%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : RNG038-2 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n014.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:23:58 AM UTC 2025
% Result : Unsatisfiable 0.47s 0.63s
% Output : Proof 0.47s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : RNG038-2 : TPTP v9.0.0. Released v1.0.0.
% 0.10/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n014.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 03:15:54 EDT 2025
% 0.17/0.33 % CPUTime :
% 0.47/0.63 % SZS status Unsatisfiable
% 0.47/0.63 % SZS output start Proof
% 0.47/0.63 tff(product_type, type, (
% 0.47/0.63 product: ( $i * $i * $i ) > $o)).
% 0.47/0.63 tff(additive_identity_type, type, (
% 0.47/0.63 additive_identity: $i)).
% 0.47/0.63 tff(h_type, type, (
% 0.47/0.63 h: ( $i * $i ) > $i)).
% 0.47/0.63 tff(b_type, type, (
% 0.47/0.63 b: $i)).
% 0.47/0.63 tff(multiply_type, type, (
% 0.47/0.63 multiply: ( $i * $i ) > $i)).
% 0.47/0.63 tff(a_type, type, (
% 0.47/0.63 a: $i)).
% 0.47/0.63 tff(equalish_type, type, (
% 0.47/0.63 equalish: ( $i * $i ) > $o)).
% 0.47/0.63 tff(sum_type, type, (
% 0.47/0.63 sum: ( $i * $i * $i ) > $o)).
% 0.47/0.63 tff(additive_inverse_type, type, (
% 0.47/0.63 additive_inverse: $i > $i)).
% 0.47/0.63 tff(1,assumption,(~product(additive_identity, a, additive_identity)), introduced(assumption)).
% 0.47/0.63 tff(2,plain,
% 0.47/0.63 (^[X: $i] : refl(product(additive_identity, X, additive_identity) <=> product(additive_identity, X, additive_identity))),
% 0.47/0.63 inference(bind,[status(th)],[])).
% 0.47/0.63 tff(3,plain,
% 0.47/0.63 (![X: $i] : product(additive_identity, X, additive_identity) <=> ![X: $i] : product(additive_identity, X, additive_identity)),
% 0.47/0.63 inference(quant_intro,[status(thm)],[2])).
% 0.47/0.63 tff(4,plain,
% 0.47/0.63 (![X: $i] : product(additive_identity, X, additive_identity) <=> ![X: $i] : product(additive_identity, X, additive_identity)),
% 0.47/0.63 inference(rewrite,[status(thm)],[])).
% 0.47/0.63 tff(5,axiom,(![X: $i] : product(additive_identity, X, additive_identity)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','multiplicative_identity1')).
% 0.47/0.63 tff(6,plain,
% 0.47/0.63 (![X: $i] : product(additive_identity, X, additive_identity)),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.47/0.63 tff(7,plain,(
% 0.47/0.63 ![X: $i] : product(additive_identity, X, additive_identity)),
% 0.47/0.63 inference(skolemize,[status(sab)],[6])).
% 0.47/0.63 tff(8,plain,
% 0.47/0.63 (![X: $i] : product(additive_identity, X, additive_identity)),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.47/0.63 tff(9,plain,
% 0.47/0.63 ((~![X: $i] : product(additive_identity, X, additive_identity)) | product(additive_identity, a, additive_identity)),
% 0.47/0.63 inference(quant_inst,[status(thm)],[])).
% 0.47/0.63 tff(10,plain,
% 0.47/0.63 ($false),
% 0.47/0.63 inference(unit_resolution,[status(thm)],[9, 8, 1])).
% 0.47/0.63 tff(11,plain,(product(additive_identity, a, additive_identity)), inference(lemma,lemma(discharge,[]))).
% 0.47/0.63 tff(12,plain,
% 0.47/0.63 (^[Y: $i, X: $i] : refl(product(X, Y, multiply(X, Y)) <=> product(X, Y, multiply(X, Y)))),
% 0.47/0.63 inference(bind,[status(th)],[])).
% 0.47/0.63 tff(13,plain,
% 0.47/0.63 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.47/0.63 inference(quant_intro,[status(thm)],[12])).
% 0.47/0.63 tff(14,plain,
% 0.47/0.63 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.47/0.63 inference(rewrite,[status(thm)],[])).
% 0.47/0.63 tff(15,axiom,(![Y: $i, X: $i] : product(X, Y, multiply(X, Y))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','closure_of_multiplication')).
% 0.47/0.63 tff(16,plain,
% 0.47/0.63 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.47/0.63 tff(17,plain,(
% 0.47/0.63 ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.47/0.63 inference(skolemize,[status(sab)],[16])).
