TSTP Solution File: RNG007-1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : RNG007-1 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n015.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 : Tue Sep 20 03:17:37 EDT 2022
% Result : Unsatisfiable 5.87s 4.00s
% Output : Proof 5.95s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.10 % Problem : RNG007-1 : TPTP v8.1.0. Released v1.0.0.
% 0.10/0.11 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.10/0.31 % Computer : n015.cluster.edu
% 0.10/0.31 % Model : x86_64 x86_64
% 0.10/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.31 % Memory : 8042.1875MB
% 0.10/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.31 % CPULimit : 300
% 0.10/0.31 % WCLimit : 300
% 0.10/0.31 % DateTime : Fri Sep 2 21:28:39 EDT 2022
% 0.16/0.32 % CPUTime :
% 0.16/0.32 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.16/0.32 Usage: tptp [options] [-file:]file
% 0.16/0.32 -h, -? prints this message.
% 0.16/0.32 -smt2 print SMT-LIB2 benchmark.
% 0.16/0.32 -m, -model generate model.
% 0.16/0.32 -p, -proof generate proof.
% 0.16/0.32 -c, -core generate unsat core of named formulas.
% 0.16/0.32 -st, -statistics display statistics.
% 0.16/0.32 -t:timeout set timeout (in second).
% 0.16/0.32 -smt2status display status in smt2 format instead of SZS.
% 0.16/0.32 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.16/0.32 -<param>:<value> configuration parameter and value.
% 0.16/0.32 -o:<output-file> file to place output in.
% 5.87/4.00 % SZS status Unsatisfiable
% 5.87/4.00 % SZS output start Proof
% 5.87/4.00 tff(sum_type, type, (
% 5.87/4.00 sum: ( $i * $i * $i ) > $o)).
% 5.87/4.00 tff(add_type, type, (
% 5.87/4.00 add: ( $i * $i ) > $i)).
% 5.87/4.00 tff(multiply_type, type, (
% 5.87/4.00 multiply: ( $i * $i ) > $i)).
% 5.87/4.00 tff(additive_identity_type, type, (
% 5.87/4.00 additive_identity: $i)).
% 5.87/4.00 tff(a_type, type, (
% 5.87/4.00 a: $i)).
% 5.87/4.00 tff(additive_inverse_type, type, (
% 5.87/4.00 additive_inverse: $i > $i)).
% 5.87/4.00 tff(product_type, type, (
% 5.87/4.00 product: ( $i * $i * $i ) > $o)).
% 5.87/4.00 tff(1,plain,
% 5.87/4.00 (^[Y: $i, X: $i] : refl(product(X, Y, multiply(X, Y)) <=> product(X, Y, multiply(X, Y)))),
% 5.87/4.00 inference(bind,[status(th)],[])).
% 5.87/4.00 tff(2,plain,
% 5.87/4.00 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 5.87/4.00 inference(quant_intro,[status(thm)],[1])).
% 5.87/4.00 tff(3,plain,
% 5.87/4.00 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 5.87/4.00 inference(rewrite,[status(thm)],[])).
% 5.87/4.00 tff(4,axiom,(![Y: $i, X: $i] : product(X, Y, multiply(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','closure_of_multiplication')).
% 5.87/4.00 tff(5,plain,
% 5.87/4.00 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[4, 3])).
% 5.87/4.00 tff(6,plain,(
% 5.87/4.00 ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 5.87/4.00 inference(skolemize,[status(sab)],[5])).
% 5.87/4.00 tff(7,plain,
% 5.87/4.00 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[6, 2])).
% 5.87/4.00 tff(8,plain,
% 5.87/4.00 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))),
% 5.87/4.00 inference(quant_inst,[status(thm)],[])).
% 5.87/4.00 tff(9,plain,
% 5.87/4.00 (product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))),
% 5.87/4.00 inference(unit_resolution,[status(thm)],[8, 7])).
% 5.87/4.00 tff(10,plain,
% 5.87/4.00 (^[Y: $i, X: $i] : refl(sum(X, Y, add(X, Y)) <=> sum(X, Y, add(X, Y)))),
% 5.87/4.00 inference(bind,[status(th)],[])).
% 5.87/4.00 tff(11,plain,
% 5.87/4.00 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 5.87/4.00 inference(quant_intro,[status(thm)],[10])).
% 5.87/4.00 tff(12,plain,
% 5.87/4.00 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 5.87/4.00 inference(rewrite,[status(thm)],[])).
% 5.87/4.00 tff(13,axiom,(![Y: $i, X: $i] : sum(X, Y, add(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','closure_of_addition')).
% 5.87/4.00 tff(14,plain,
% 5.87/4.00 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[13, 12])).
% 5.87/4.00 tff(15,plain,(
% 5.87/4.00 ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 5.87/4.00 inference(skolemize,[status(sab)],[14])).
% 5.87/4.00 tff(16,plain,
% 5.87/4.00 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[15, 11])).
% 5.87/4.00 tff(17,plain,
% 5.87/4.00 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))),
% 5.87/4.00 inference(quant_inst,[status(thm)],[])).
% 5.87/4.00 tff(18,plain,
% 5.87/4.00 (sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))),
% 5.87/4.00 inference(unit_resolution,[status(thm)],[17, 16])).
% 5.87/4.00 tff(19,plain,
% 5.87/4.00 (^[X: $i] : refl(product(X, X, X) <=> product(X, X, X))),
% 5.87/4.00 inference(bind,[status(th)],[])).
% 5.87/4.00 tff(20,plain,
% 5.87/4.00 (![X: $i] : product(X, X, X) <=> ![X: $i] : product(X, X, X)),
% 5.87/4.00 inference(quant_intro,[status(thm)],[19])).
% 5.87/4.00 tff(21,plain,
% 5.87/4.00 (![X: $i] : product(X, X, X) <=> ![X: $i] : product(X, X, X)),
% 5.87/4.00 inference(rewrite,[status(thm)],[])).
% 5.87/4.00 tff(22,axiom,(![X: $i] : product(X, X, X)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','x_squared_is_x')).
% 5.87/4.00 tff(23,plain,
% 5.87/4.00 (![X: $i] : product(X, X, X)),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[22, 21])).
% 5.87/4.00 tff(24,plain,(
% 5.87/4.00 ![X: $i] : product(X, X, X)),
% 5.87/4.00 inference(skolemize,[status(sab)],[23])).
