplicative_identity)),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[63, 54])).
% 0.21/0.44 tff(65,plain,
% 0.21/0.44 ((~(inverse(additive_identity) = multiplicative_identity)) <=> (~(inverse(additive_identity) = multiplicative_identity))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(66,axiom,(~(inverse(additive_identity) = multiplicative_identity)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_equation')).
% 0.21/0.44 tff(67,plain,
% 0.21/0.44 (~(inverse(additive_identity) = multiplicative_identity)),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.21/0.44 tff(68,plain,
% 0.21/0.44 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((inverse(additive_identity) = multiplicative_identity) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), multiplicative_identity)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (inverse(additive_identity) = multiplicative_identity) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), multiplicative_identity)))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(69,plain,
% 0.21/0.44 (((inverse(additive_identity) = multiplicative_identity) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), multiplicative_identity)) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), inverse(additive_identity)))) <=> ((inverse(additive_identity) = multiplicative_identity) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), multiplicative_identity)))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(70,plain,
% 0.21/0.44 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((inverse(additive_identity) = multiplicative_identity) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), multiplicative_identity)) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), inverse(additive_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((inverse(additive_identity) = multiplicative_identity) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), multiplicative_identity))))),
% 0.21/0.44 inference(monotonicity,[status(thm)],[69])).
% 0.21/0.44 tff(71,plain,
% 0.21/0.44 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((inverse(additive_identity) = multiplicative_identity) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), multiplicative_identity)) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), inverse(additive_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (inverse(additive_identity) = multiplicative_identity) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), multiplicative_identity)))),
% 0.21/0.44 inference(transitivity,[status(thm)],[70, 68])).
% 0.21/0.44 tff(72,plain,
% 0.21/0.44 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((inverse(additive_identity) = multiplicative_identity) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), multiplicative_identity)) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), inverse(additive_identity))))),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(73,plain,
% 0.21/0.44 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (inverse(additive_identity) = multiplicative_identity) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), multiplicative_identity))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[72, 71])).
% 0.21/0.44 tff(74,plain,
% 0.21/0.44 ((~sum(add(additive_identity, additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~sum(add(additive_identity, additive_identity), inverse(additive_identity), multiplicative_identity))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[73, 30, 67])).
% 0.21/0.44 tff(75,plain,
% 0.21/0.44 ($false),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[74, 64, 52])).
% 0.21/0.44 % SZS output end Proof
% 0.21/0.44 % E exiting
%------------------------------------------------------------------------------