% 0.47/0.63 tff(18,plain,
% 0.47/0.63 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.47/0.63 tff(19,plain,
% 0.47/0.63 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(additive_identity, a, multiply(additive_identity, a))),
% 0.47/0.63 inference(quant_inst,[status(thm)],[])).
% 0.47/0.63 tff(20,plain,
% 0.47/0.63 (product(additive_identity, a, multiply(additive_identity, a))),
% 0.47/0.63 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.47/0.63 tff(21,plain,
% 0.47/0.63 (^[V: $i, Y: $i, U: $i, X: $i] : refl(((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V)) <=> ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V)))),
% 0.47/0.63 inference(bind,[status(th)],[])).
% 0.47/0.63 tff(22,plain,
% 0.47/0.63 (![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))),
% 0.47/0.63 inference(quant_intro,[status(thm)],[21])).
% 0.47/0.63 tff(23,plain,
% 0.47/0.63 (![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))),
% 0.47/0.63 inference(rewrite,[status(thm)],[])).
% 0.47/0.63 tff(24,plain,
% 0.47/0.63 (^[V: $i, Y: $i, U: $i, X: $i] : rewrite((((~product(X, Y, U)) | (~product(X, Y, V))) | equalish(U, V)) <=> ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V)))),
% 0.47/0.63 inference(bind,[status(th)],[])).
% 0.47/0.63 tff(25,plain,
% 0.47/0.63 (![V: $i, Y: $i, U: $i, X: $i] : (((~product(X, Y, U)) | (~product(X, Y, V))) | equalish(U, V)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))),
% 0.47/0.63 inference(quant_intro,[status(thm)],[24])).
% 0.47/0.63 tff(26,axiom,(![V: $i, Y: $i, U: $i, X: $i] : (((~product(X, Y, U)) | (~product(X, Y, V))) | equalish(U, V))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','multiplication_is_well_defined')).
% 0.47/0.63 tff(27,plain,
% 0.47/0.63 (![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.47/0.63 tff(28,plain,
% 0.47/0.63 (![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[27, 23])).
% 0.47/0.63 tff(29,plain,(
% 0.47/0.63 ![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))),
% 0.47/0.63 inference(skolemize,[status(sab)],[28])).
% 0.47/0.63 tff(30,plain,
% 0.47/0.63 (![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[29, 22])).
% 0.47/0.63 tff(31,plain,
% 0.47/0.63 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))) | ((~product(additive_identity, a, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | equalish(multiply(additive_identity, a), additive_identity))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))) | (~product(additive_identity, a, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | equalish(multiply(additive_identity, a), additive_identity))),
% 0.47/0.63 inference(rewrite,[status(thm)],[])).
% 0.47/0.63 tff(32,plain,
% 0.47/0.63 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))) | ((~product(additive_identity, a, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | equalish(multiply(additive_identity, a), additive_identity))),
% 0.47/0.63 inference(quant_inst,[status(thm)],[])).
% 0.47/0.63 tff(33,plain,
% 0.47/0.63 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))) | (~product(additive_identity, a, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | equalish(multiply(additive_identity, a), additive_identity)),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.47/0.63 tff(34,plain,
% 0.47/0.63 (equalish(multiply(additive_identity, a), additive_identity)),
% 0.47/0.63 inference(unit_resolution,[status(thm)],[33, 30, 20, 11])).
% 0.47/0.63 tff(35,plain,
% 0.47/0.63 (^[Y: $i, X: $i] : refl(((~equalish(X, additive_identity)) | product(X, h(X, Y), Y)) <=> ((~equalish(X, additive_identity)) | product(X, h(X, Y), Y)))),
% 0.47/0.63 inference(bind,[status(th)],[])).
% 0.47/0.63 tff(36,plain,
% 0.47/0.63 (![Y: $i, X: $i] : ((~equalish(X, additive_identity)) | product(X, h(X, Y), Y)) <=> ![Y: $i, X: $i] : ((~equalish(X, additive_identity)) | product(X, h(X, Y), Y))),
% 0.47/0.63 inference(quant_intro,[status(thm)],[35])).
% 0.47/0.63 tff(37,plain,
% 0.47/0.63 (![Y: $i, X: $i] : ((~equalish(X, additive_identity)) | product(X, h(X, Y), Y)) <=> ![Y: $i, X: $i] : ((~equalish(X, additive_identity)) | product(X, h(X, Y), Y))),
% 0.47/0.63 inference(rewrite,[status(thm)],[])).
% 0.47/0.63 tff(38,axiom,(![Y: $i, X: $i] : ((~equalish(X, additive_identity)) | product(X, h(X, Y), Y))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','some_property')).