% 5.87/4.00 tff(25,plain,
% 5.87/4.00 (![X: $i] : product(X, X, X)),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[24, 20])).
% 5.87/4.00 tff(26,plain,
% 5.87/4.00 ((~![X: $i] : product(X, X, X)) | product(a, a, a)),
% 5.87/4.00 inference(quant_inst,[status(thm)],[])).
% 5.87/4.00 tff(27,plain,
% 5.87/4.00 (product(a, a, a)),
% 5.87/4.00 inference(unit_resolution,[status(thm)],[26, 25])).
% 5.87/4.00 tff(28,plain,
% 5.87/4.00 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(a, additive_identity, multiply(a, additive_identity))),
% 5.87/4.00 inference(quant_inst,[status(thm)],[])).
% 5.87/4.00 tff(29,plain,
% 5.87/4.00 (product(a, additive_identity, multiply(a, additive_identity))),
% 5.87/4.00 inference(unit_resolution,[status(thm)],[28, 7])).
% 5.87/4.00 tff(30,plain,
% 5.87/4.00 (^[X: $i] : refl(sum(X, additive_identity, X) <=> sum(X, additive_identity, X))),
% 5.87/4.00 inference(bind,[status(th)],[])).
% 5.87/4.00 tff(31,plain,
% 5.87/4.00 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 5.87/4.00 inference(quant_intro,[status(thm)],[30])).
% 5.87/4.00 tff(32,plain,
% 5.87/4.00 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 5.87/4.00 inference(rewrite,[status(thm)],[])).
% 5.87/4.00 tff(33,axiom,(![X: $i] : sum(X, additive_identity, X)), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','additive_identity2')).
% 5.87/4.00 tff(34,plain,
% 5.87/4.00 (![X: $i] : sum(X, additive_identity, X)),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[33, 32])).
% 5.87/4.00 tff(35,plain,(
% 5.87/4.00 ![X: $i] : sum(X, additive_identity, X)),
% 5.87/4.00 inference(skolemize,[status(sab)],[34])).
% 5.87/4.00 tff(36,plain,
% 5.87/4.00 (![X: $i] : sum(X, additive_identity, X)),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[35, 31])).
% 5.87/4.00 tff(37,plain,
% 5.87/4.00 ((~![X: $i] : sum(X, additive_identity, X)) | sum(a, additive_identity, a)),
% 5.87/4.00 inference(quant_inst,[status(thm)],[])).
% 5.87/4.00 tff(38,plain,
% 5.87/4.00 (sum(a, additive_identity, a)),
% 5.87/4.00 inference(unit_resolution,[status(thm)],[37, 36])).
% 5.87/4.00 tff(39,plain,
% 5.87/4.00 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) <=> (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))))),
% 5.87/4.00 inference(bind,[status(th)],[])).
% 5.87/4.00 tff(40,plain,
% 5.87/4.00 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 5.87/4.00 inference(quant_intro,[status(thm)],[39])).
% 5.87/4.00 tff(41,plain,
% 5.87/4.00 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 5.87/4.00 inference(rewrite,[status(thm)],[])).
% 5.87/4.00 tff(42,plain,
% 5.87/4.00 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) <=> (((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~product(X, V3, V4))))), rewrite((((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~product(X, V3, V4))) <=> ((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) <=> ((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4)) <=> (((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | sum(V1, V2, V4)))), rewrite((((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | sum(V1, V2, V4)) <=> (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4)) <=> (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))))),
% 5.87/4.00 inference(bind,[status(th)],[])).
% 5.87/4.00 tff(43,plain,
% 5.87/4.00 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 5.87/4.00 inference(quant_intro,[status(thm)],[42])).
% 5.87/4.00 tff(44,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','distributivity1')).
% 5.87/4.00 tff(45,plain,
% 5.87/4.00 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[44, 43])).
% 5.87/4.00 tff(46,plain,
% 5.87/4.00 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[45, 41])).
% 5.87/4.00 tff(47,plain,(
% 5.87/4.00 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 5.87/4.00 inference(skolemize,[status(sab)],[46])).
% 5.87/4.00 tff(48,plain,
% 5.87/4.00 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[47, 40])).
% 5.87/4.00 tff(49,plain,
% 5.87/4.00 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~sum(a, additive_identity, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(a, multiply(a, additive_identity), a) | (~product(a, a, a)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(a, additive_identity, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(a, multiply(a, additive_identity), a) | (~product(a, a, a)))),
% 5.87/4.00 inference(rewrite,[status(thm)],[])).
% 5.87/4.00 tff(50,plain,
% 5.87/4.00 ((sum(a, multiply(a, additive_identity), a) | (~product(a, a, a)) | (~sum(a, additive_identity, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, a, a))) <=> ((~sum(a, additive_identity, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(a, multiply(a, additive_identity), a) | (~product(a, a, a)))),
% 5.87/4.00 inference(rewrite,[status(thm)],[])).
% 5.87/4.00 tff(51,plain,
% 5.87/4.00 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(a, multiply(a, additive_identity), a) | (~product(a, a, a)) | (~sum(a, additive_identity, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, a, a)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~sum(a, additive_identity, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(a, multiply(a, additive_identity), a) | (~product(a, a, a))))),
% 5.87/4.00 inference(monotonicity,[status(thm)],[50])).
% 5.87/4.00 tff(52,plain,
% 5.87/4.00 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(a, multiply(a, additive_identity), a) | (~product(a, a, a)) | (~sum(a, additive_identity, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, a, a)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(a, additive_identity, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(a, multiply(a, additive_identity), a) | (~product(a, a, a)))),
% 5.87/4.00 inference(transitivity,[status(thm)],[51, 49])).
% 5.87/4.00 tff(53,plain,
% 5.87/4.00 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(a, multiply(a, additive_identity), a) | (~product(a, a, a)) | (~sum(a, additive_identity, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, a, a)))),
% 5.87/4.00 inference(quant_inst,[status(thm)],[])).
% 5.87/4.00 tff(54,plain,
% 5.87/4.00 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(a, additive_identity, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(a, multiply(a, additive_identity), a) | (~product(a, a, a))),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[53, 52])).
% 5.87/4.00 tff(55,plain,
% 5.87/4.00 (sum(a, multiply(a, additive_identity), a)),
% 5.87/4.00 inference(unit_resolution,[status(thm)],[54, 48, 38, 29, 27])).
% 5.87/4.00 tff(56,plain,
% 5.87/4.00 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(a, multiply(a, additive_identity), add(a, multiply(a, additive_identity)))),
% 5.87/4.00 inference(quant_inst,[status(thm)],[])).
% 5.87/4.00 tff(57,plain,
% 5.87/4.00 (sum(a, multiply(a, additive_identity), add(a, multiply(a, additive_identity)))),
% 5.87/4.00 inference(unit_resolution,[status(thm)],[56, 16])).
% 5.87/4.00 tff(58,plain,
% 5.87/4.00 (^[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)))),
% 5.87/4.00 inference(bind,[status(th)],[])).
% 5.87/4.00 tff(59,plain,
% 5.87/4.00 (![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))),
% 5.87/4.00 inference(quant_intro,[status(thm)],[58])).
% 5.87/4.00 tff(60,plain,
% 5.87/4.00 (![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))),
% 5.87/4.00 inference(rewrite,[status(thm)],[])).
% 5.87/4.00 tff(61,plain,
% 5.87/4.00 (^[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)))),
% 5.87/4.00 inference(bind,[status(th)],[])).
% 5.87/4.00 tff(62,plain,
% 5.87/4.00 (![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))),
% 5.87/4.00 inference(quant_intro,[status(thm)],[61])).
% 5.87/4.00 tff(63,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')).
% 5.87/4.00 tff(64,plain,
% 5.87/4.00 (![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[63, 62])).
% 5.87/4.00 tff(65,plain,
% 5.87/4.00 (![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[64, 60])).
% 5.87/4.00 tff(66,plain,(
% 5.87/4.00 ![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 5.87/4.00 inference(skolemize,[status(sab)],[65])).
% 5.87/4.00 tff(67,plain,
% 5.87/4.00 (![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))),
% 5.87/4.00 inference(modus_ponens,[status(thm)],[66, 59])).
% 5.87/4.00 tff(68,plain,
% 5.87/4.00 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(a, multiply(a, additive_identity), a)) | (~sum(a, multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) | (a = add(a, multiply(a, additive_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(a, multiply(a, additive_identity), a)) | (~sum(a, multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) | (a = add(a, multiply(a, additive_identity))))),
% 5.87/4.01 inference(rewrite,[status(thm)],[])).
% 5.87/4.01 tff(69,plain,
% 5.87/4.01 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(a, multiply(a, additive_identity), a)) | (~sum(a, multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) | (a = add(a, multiply(a, additive_identity))))),
% 5.87/4.01 inference(quant_inst,[status(thm)],[])).
% 5.87/4.01 tff(70,plain,
% 5.87/4.01 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(a, multiply(a, additive_identity), a)) | (~sum(a, multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) | (a = add(a, multiply(a, additive_identity)))),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[69, 68])).
% 5.87/4.01 tff(71,plain,
% 5.87/4.01 (a = add(a, multiply(a, additive_identity))),
% 5.87/4.01 inference(unit_resolution,[status(thm)],[70, 67, 57, 55])).
% 5.87/4.01 tff(72,plain,
% 5.87/4.01 (add(a, multiply(a, additive_identity)) = a),
% 5.87/4.01 inference(symmetry,[status(thm)],[71])).
% 5.87/4.01 tff(73,plain,
% 5.87/4.01 (sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) <=> sum(multiply(a, additive_identity), a, a)),
% 5.87/4.01 inference(monotonicity,[status(thm)],[72, 72])).
% 5.87/4.01 tff(74,plain,
% 5.87/4.01 (sum(multiply(a, additive_identity), a, a) <=> sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))),
% 5.87/4.01 inference(symmetry,[status(thm)],[73])).
% 5.87/4.01 tff(75,plain,
% 5.87/4.01 (^[Z: $i, Y: $i, X: $i] : refl(((~sum(X, Y, Z)) | sum(Y, X, Z)) <=> ((~sum(X, Y, Z)) | sum(Y, X, Z)))),
% 5.87/4.01 inference(bind,[status(th)],[])).
% 5.87/4.01 tff(76,plain,
% 5.87/4.01 (![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))),
% 5.87/4.01 inference(quant_intro,[status(thm)],[75])).
% 5.87/4.01 tff(77,plain,
% 5.87/4.01 (![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))),
% 5.87/4.01 inference(rewrite,[status(thm)],[])).
% 5.87/4.01 tff(78,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')).
% 5.87/4.01 tff(79,plain,
% 5.87/4.01 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[78, 77])).
% 5.87/4.01 tff(80,plain,(
% 5.87/4.01 ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 5.87/4.01 inference(skolemize,[status(sab)],[79])).
% 5.87/4.01 tff(81,plain,
% 5.87/4.01 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[80, 76])).
% 5.87/4.01 tff(82,plain,
% 5.87/4.01 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(a, additive_identity, a)) | sum(additive_identity, a, a))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(a, additive_identity, a)) | sum(additive_identity, a, a))),
% 5.87/4.01 inference(rewrite,[status(thm)],[])).
% 5.87/4.01 tff(83,plain,
% 5.87/4.01 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(a, additive_identity, a)) | sum(additive_identity, a, a))),
% 5.87/4.01 inference(quant_inst,[status(thm)],[])).
% 5.87/4.01 tff(84,plain,
% 5.87/4.01 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(a, additive_identity, a)) | sum(additive_identity, a, a)),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[83, 82])).
% 5.87/4.01 tff(85,plain,
% 5.87/4.01 (sum(additive_identity, a, a)),
% 5.87/4.01 inference(unit_resolution,[status(thm)],[84, 81, 38])).
% 5.87/4.01 tff(86,plain,
% 5.87/4.01 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~sum(additive_identity, a, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(multiply(a, additive_identity), a, a) | (~product(a, a, a)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(additive_identity, a, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(multiply(a, additive_identity), a, a) | (~product(a, a, a)))),
% 5.87/4.01 inference(rewrite,[status(thm)],[])).
% 5.87/4.01 tff(87,plain,
% 5.87/4.01 ((sum(multiply(a, additive_identity), a, a) | (~product(a, a, a)) | (~sum(additive_identity, a, a)) | (~product(a, a, a)) | (~product(a, additive_identity, multiply(a, additive_identity)))) <=> ((~sum(additive_identity, a, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(multiply(a, additive_identity), a, a) | (~product(a, a, a)))),
% 5.87/4.01 inference(rewrite,[status(thm)],[])).