% 0.47/0.63 tff(39,plain,
% 0.47/0.63 (![Y: $i, X: $i] : ((~equalish(X, additive_identity)) | product(X, h(X, Y), Y))),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.47/0.63 tff(40,plain,(
% 0.47/0.63 ![Y: $i, X: $i] : ((~equalish(X, additive_identity)) | product(X, h(X, Y), Y))),
% 0.47/0.63 inference(skolemize,[status(sab)],[39])).
% 0.47/0.63 tff(41,plain,
% 0.47/0.63 (![Y: $i, X: $i] : ((~equalish(X, additive_identity)) | product(X, h(X, Y), Y))),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[40, 36])).
% 0.47/0.63 tff(42,plain,
% 0.47/0.63 (((~![Y: $i, X: $i] : ((~equalish(X, additive_identity)) | product(X, h(X, Y), Y))) | ((~equalish(multiply(additive_identity, a), additive_identity)) | product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b))) <=> ((~![Y: $i, X: $i] : ((~equalish(X, additive_identity)) | product(X, h(X, Y), Y))) | (~equalish(multiply(additive_identity, a), additive_identity)) | product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b))),
% 0.47/0.63 inference(rewrite,[status(thm)],[])).
% 0.47/0.63 tff(43,plain,
% 0.47/0.63 ((~![Y: $i, X: $i] : ((~equalish(X, additive_identity)) | product(X, h(X, Y), Y))) | ((~equalish(multiply(additive_identity, a), additive_identity)) | product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b))),
% 0.47/0.63 inference(quant_inst,[status(thm)],[])).
% 0.47/0.63 tff(44,plain,
% 0.47/0.63 ((~![Y: $i, X: $i] : ((~equalish(X, additive_identity)) | product(X, h(X, Y), Y))) | (~equalish(multiply(additive_identity, a), additive_identity)) | product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b)),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.47/0.63 tff(45,plain,
% 0.47/0.63 (product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b)),
% 0.47/0.63 inference(unit_resolution,[status(thm)],[44, 41, 34])).
% 0.47/0.63 tff(46,plain,
% 0.47/0.63 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))) | ((~product(additive_identity, a, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | equalish(additive_identity, multiply(additive_identity, a)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))) | (~product(additive_identity, a, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | equalish(additive_identity, multiply(additive_identity, a)))),
% 0.47/0.63 inference(rewrite,[status(thm)],[])).
% 0.47/0.63 tff(47,plain,
% 0.47/0.63 (((~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, a, additive_identity)) | equalish(additive_identity, multiply(additive_identity, a))) <=> ((~product(additive_identity, a, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | equalish(additive_identity, multiply(additive_identity, a)))),
% 0.47/0.63 inference(rewrite,[status(thm)],[])).
% 0.47/0.63 tff(48,plain,
% 0.47/0.63 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))) | ((~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, a, additive_identity)) | equalish(additive_identity, multiply(additive_identity, a)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))) | ((~product(additive_identity, a, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | equalish(additive_identity, multiply(additive_identity, a))))),
% 0.47/0.63 inference(monotonicity,[status(thm)],[47])).
% 0.47/0.63 tff(49,plain,
% 0.47/0.63 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))) | ((~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, a, additive_identity)) | equalish(additive_identity, multiply(additive_identity, a)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))) | (~product(additive_identity, a, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | equalish(additive_identity, multiply(additive_identity, a)))),
% 0.47/0.63 inference(transitivity,[status(thm)],[48, 46])).
% 0.47/0.63 tff(50,plain,
% 0.47/0.63 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))) | ((~product(additive_identity, a, multiply(additive_identity, a))) | (~product(additive_identity, a, additive_identity)) | equalish(additive_identity, multiply(additive_identity, a)))),
% 0.47/0.63 inference(quant_inst,[status(thm)],[])).
% 0.47/0.63 tff(51,plain,
% 0.47/0.63 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | equalish(U, V))) | (~product(additive_identity, a, additive_identity)) | (~product(additive_identity, a, multiply(additive_identity, a))) | equalish(additive_identity, multiply(additive_identity, a))),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.47/0.63 tff(52,plain,
% 0.47/0.63 (equalish(additive_identity, multiply(additive_identity, a))),
% 0.47/0.63 inference(unit_resolution,[status(thm)],[51, 30, 20, 11])).
% 0.47/0.63 tff(53,assumption,(~sum(additive_identity, additive_identity, additive_identity)), introduced(assumption)).
% 0.47/0.63 tff(54,plain,
% 0.47/0.63 (^[X: $i] : refl(sum(X, additive_identity, X) <=> sum(X, additive_identity, X))),
% 0.47/0.63 inference(bind,[status(th)],[])).
% 0.47/0.63 tff(55,plain,
% 0.47/0.63 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 0.47/0.63 inference(quant_intro,[status(thm)],[54])).
% 0.47/0.63 tff(56,plain,
% 0.47/0.63 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 0.47/0.63 inference(rewrite,[status(thm)],[])).
% 0.47/0.63 tff(57,axiom,(![X: $i] : sum(X, additive_identity, X)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','additive_identity2')).
% 0.47/0.63 tff(58,plain,
% 0.47/0.63 (![X: $i] : sum(X, additive_identity, X)),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.47/0.63 tff(59,plain,(
% 0.47/0.63 ![X: $i] : sum(X, additive_identity, X)),
% 0.47/0.63 inference(skolemize,[status(sab)],[58])).
% 0.47/0.63 tff(60,plain,
% 0.47/0.63 (![X: $i] : sum(X, additive_identity, X)),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[59, 55])).
% 0.47/0.63 tff(61,plain,
% 0.47/0.63 ((~![X: $i] : sum(X, additive_identity, X)) | sum(additive_identity, additive_identity, additive_identity)),
% 0.47/0.63 inference(quant_inst,[status(thm)],[])).
% 0.47/0.63 tff(62,plain,
% 0.47/0.63 ($false),
% 0.47/0.63 inference(unit_resolution,[status(thm)],[61, 60, 53])).
% 0.47/0.63 tff(63,plain,(sum(additive_identity, additive_identity, additive_identity)), inference(lemma,lemma(discharge,[]))).
% 0.47/0.63 tff(64,plain,
% 0.47/0.63 (^[W: $i, Z: $i, Y: $i, X: $i] : refl(((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X))) <=> ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X))))),
% 0.47/0.63 inference(bind,[status(th)],[])).
% 0.47/0.63 tff(65,plain,
% 0.47/0.63 (![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))),
% 0.47/0.63 inference(quant_intro,[status(thm)],[64])).
% 0.47/0.63 tff(66,plain,
% 0.47/0.63 (![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))),
% 0.47/0.63 inference(rewrite,[status(thm)],[])).
% 0.47/0.63 tff(67,plain,
% 0.47/0.63 (^[W: $i, Z: $i, Y: $i, X: $i] : rewrite((((~equalish(X, Y)) | (~sum(W, Z, X))) | sum(W, Z, Y)) <=> ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X))))),
% 0.47/0.63 inference(bind,[status(th)],[])).
% 0.47/0.63 tff(68,plain,
% 0.47/0.63 (![W: $i, Z: $i, Y: $i, X: $i] : (((~equalish(X, Y)) | (~sum(W, Z, X))) | sum(W, Z, Y)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))),
% 0.47/0.63 inference(quant_intro,[status(thm)],[67])).
% 0.47/0.63 tff(69,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~equalish(X, Y)) | (~sum(W, Z, X))) | sum(W, Z, Y))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','sum_substitution3')).
% 0.47/0.63 tff(70,plain,
% 0.47/0.63 (![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.47/0.63 tff(71,plain,
% 0.47/0.63 (![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[70, 66])).
% 0.47/0.63 tff(72,plain,(
% 0.47/0.63 ![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))),
% 0.47/0.63 inference(skolemize,[status(sab)],[71])).
% 0.47/0.63 tff(73,plain,
% 0.47/0.63 (![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[72, 65])).
% 0.47/0.63 tff(74,plain,
% 0.47/0.63 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))) | ((~sum(additive_identity, additive_identity, additive_identity)) | sum(additive_identity, additive_identity, multiply(additive_identity, a)) | (~equalish(additive_identity, multiply(additive_identity, a))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))) | (~sum(additive_identity, additive_identity, additive_identity)) | sum(additive_identity, additive_identity, multiply(additive_identity, a)) | (~equalish(additive_identity, multiply(additive_identity, a))))),
% 0.47/0.63 inference(rewrite,[status(thm)],[])).
% 0.47/0.63 tff(75,plain,
% 0.47/0.63 (((~equalish(additive_identity, multiply(additive_identity, a))) | sum(additive_identity, additive_identity, multiply(additive_identity, a)) | (~sum(additive_identity, additive_identity, additive_identity))) <=> ((~sum(additive_identity, additive_identity, additive_identity)) | sum(additive_identity, additive_identity, multiply(additive_identity, a)) | (~equalish(additive_identity, multiply(additive_identity, a))))),
% 0.47/0.63 inference(rewrite,[status(thm)],[])).