% 5.87/4.01 tff(88,plain,
% 5.87/4.01 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(multiply(a, additive_identity), a, a) | (~product(a, a, a)) | (~sum(additive_identity, a, a)) | (~product(a, a, a)) | (~product(a, additive_identity, multiply(a, additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~sum(additive_identity, a, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(multiply(a, additive_identity), a, a) | (~product(a, a, a))))),
% 5.87/4.01 inference(monotonicity,[status(thm)],[87])).
% 5.87/4.01 tff(89,plain,
% 5.87/4.01 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(multiply(a, additive_identity), a, a) | (~product(a, a, a)) | (~sum(additive_identity, a, a)) | (~product(a, a, a)) | (~product(a, additive_identity, multiply(a, additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(additive_identity, a, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(multiply(a, additive_identity), a, a) | (~product(a, a, a)))),
% 5.87/4.01 inference(transitivity,[status(thm)],[88, 86])).
% 5.87/4.01 tff(90,plain,
% 5.87/4.01 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(multiply(a, additive_identity), a, a) | (~product(a, a, a)) | (~sum(additive_identity, a, a)) | (~product(a, a, a)) | (~product(a, additive_identity, multiply(a, additive_identity))))),
% 5.87/4.01 inference(quant_inst,[status(thm)],[])).
% 5.87/4.01 tff(91,plain,
% 5.87/4.01 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(additive_identity, a, a)) | (~product(a, additive_identity, multiply(a, additive_identity))) | sum(multiply(a, additive_identity), a, a) | (~product(a, a, a))),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[90, 89])).
% 5.87/4.01 tff(92,plain,
% 5.87/4.01 (sum(multiply(a, additive_identity), a, a)),
% 5.87/4.01 inference(unit_resolution,[status(thm)],[91, 48, 29, 27, 85])).
% 5.87/4.01 tff(93,plain,
% 5.87/4.01 (sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[92, 74])).
% 5.87/4.01 tff(94,plain,
% 5.87/4.01 (sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity) <=> sum(a, additive_inverse(a), additive_identity)),
% 5.87/4.01 inference(monotonicity,[status(thm)],[72])).
% 5.87/4.01 tff(95,plain,
% 5.87/4.01 (sum(a, additive_inverse(a), additive_identity) <=> sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)),
% 5.87/4.01 inference(symmetry,[status(thm)],[94])).
% 5.87/4.01 tff(96,plain,
% 5.87/4.01 (^[X: $i] : refl(sum(X, additive_inverse(X), additive_identity) <=> sum(X, additive_inverse(X), additive_identity))),
% 5.87/4.01 inference(bind,[status(th)],[])).
% 5.87/4.01 tff(97,plain,
% 5.87/4.01 (![X: $i] : sum(X, additive_inverse(X), additive_identity) <=> ![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 5.87/4.01 inference(quant_intro,[status(thm)],[96])).
% 5.87/4.01 tff(98,plain,
% 5.87/4.01 (![X: $i] : sum(X, additive_inverse(X), additive_identity) <=> ![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 5.87/4.01 inference(rewrite,[status(thm)],[])).
% 5.87/4.01 tff(99,axiom,(![X: $i] : sum(X, additive_inverse(X), additive_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','right_inverse')).
% 5.87/4.01 tff(100,plain,
% 5.87/4.01 (![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[99, 98])).
% 5.87/4.01 tff(101,plain,(
% 5.87/4.01 ![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 5.87/4.01 inference(skolemize,[status(sab)],[100])).
% 5.87/4.01 tff(102,plain,
% 5.87/4.01 (![X: $i] : sum(X, additive_inverse(X), additive_identity)),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[101, 97])).
% 5.87/4.01 tff(103,plain,
% 5.87/4.01 ((~![X: $i] : sum(X, additive_inverse(X), additive_identity)) | sum(a, additive_inverse(a), additive_identity)),
% 5.87/4.01 inference(quant_inst,[status(thm)],[])).
% 5.87/4.01 tff(104,plain,
% 5.87/4.01 (sum(a, additive_inverse(a), additive_identity)),
% 5.87/4.01 inference(unit_resolution,[status(thm)],[103, 102])).
% 5.87/4.01 tff(105,plain,
% 5.87/4.01 (sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[104, 95])).
% 5.87/4.01 tff(106,plain,
% 5.87/4.01 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : refl((sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U))) <=> (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U))))),
% 5.87/4.01 inference(bind,[status(th)],[])).
% 5.87/4.01 tff(107,plain,
% 5.87/4.01 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))),
% 5.87/4.01 inference(quant_intro,[status(thm)],[106])).
% 5.87/4.01 tff(108,plain,
% 5.87/4.01 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))),
% 5.87/4.01 inference(rewrite,[status(thm)],[])).
% 5.87/4.01 tff(109,plain,
% 5.87/4.01 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : trans(monotonicity(rewrite((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(U, Z, W))) <=> ((~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))), (((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(U, Z, W))) | sum(X, V, W)) <=> (((~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U))) | sum(X, V, W)))), rewrite((((~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U))) | sum(X, V, W)) <=> (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))), (((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(U, Z, W))) | sum(X, V, W)) <=> (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))))),
% 5.87/4.01 inference(bind,[status(th)],[])).
% 5.87/4.01 tff(110,plain,
% 5.87/4.01 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(U, Z, W))) | sum(X, V, W)) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))),
% 5.87/4.01 inference(quant_intro,[status(thm)],[109])).
% 5.87/4.01 tff(111,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(U, Z, W))) | sum(X, V, W))), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','associativity_of_addition1')).
% 5.87/4.01 tff(112,plain,
% 5.87/4.01 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[111, 110])).
% 5.87/4.01 tff(113,plain,
% 5.87/4.01 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[112, 108])).
% 5.87/4.01 tff(114,plain,(
% 5.87/4.01 ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))),
% 5.87/4.01 inference(skolemize,[status(sab)],[113])).
% 5.87/4.01 tff(115,plain,
% 5.87/4.01 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[114, 107])).