% 0.47/0.63 tff(76,plain,
% 0.47/0.63 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))) | ((~equalish(additive_identity, multiply(additive_identity, a))) | sum(additive_identity, additive_identity, multiply(additive_identity, a)) | (~sum(additive_identity, additive_identity, additive_identity)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))) | ((~sum(additive_identity, additive_identity, additive_identity)) | sum(additive_identity, additive_identity, multiply(additive_identity, a)) | (~equalish(additive_identity, multiply(additive_identity, a)))))),
% 0.47/0.63 inference(monotonicity,[status(thm)],[75])).
% 0.47/0.63 tff(77,plain,
% 0.47/0.63 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))) | ((~equalish(additive_identity, multiply(additive_identity, a))) | sum(additive_identity, additive_identity, multiply(additive_identity, a)) | (~sum(additive_identity, additive_identity, additive_identity)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))) | (~sum(additive_identity, additive_identity, additive_identity)) | sum(additive_identity, additive_identity, multiply(additive_identity, a)) | (~equalish(additive_identity, multiply(additive_identity, a))))),
% 0.47/0.63 inference(transitivity,[status(thm)],[76, 74])).
% 0.47/0.63 tff(78,plain,
% 0.47/0.63 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))) | ((~equalish(additive_identity, multiply(additive_identity, a))) | sum(additive_identity, additive_identity, multiply(additive_identity, a)) | (~sum(additive_identity, additive_identity, additive_identity)))),
% 0.47/0.63 inference(quant_inst,[status(thm)],[])).
% 0.47/0.63 tff(79,plain,
% 0.47/0.63 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))) | (~sum(additive_identity, additive_identity, additive_identity)) | sum(additive_identity, additive_identity, multiply(additive_identity, a)) | (~equalish(additive_identity, multiply(additive_identity, a)))),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[78, 77])).
% 0.47/0.63 tff(80,plain,
% 0.47/0.63 (sum(additive_identity, additive_identity, multiply(additive_identity, a))),
% 0.47/0.63 inference(unit_resolution,[status(thm)],[79, 73, 63, 52])).
% 0.47/0.63 tff(81,plain,
% 0.47/0.63 (^[X: $i] : refl(sum(X, additive_inverse(X), additive_identity) <=> sum(X, additive_inverse(X), additive_identity))),
% 0.47/0.63 inference(bind,[status(th)],[])).
% 0.47/0.63 tff(82,plain,
% 0.47/0.63 (![X: $i] : sum(X, additive_inverse(X), additive_identity) <=> ![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 0.47/0.63 inference(quant_intro,[status(thm)],[81])).
% 0.47/0.63 tff(83,plain,
% 0.47/0.63 (![X: $i] : sum(X, additive_inverse(X), additive_identity) <=> ![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 0.47/0.63 inference(rewrite,[status(thm)],[])).
% 0.47/0.63 tff(84,axiom,(![X: $i] : sum(X, additive_inverse(X), additive_identity)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','additive_inverse2')).
% 0.47/0.63 tff(85,plain,
% 0.47/0.63 (![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 0.47/0.63 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.47/0.63 tff(86,plain,(
% 0.47/0.63 ![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 0.47/0.63 inference(skolemize,[status(sab)],[85])).
% 0.47/0.63 tff(87,plain,
% 0.47/0.63 (![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[86, 82])).
% 0.47/0.64 tff(88,plain,
% 0.47/0.64 ((~![X: $i] : sum(X, additive_inverse(X), additive_identity)) | sum(additive_identity, additive_inverse(additive_identity), additive_identity)),
% 0.47/0.64 inference(quant_inst,[status(thm)],[])).
% 0.47/0.64 tff(89,plain,
% 0.47/0.64 (sum(additive_identity, additive_inverse(additive_identity), additive_identity)),
% 0.47/0.64 inference(unit_resolution,[status(thm)],[88, 87])).
% 0.47/0.64 tff(90,plain,
% 0.47/0.64 (^[Z: $i, Y: $i, X: $i] : refl(((~sum(X, Y, Z)) | sum(Y, X, Z)) <=> ((~sum(X, Y, Z)) | sum(Y, X, Z)))),
% 0.47/0.64 inference(bind,[status(th)],[])).
% 0.47/0.64 tff(91,plain,
% 0.47/0.64 (![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))),
% 0.47/0.64 inference(quant_intro,[status(thm)],[90])).
% 0.47/0.64 tff(92,plain,
% 0.47/0.64 (![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))),
% 0.47/0.64 inference(rewrite,[status(thm)],[])).
% 0.47/0.64 tff(93,axiom,(![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','commutativity_of_addition')).
% 0.47/0.64 tff(94,plain,
% 0.47/0.64 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.47/0.64 tff(95,plain,(
% 0.47/0.64 ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 0.47/0.64 inference(skolemize,[status(sab)],[94])).
% 0.47/0.64 tff(96,plain,
% 0.47/0.64 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[95, 91])).
% 0.47/0.64 tff(97,plain,
% 0.47/0.64 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | sum(additive_inverse(additive_identity), additive_identity, additive_identity))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | sum(additive_inverse(additive_identity), additive_identity, additive_identity))),
% 0.47/0.64 inference(rewrite,[status(thm)],[])).
% 0.47/0.64 tff(98,plain,
% 0.47/0.64 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | sum(additive_inverse(additive_identity), additive_identity, additive_identity))),
% 0.47/0.64 inference(quant_inst,[status(thm)],[])).
% 0.47/0.64 tff(99,plain,
% 0.47/0.64 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | sum(additive_inverse(additive_identity), additive_identity, additive_identity)),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[98, 97])).
% 0.47/0.64 tff(100,plain,
% 0.47/0.64 (sum(additive_inverse(additive_identity), additive_identity, additive_identity)),
% 0.47/0.64 inference(unit_resolution,[status(thm)],[99, 96, 89])).
% 0.47/0.64 tff(101,plain,
% 0.47/0.64 ((~equalish(b, additive_identity)) <=> (~equalish(b, additive_identity))),
% 0.47/0.64 inference(rewrite,[status(thm)],[])).
% 0.47/0.64 tff(102,axiom,(~equalish(b, additive_identity)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_b_is_additive_identity')).
% 0.47/0.64 tff(103,plain,
% 0.47/0.64 (~equalish(b, additive_identity)),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.47/0.64 tff(104,plain,
% 0.47/0.64 (^[V: $i, Y: $i, U: $i, X: $i] : refl(((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V)) <=> ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V)))),
% 0.47/0.64 inference(bind,[status(th)],[])).
% 0.47/0.64 tff(105,plain,
% 0.47/0.64 (![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))),
% 0.47/0.64 inference(quant_intro,[status(thm)],[104])).
% 0.47/0.64 tff(106,plain,
% 0.47/0.64 (![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))),
% 0.47/0.64 inference(rewrite,[status(thm)],[])).
% 0.47/0.64 tff(107,plain,
% 0.47/0.64 (^[V: $i, Y: $i, U: $i, X: $i] : rewrite((((~sum(X, Y, U)) | (~sum(X, Y, V))) | equalish(U, V)) <=> ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V)))),
% 0.47/0.64 inference(bind,[status(th)],[])).
% 0.47/0.64 tff(108,plain,
% 0.47/0.64 (![V: $i, Y: $i, U: $i, X: $i] : (((~sum(X, Y, U)) | (~sum(X, Y, V))) | equalish(U, V)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))),
% 0.47/0.64 inference(quant_intro,[status(thm)],[107])).
% 0.47/0.64 tff(109,axiom,(![V: $i, Y: $i, U: $i, X: $i] : (((~sum(X, Y, U)) | (~sum(X, Y, V))) | equalish(U, V))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','addition_is_well_defined')).
% 0.47/0.64 tff(110,plain,
% 0.47/0.64 (![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[109, 108])).
% 0.47/0.64 tff(111,plain,
% 0.47/0.64 (![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[110, 106])).
% 0.47/0.64 tff(112,plain,(
% 0.47/0.64 ![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))),
% 0.47/0.64 inference(skolemize,[status(sab)],[111])).
% 0.47/0.64 tff(113,plain,
% 0.47/0.64 (![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[112, 105])).
% 0.47/0.64 tff(114,plain,
% 0.47/0.64 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))) | (equalish(b, additive_identity) | (~sum(additive_inverse(additive_identity), additive_identity, b)) | (~sum(additive_inverse(additive_identity), additive_identity, additive_identity)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))) | equalish(b, additive_identity) | (~sum(additive_inverse(additive_identity), additive_identity, b)) | (~sum(additive_inverse(additive_identity), additive_identity, additive_identity)))),
% 0.47/0.64 inference(rewrite,[status(thm)],[])).
% 0.47/0.64 tff(115,plain,
% 0.47/0.64 (((~sum(additive_inverse(additive_identity), additive_identity, b)) | (~sum(additive_inverse(additive_identity), additive_identity, additive_identity)) | equalish(b, additive_identity)) <=> (equalish(b, additive_identity) | (~sum(additive_inverse(additive_identity), additive_identity, b)) | (~sum(additive_inverse(additive_identity), additive_identity, additive_identity)))),
% 0.47/0.64 inference(rewrite,[status(thm)],[])).