% 5.87/4.01 tff(116,plain,
% 5.87/4.01 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | ((~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | sum(multiply(a, additive_identity), additive_identity, additive_identity))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | sum(multiply(a, additive_identity), additive_identity, additive_identity))),
% 5.87/4.01 inference(rewrite,[status(thm)],[])).
% 5.87/4.01 tff(117,plain,
% 5.87/4.01 ((sum(multiply(a, additive_identity), additive_identity, additive_identity) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | (~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))))) <=> ((~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | sum(multiply(a, additive_identity), additive_identity, additive_identity))),
% 5.87/4.01 inference(rewrite,[status(thm)],[])).
% 5.87/4.01 tff(118,plain,
% 5.87/4.01 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(multiply(a, additive_identity), additive_identity, additive_identity) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | (~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | ((~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | sum(multiply(a, additive_identity), additive_identity, additive_identity)))),
% 5.87/4.01 inference(monotonicity,[status(thm)],[117])).
% 5.87/4.01 tff(119,plain,
% 5.87/4.01 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(multiply(a, additive_identity), additive_identity, additive_identity) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | (~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | sum(multiply(a, additive_identity), additive_identity, additive_identity))),
% 5.87/4.01 inference(transitivity,[status(thm)],[118, 116])).
% 5.87/4.01 tff(120,plain,
% 5.87/4.01 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(multiply(a, additive_identity), additive_identity, additive_identity) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | (~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))))),
% 5.87/4.01 inference(quant_inst,[status(thm)],[])).
% 5.87/4.01 tff(121,plain,
% 5.87/4.01 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | sum(multiply(a, additive_identity), additive_identity, additive_identity)),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[120, 119])).
% 5.87/4.01 tff(122,plain,
% 5.87/4.01 ((~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(a), additive_identity)) | sum(multiply(a, additive_identity), additive_identity, additive_identity)),
% 5.87/4.01 inference(unit_resolution,[status(thm)],[121, 115])).
% 5.87/4.01 tff(123,plain,
% 5.87/4.01 (sum(multiply(a, additive_identity), additive_identity, additive_identity)),
% 5.87/4.01 inference(unit_resolution,[status(thm)],[122, 105, 93])).
% 5.87/4.01 tff(124,plain,
% 5.87/4.01 ((~![X: $i] : sum(X, additive_identity, X)) | sum(multiply(a, additive_identity), additive_identity, multiply(a, additive_identity))),
% 5.87/4.01 inference(quant_inst,[status(thm)],[])).
% 5.87/4.01 tff(125,plain,
% 5.87/4.01 (sum(multiply(a, additive_identity), additive_identity, multiply(a, additive_identity))),
% 5.87/4.01 inference(unit_resolution,[status(thm)],[124, 36])).
% 5.87/4.01 tff(126,plain,
% 5.87/4.01 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(multiply(a, additive_identity), additive_identity, multiply(a, additive_identity))) | (multiply(a, additive_identity) = additive_identity) | (~sum(multiply(a, additive_identity), additive_identity, additive_identity)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(multiply(a, additive_identity), additive_identity, multiply(a, additive_identity))) | (multiply(a, additive_identity) = additive_identity) | (~sum(multiply(a, additive_identity), additive_identity, additive_identity)))),
% 5.87/4.01 inference(rewrite,[status(thm)],[])).
% 5.87/4.01 tff(127,plain,
% 5.87/4.01 (((~sum(multiply(a, additive_identity), additive_identity, multiply(a, additive_identity))) | (~sum(multiply(a, additive_identity), additive_identity, additive_identity)) | (multiply(a, additive_identity) = additive_identity)) <=> ((~sum(multiply(a, additive_identity), additive_identity, multiply(a, additive_identity))) | (multiply(a, additive_identity) = additive_identity) | (~sum(multiply(a, additive_identity), additive_identity, additive_identity)))),
% 5.87/4.01 inference(rewrite,[status(thm)],[])).
% 5.87/4.01 tff(128,plain,
% 5.87/4.01 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(multiply(a, additive_identity), additive_identity, multiply(a, additive_identity))) | (~sum(multiply(a, additive_identity), additive_identity, additive_identity)) | (multiply(a, additive_identity) = additive_identity))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(multiply(a, additive_identity), additive_identity, multiply(a, additive_identity))) | (multiply(a, additive_identity) = additive_identity) | (~sum(multiply(a, additive_identity), additive_identity, additive_identity))))),
% 5.87/4.01 inference(monotonicity,[status(thm)],[127])).
% 5.87/4.01 tff(129,plain,
% 5.87/4.01 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(multiply(a, additive_identity), additive_identity, multiply(a, additive_identity))) | (~sum(multiply(a, additive_identity), additive_identity, additive_identity)) | (multiply(a, additive_identity) = additive_identity))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(multiply(a, additive_identity), additive_identity, multiply(a, additive_identity))) | (multiply(a, additive_identity) = additive_identity) | (~sum(multiply(a, additive_identity), additive_identity, additive_identity)))),
% 5.87/4.01 inference(transitivity,[status(thm)],[128, 126])).
% 5.87/4.01 tff(130,plain,
% 5.87/4.01 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(multiply(a, additive_identity), additive_identity, multiply(a, additive_identity))) | (~sum(multiply(a, additive_identity), additive_identity, additive_identity)) | (multiply(a, additive_identity) = additive_identity))),
% 5.87/4.01 inference(quant_inst,[status(thm)],[])).
% 5.87/4.01 tff(131,plain,
% 5.87/4.01 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(multiply(a, additive_identity), additive_identity, multiply(a, additive_identity))) | (multiply(a, additive_identity) = additive_identity) | (~sum(multiply(a, additive_identity), additive_identity, additive_identity))),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[130, 129])).
% 5.87/4.01 tff(132,plain,
% 5.87/4.01 (multiply(a, additive_identity) = additive_identity),
% 5.87/4.01 inference(unit_resolution,[status(thm)],[131, 67, 125, 123])).
% 5.87/4.01 tff(133,plain,
% 5.87/4.01 (product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity)) <=> product(a, multiply(a, additive_identity), multiply(a, additive_identity))),
% 5.87/4.01 inference(monotonicity,[status(thm)],[72])).
% 5.87/4.01 tff(134,plain,
% 5.87/4.01 (product(a, multiply(a, additive_identity), multiply(a, additive_identity)) <=> product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity))),
% 5.87/4.01 inference(symmetry,[status(thm)],[133])).