% 0.47/0.64 tff(116,plain,
% 0.47/0.64 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))) | ((~sum(additive_inverse(additive_identity), additive_identity, b)) | (~sum(additive_inverse(additive_identity), additive_identity, additive_identity)) | equalish(b, additive_identity))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))) | (equalish(b, additive_identity) | (~sum(additive_inverse(additive_identity), additive_identity, b)) | (~sum(additive_inverse(additive_identity), additive_identity, additive_identity))))),
% 0.47/0.64 inference(monotonicity,[status(thm)],[115])).
% 0.47/0.64 tff(117,plain,
% 0.47/0.64 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))) | ((~sum(additive_inverse(additive_identity), additive_identity, b)) | (~sum(additive_inverse(additive_identity), additive_identity, additive_identity)) | equalish(b, additive_identity))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))) | equalish(b, additive_identity) | (~sum(additive_inverse(additive_identity), additive_identity, b)) | (~sum(additive_inverse(additive_identity), additive_identity, additive_identity)))),
% 0.47/0.64 inference(transitivity,[status(thm)],[116, 114])).
% 0.47/0.64 tff(118,plain,
% 0.47/0.64 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))) | ((~sum(additive_inverse(additive_identity), additive_identity, b)) | (~sum(additive_inverse(additive_identity), additive_identity, additive_identity)) | equalish(b, additive_identity))),
% 0.47/0.64 inference(quant_inst,[status(thm)],[])).
% 0.47/0.64 tff(119,plain,
% 0.47/0.64 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))) | equalish(b, additive_identity) | (~sum(additive_inverse(additive_identity), additive_identity, b)) | (~sum(additive_inverse(additive_identity), additive_identity, additive_identity))),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[118, 117])).
% 0.47/0.64 tff(120,plain,
% 0.47/0.64 (~sum(additive_inverse(additive_identity), additive_identity, b)),
% 0.47/0.64 inference(unit_resolution,[status(thm)],[119, 113, 103, 100])).
% 0.47/0.64 tff(121,plain,
% 0.47/0.64 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))) | ((~equalish(additive_identity, b)) | sum(additive_inverse(additive_identity), additive_identity, b) | (~sum(additive_inverse(additive_identity), additive_identity, additive_identity)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))) | (~equalish(additive_identity, b)) | sum(additive_inverse(additive_identity), additive_identity, b) | (~sum(additive_inverse(additive_identity), additive_identity, additive_identity)))),
% 0.47/0.64 inference(rewrite,[status(thm)],[])).
% 0.47/0.64 tff(122,plain,
% 0.47/0.64 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))) | ((~equalish(additive_identity, b)) | sum(additive_inverse(additive_identity), additive_identity, b) | (~sum(additive_inverse(additive_identity), additive_identity, additive_identity)))),
% 0.47/0.64 inference(quant_inst,[status(thm)],[])).
% 0.47/0.64 tff(123,plain,
% 0.47/0.64 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~equalish(X, Y)) | sum(W, Z, Y) | (~sum(W, Z, X)))) | (~equalish(additive_identity, b)) | sum(additive_inverse(additive_identity), additive_identity, b) | (~sum(additive_inverse(additive_identity), additive_identity, additive_identity))),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[122, 121])).
% 0.47/0.64 tff(124,plain,
% 0.47/0.64 (~equalish(additive_identity, b)),
% 0.47/0.64 inference(unit_resolution,[status(thm)],[123, 73, 100, 120])).
% 0.47/0.64 tff(125,plain,
% 0.47/0.64 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))) | ((~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, b)) | equalish(additive_identity, b))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, b)) | equalish(additive_identity, b))),
% 0.47/0.64 inference(rewrite,[status(thm)],[])).
% 0.47/0.64 tff(126,plain,
% 0.47/0.64 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))) | ((~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, b)) | equalish(additive_identity, b))),
% 0.47/0.64 inference(quant_inst,[status(thm)],[])).
% 0.47/0.64 tff(127,plain,
% 0.47/0.64 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | equalish(U, V))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, b)) | equalish(additive_identity, b)),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[126, 125])).
% 0.47/0.64 tff(128,plain,
% 0.47/0.64 ((~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, b)) | equalish(additive_identity, b)),
% 0.47/0.64 inference(unit_resolution,[status(thm)],[127, 113])).
% 0.47/0.64 tff(129,plain,
% 0.47/0.64 ((~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, b))),
% 0.47/0.64 inference(unit_resolution,[status(thm)],[128, 124])).
% 0.47/0.64 tff(130,plain,
% 0.47/0.64 (~sum(additive_identity, additive_identity, b)),
% 0.47/0.64 inference(unit_resolution,[status(thm)],[129, 63])).
% 0.47/0.64 tff(131,plain,
% 0.47/0.64 (^[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))))),
% 0.47/0.64 inference(bind,[status(th)],[])).
% 0.47/0.64 tff(132,plain,
% 0.47/0.64 (![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)))),
% 0.47/0.64 inference(quant_intro,[status(thm)],[131])).
% 0.47/0.64 tff(133,plain,
% 0.47/0.64 (![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)))),
% 0.47/0.64 inference(rewrite,[status(thm)],[])).
% 0.47/0.64 tff(134,plain,
% 0.47/0.64 (^[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)))))),
% 0.47/0.64 inference(bind,[status(th)],[])).
% 0.47/0.64 tff(135,plain,
% 0.47/0.64 (![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)))),
% 0.47/0.64 inference(quant_intro,[status(thm)],[134])).
% 0.47/0.64 tff(136,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/theBenchmark.p','distributivity3')).
% 0.47/0.64 tff(137,plain,
% 0.47/0.64 (![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)))),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[136, 135])).
% 0.47/0.64 tff(138,plain,
% 0.47/0.64 (![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)))),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[137, 133])).
% 0.47/0.64 tff(139,plain,(
% 0.47/0.64 ![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)))),
% 0.47/0.64 inference(skolemize,[status(sab)],[138])).
% 0.47/0.64 tff(140,plain,
% 0.47/0.64 (![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)))),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[139, 132])).
% 0.47/0.64 tff(141,plain,
% 0.47/0.64 (((~![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, b) | (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)) | (~product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b)) | (~sum(additive_identity, additive_identity, 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, b) | (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)) | (~product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b)) | (~sum(additive_identity, additive_identity, multiply(additive_identity, a))))),
% 0.47/0.64 inference(rewrite,[status(thm)],[])).
% 0.47/0.64 tff(142,plain,
% 0.47/0.64 ((sum(additive_identity, additive_identity, b) | (~sum(additive_identity, additive_identity, multiply(additive_identity, a))) | (~product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b)) | (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)) | (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity))) <=> (sum(additive_identity, additive_identity, b) | (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)) | (~product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b)) | (~sum(additive_identity, additive_identity, multiply(additive_identity, a))))),
% 0.47/0.64 inference(rewrite,[status(thm)],[])).
% 0.47/0.64 tff(143,plain,
% 0.47/0.64 (((~![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, b) | (~sum(additive_identity, additive_identity, multiply(additive_identity, a))) | (~product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b)) | (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)) | (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)))) <=> ((~![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, b) | (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)) | (~product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b)) | (~sum(additive_identity, additive_identity, multiply(additive_identity, a)))))),
% 0.47/0.64 inference(monotonicity,[status(thm)],[142])).
% 0.47/0.64 tff(144,plain,
% 0.47/0.64 (((~![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, b) | (~sum(additive_identity, additive_identity, multiply(additive_identity, a))) | (~product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b)) | (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)) | (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)))) <=> ((~![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, b) | (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)) | (~product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b)) | (~sum(additive_identity, additive_identity, multiply(additive_identity, a))))),
% 0.47/0.64 inference(transitivity,[status(thm)],[143, 141])).
% 0.47/0.64 tff(145,plain,
% 0.47/0.64 ((~![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, b) | (~sum(additive_identity, additive_identity, multiply(additive_identity, a))) | (~product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b)) | (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)) | (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)))),
% 0.47/0.64 inference(quant_inst,[status(thm)],[])).
% 0.47/0.64 tff(146,plain,
% 0.47/0.64 ((~![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, b) | (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)) | (~product(multiply(additive_identity, a), h(multiply(additive_identity, a), b), b)) | (~sum(additive_identity, additive_identity, multiply(additive_identity, a)))),
% 0.47/0.64 inference(modus_ponens,[status(thm)],[145, 144])).
% 0.47/0.64 tff(147,plain,
% 0.47/0.64 (~product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)),
% 0.47/0.64 inference(unit_resolution,[status(thm)],[146, 140, 130, 80, 45])).
% 0.47/0.64 tff(148,plain,
% 0.47/0.64 ((~![X: $i] : product(additive_identity, X, additive_identity)) | product(additive_identity, h(multiply(additive_identity, a), b), additive_identity)),
% 0.47/0.64 inference(quant_inst,[status(thm)],[])).
% 0.47/0.64 tff(149,plain,
% 0.47/0.64 ($false),
% 0.47/0.64 inference(unit_resolution,[status(thm)],[148, 8, 147])).
% 0.47/0.64 % SZS output end Proof
% 0.47/0.65 % E exiting
%------------------------------------------------------------------------------