% 5.87/4.01 tff(135,plain,
% 5.87/4.01 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : refl((product(X, V, W) | (~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))))),
% 5.87/4.01 inference(bind,[status(th)],[])).
% 5.87/4.01 tff(136,plain,
% 5.87/4.01 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 5.87/4.01 inference(quant_intro,[status(thm)],[135])).
% 5.87/4.01 tff(137,plain,
% 5.87/4.01 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 5.87/4.01 inference(rewrite,[status(thm)],[])).
% 5.87/4.01 tff(138,plain,
% 5.87/4.01 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : trans(monotonicity(rewrite((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) <=> ((~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))), (((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) | product(X, V, W)) <=> (((~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))) | product(X, V, W)))), rewrite((((~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))) | product(X, V, W)) <=> (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))), (((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) | product(X, V, W)) <=> (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))))),
% 5.87/4.01 inference(bind,[status(th)],[])).
% 5.87/4.01 tff(139,plain,
% 5.87/4.01 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) | product(X, V, W)) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 5.87/4.01 inference(quant_intro,[status(thm)],[138])).
% 5.87/4.01 tff(140,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) | product(X, V, W))), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','associativity_of_multiplication1')).
% 5.87/4.01 tff(141,plain,
% 5.87/4.01 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[140, 139])).
% 5.87/4.01 tff(142,plain,
% 5.87/4.01 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 5.87/4.01 inference(modus_ponens,[status(thm)],[141, 137])).
% 5.87/4.02 tff(143,plain,(
% 5.87/4.02 ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 5.87/4.02 inference(skolemize,[status(sab)],[142])).
% 5.87/4.02 tff(144,plain,
% 5.87/4.02 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 5.87/4.02 inference(modus_ponens,[status(thm)],[143, 136])).
% 5.87/4.02 tff(145,plain,
% 5.87/4.02 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(a, additive_identity, multiply(a, additive_identity))) | product(a, multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(a, a, a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(a, additive_identity, multiply(a, additive_identity))) | product(a, multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(a, a, a)))),
% 5.87/4.02 inference(rewrite,[status(thm)],[])).
% 5.87/4.02 tff(146,plain,
% 5.87/4.02 ((product(a, multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, a, a))) <=> ((~product(a, additive_identity, multiply(a, additive_identity))) | product(a, multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(a, a, a)))),
% 5.87/4.02 inference(rewrite,[status(thm)],[])).
% 5.87/4.02 tff(147,plain,
% 5.87/4.02 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(a, multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, a, a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(a, additive_identity, multiply(a, additive_identity))) | product(a, multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(a, a, a))))),
% 5.87/4.02 inference(monotonicity,[status(thm)],[146])).
% 5.87/4.02 tff(148,plain,
% 5.87/4.02 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(a, multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, a, a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(a, additive_identity, multiply(a, additive_identity))) | product(a, multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(a, a, a)))),
% 5.87/4.02 inference(transitivity,[status(thm)],[147, 145])).
% 5.87/4.02 tff(149,plain,
% 5.87/4.02 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(a, multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, additive_identity, multiply(a, additive_identity))) | (~product(a, a, a)))),
% 5.87/4.02 inference(quant_inst,[status(thm)],[])).
% 5.87/4.02 tff(150,plain,
% 5.87/4.02 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(a, additive_identity, multiply(a, additive_identity))) | product(a, multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(a, a, a))),
% 5.87/4.02 inference(modus_ponens,[status(thm)],[149, 148])).
% 5.87/4.02 tff(151,plain,
% 5.87/4.02 (product(a, multiply(a, additive_identity), multiply(a, additive_identity))),
% 5.87/4.02 inference(unit_resolution,[status(thm)],[150, 144, 29, 27])).
% 5.87/4.02 tff(152,plain,
% 5.87/4.02 (product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity))),
% 5.87/4.02 inference(modus_ponens,[status(thm)],[151, 134])).
% 5.87/4.02 tff(153,plain,
% 5.87/4.02 ((~![X: $i] : product(X, X, X)) | product(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))),
% 5.87/4.02 inference(quant_inst,[status(thm)],[])).
% 5.87/4.02 tff(154,plain,
% 5.87/4.02 (product(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))),
% 5.87/4.02 inference(unit_resolution,[status(thm)],[153, 25])).
% 5.87/4.02 tff(155,plain,
% 5.87/4.02 (^[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))))),
% 5.87/4.02 inference(bind,[status(th)],[])).
% 5.87/4.02 tff(156,plain,
% 5.87/4.02 (![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)))),
% 5.87/4.02 inference(quant_intro,[status(thm)],[155])).
% 5.87/4.02 tff(157,plain,
% 5.87/4.02 (![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)))),
% 5.87/4.02 inference(rewrite,[status(thm)],[])).
% 5.87/4.02 tff(158,plain,
% 5.87/4.02 (^[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)))))),
% 5.87/4.02 inference(bind,[status(th)],[])).
% 5.87/4.02 tff(159,plain,
% 5.87/4.02 (![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)))),
% 5.87/4.02 inference(quant_intro,[status(thm)],[158])).
% 5.87/4.02 tff(160,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/sandbox2/benchmark/Axioms/RNG001-0.ax','distributivity3')).
% 5.87/4.02 tff(161,plain,
% 5.87/4.02 (![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)))),
% 5.87/4.02 inference(modus_ponens,[status(thm)],[160, 159])).
% 5.87/4.02 tff(162,plain,
% 5.87/4.02 (![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)))),
% 5.87/4.02 inference(modus_ponens,[status(thm)],[161, 157])).
% 5.87/4.02 tff(163,plain,(
% 5.87/4.02 ![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)))),
% 5.87/4.02 inference(skolemize,[status(sab)],[162])).
% 5.87/4.02 tff(164,plain,
% 5.87/4.02 (![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)))),
% 5.87/4.02 inference(modus_ponens,[status(thm)],[163, 156])).
% 5.87/4.02 tff(165,plain,
% 5.87/4.02 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, 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(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity))))),
% 5.87/4.02 inference(rewrite,[status(thm)],[])).
% 5.87/4.02 tff(166,plain,
% 5.87/4.02 ((sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)))) <=> ((~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity))))),
% 5.87/4.02 inference(rewrite,[status(thm)],[])).
% 5.87/4.02 tff(167,plain,
% 5.87/4.02 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, 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(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity)))))),
% 5.87/4.02 inference(monotonicity,[status(thm)],[166])).
% 5.87/4.02 tff(168,plain,
% 5.87/4.02 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, 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(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity))))),
% 5.87/4.02 inference(transitivity,[status(thm)],[167, 165])).
% 5.87/4.02 tff(169,plain,
% 5.87/4.02 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))))),
% 5.87/4.02 inference(quant_inst,[status(thm)],[])).
% 5.87/4.02 tff(170,plain,
% 5.87/4.02 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.02 inference(modus_ponens,[status(thm)],[169, 168])).
% 5.87/4.02 tff(171,plain,
% 5.87/4.02 ((~sum(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)) | (~product(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.02 inference(unit_resolution,[status(thm)],[170, 164, 154])).
% 5.87/4.02 tff(172,plain,
% 5.87/4.02 (sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))),
% 5.87/4.02 inference(unit_resolution,[status(thm)],[171, 93, 152])).
% 5.87/4.02 tff(173,plain,
% 5.87/4.02 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.02 inference(quant_inst,[status(thm)],[])).
% 5.87/4.02 tff(174,plain,
% 5.87/4.02 (sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.02 inference(unit_resolution,[status(thm)],[173, 16])).
% 5.87/4.02 tff(175,plain,
% 5.87/4.02 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (add(multiply(a, additive_identity), multiply(a, additive_identity)) = multiply(a, additive_identity)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (add(multiply(a, additive_identity), multiply(a, additive_identity)) = multiply(a, additive_identity)))),
% 5.87/4.02 inference(rewrite,[status(thm)],[])).
% 5.87/4.02 tff(176,plain,
% 5.87/4.02 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (add(multiply(a, additive_identity), multiply(a, additive_identity)) = multiply(a, additive_identity)))),
% 5.87/4.02 inference(quant_inst,[status(thm)],[])).
% 5.87/4.02 tff(177,plain,
% 5.87/4.02 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity))) | (add(multiply(a, additive_identity), multiply(a, additive_identity)) = multiply(a, additive_identity))),
% 5.87/4.02 inference(modus_ponens,[status(thm)],[176, 175])).
% 5.87/4.02 tff(178,plain,
% 5.87/4.02 (add(multiply(a, additive_identity), multiply(a, additive_identity)) = multiply(a, additive_identity)),
% 5.87/4.02 inference(unit_resolution,[status(thm)],[177, 67, 174, 172])).
% 5.87/4.02 tff(179,plain,
% 5.87/4.02 (add(multiply(a, additive_identity), multiply(a, additive_identity)) = additive_identity),
% 5.87/4.02 inference(transitivity,[status(thm)],[178, 132])).
% 5.87/4.02 tff(180,plain,
% 5.87/4.02 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(additive_identity, additive_inverse(a), add(additive_identity, additive_inverse(a)))),
% 5.87/4.02 inference(quant_inst,[status(thm)],[])).
% 5.87/4.02 tff(181,plain,
% 5.87/4.02 (sum(additive_identity, additive_inverse(a), add(additive_identity, additive_inverse(a)))),
% 5.87/4.02 inference(unit_resolution,[status(thm)],[180, 16])).
% 5.87/4.02 tff(182,plain,
% 5.87/4.02 (^[X: $i] : refl(sum(additive_identity, X, X) <=> sum(additive_identity, X, X))),
% 5.87/4.02 inference(bind,[status(th)],[])).
% 5.87/4.02 tff(183,plain,
% 5.87/4.02 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 5.87/4.02 inference(quant_intro,[status(thm)],[182])).
% 5.87/4.02 tff(184,plain,
% 5.87/4.02 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 5.87/4.02 inference(rewrite,[status(thm)],[])).
% 5.87/4.02 tff(185,axiom,(![X: $i] : sum(additive_identity, X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','additive_identity1')).
% 5.87/4.02 tff(186,plain,
% 5.87/4.02 (![X: $i] : sum(additive_identity, X, X)),
% 5.87/4.02 inference(modus_ponens,[status(thm)],[185, 184])).
% 5.87/4.02 tff(187,plain,(
% 5.87/4.02 ![X: $i] : sum(additive_identity, X, X)),
% 5.87/4.02 inference(skolemize,[status(sab)],[186])).
% 5.87/4.02 tff(188,plain,
% 5.87/4.02 (![X: $i] : sum(additive_identity, X, X)),
% 5.87/4.02 inference(modus_ponens,[status(thm)],[187, 183])).
% 5.87/4.02 tff(189,plain,
% 5.87/4.02 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, additive_inverse(a), additive_inverse(a))),
% 5.87/4.02 inference(quant_inst,[status(thm)],[])).
% 5.87/4.02 tff(190,plain,
% 5.87/4.02 (sum(additive_identity, additive_inverse(a), additive_inverse(a))),
% 5.87/4.02 inference(unit_resolution,[status(thm)],[189, 188])).
% 5.87/4.02 tff(191,plain,
% 5.87/4.02 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(additive_identity, additive_inverse(a), additive_inverse(a))) | (~sum(additive_identity, additive_inverse(a), add(additive_identity, additive_inverse(a)))) | (additive_inverse(a) = add(additive_identity, additive_inverse(a))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(additive_identity, additive_inverse(a), additive_inverse(a))) | (~sum(additive_identity, additive_inverse(a), add(additive_identity, additive_inverse(a)))) | (additive_inverse(a) = add(additive_identity, additive_inverse(a))))),
% 5.87/4.02 inference(rewrite,[status(thm)],[])).
% 5.87/4.02 tff(192,plain,
% 5.87/4.02 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(additive_identity, additive_inverse(a), additive_inverse(a))) | (~sum(additive_identity, additive_inverse(a), add(additive_identity, additive_inverse(a)))) | (additive_inverse(a) = add(additive_identity, additive_inverse(a))))),
% 5.87/4.02 inference(quant_inst,[status(thm)],[])).
% 5.87/4.02 tff(193,plain,
% 5.87/4.02 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(additive_identity, additive_inverse(a), additive_inverse(a))) | (~sum(additive_identity, additive_inverse(a), add(additive_identity, additive_inverse(a)))) | (additive_inverse(a) = add(additive_identity, additive_inverse(a)))),
% 5.87/4.02 inference(modus_ponens,[status(thm)],[192, 191])).
% 5.87/4.02 tff(194,plain,
% 5.87/4.02 (additive_inverse(a) = add(additive_identity, additive_inverse(a))),
% 5.87/4.02 inference(unit_resolution,[status(thm)],[193, 67, 190, 181])).
% 5.87/4.02 tff(195,plain,
% 5.87/4.02 (add(additive_identity, additive_inverse(a)) = additive_inverse(a)),
% 5.87/4.02 inference(symmetry,[status(thm)],[194])).
% 5.87/4.02 tff(196,plain,
% 5.87/4.02 (product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity))) <=> product(additive_inverse(a), additive_identity, additive_identity)),
% 5.87/4.02 inference(monotonicity,[status(thm)],[195, 179, 179])).
% 5.87/4.02 tff(197,plain,
% 5.87/4.02 (product(additive_inverse(a), additive_identity, additive_identity) <=> product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.02 inference(symmetry,[status(thm)],[196])).
% 5.87/4.02 tff(198,assumption,(~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))), introduced(assumption)).
% 5.87/4.02 tff(199,plain,
% 5.87/4.02 (sum(multiply(additive_inverse(a), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))) <=> sum(multiply(additive_inverse(a), multiply(a, additive_identity)), additive_identity, multiply(additive_inverse(a), multiply(a, additive_identity)))),
% 5.87/4.02 inference(monotonicity,[status(thm)],[179])).
% 5.87/4.02 tff(200,plain,
% 5.87/4.02 (sum(multiply(additive_inverse(a), multiply(a, additive_identity)), additive_identity, multiply(additive_inverse(a), multiply(a, additive_identity))) <=> sum(multiply(additive_inverse(a), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))),
% 5.87/4.02 inference(symmetry,[status(thm)],[199])).
% 5.87/4.02 tff(201,plain,
% 5.87/4.02 ((~![X: $i] : sum(X, additive_identity, X)) | sum(multiply(additive_inverse(a), multiply(a, additive_identity)), additive_identity, multiply(additive_inverse(a), multiply(a, additive_identity)))),
% 5.87/4.02 inference(quant_inst,[status(thm)],[])).
% 5.87/4.02 tff(202,plain,
% 5.87/4.02 (sum(multiply(additive_inverse(a), multiply(a, additive_identity)), additive_identity, multiply(additive_inverse(a), multiply(a, additive_identity)))),
% 5.87/4.02 inference(unit_resolution,[status(thm)],[201, 36])).
% 5.87/4.02 tff(203,plain,
% 5.87/4.02 (sum(multiply(additive_inverse(a), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))),
% 5.87/4.02 inference(modus_ponens,[status(thm)],[202, 200])).
% 5.87/4.02 tff(204,plain,
% 5.87/4.02 (product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))) <=> product(additive_inverse(a), multiply(a, additive_identity), multiply(additive_inverse(a), multiply(a, additive_identity)))),
% 5.87/4.02 inference(monotonicity,[status(thm)],[195, 178])).
% 5.87/4.02 tff(205,plain,
% 5.87/4.02 (product(additive_inverse(a), multiply(a, additive_identity), multiply(additive_inverse(a), multiply(a, additive_identity))) <=> product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))),
% 5.87/4.02 inference(symmetry,[status(thm)],[204])).
% 5.87/4.02 tff(206,plain,
% 5.87/4.02 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(additive_inverse(a), multiply(a, additive_identity), multiply(additive_inverse(a), multiply(a, additive_identity)))),
% 5.87/4.02 inference(quant_inst,[status(thm)],[])).
% 5.87/4.02 tff(207,plain,
% 5.87/4.02 (product(additive_inverse(a), multiply(a, additive_identity), multiply(additive_inverse(a), multiply(a, additive_identity)))),
% 5.87/4.02 inference(unit_resolution,[status(thm)],[206, 7])).
% 5.87/4.02 tff(208,plain,
% 5.87/4.02 (product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))),
% 5.87/4.02 inference(modus_ponens,[status(thm)],[207, 205])).
% 5.87/4.02 tff(209,plain,
% 5.87/4.02 (product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity))) <=> product(a, multiply(a, additive_identity), multiply(a, additive_identity))),
% 5.87/4.02 inference(monotonicity,[status(thm)],[72, 178, 178])).
% 5.87/4.02 tff(210,plain,
% 5.87/4.02 (product(a, multiply(a, additive_identity), multiply(a, additive_identity)) <=> product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.02 inference(symmetry,[status(thm)],[209])).
% 5.87/4.02 tff(211,plain,
% 5.87/4.02 (product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.02 inference(modus_ponens,[status(thm)],[151, 210])).
% 5.87/4.02 tff(212,plain,
% 5.87/4.02 (sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity))) <=> sum(additive_inverse(a), a, additive_identity)),
% 5.87/4.02 inference(monotonicity,[status(thm)],[195, 72, 179])).
% 5.87/4.02 tff(213,plain,
% 5.87/4.02 (sum(additive_inverse(a), a, additive_identity) <=> sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.02 inference(symmetry,[status(thm)],[212])).
% 5.87/4.02 tff(214,plain,
% 5.87/4.02 (^[X: $i] : refl(sum(additive_inverse(X), X, additive_identity) <=> sum(additive_inverse(X), X, additive_identity))),
% 5.87/4.02 inference(bind,[status(th)],[])).
% 5.87/4.02 tff(215,plain,
% 5.87/4.02 (![X: $i] : sum(additive_inverse(X), X, additive_identity) <=> ![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 5.87/4.02 inference(quant_intro,[status(thm)],[214])).
% 5.87/4.02 tff(216,plain,
% 5.87/4.02 (![X: $i] : sum(additive_inverse(X), X, additive_identity) <=> ![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 5.87/4.02 inference(rewrite,[status(thm)],[])).
% 5.87/4.02 tff(217,axiom,(![X: $i] : sum(additive_inverse(X), X, additive_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','left_inverse')).
% 5.87/4.03 tff(218,plain,
% 5.87/4.03 (![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 5.87/4.03 inference(modus_ponens,[status(thm)],[217, 216])).
% 5.87/4.03 tff(219,plain,(
% 5.87/4.03 ![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 5.87/4.03 inference(skolemize,[status(sab)],[218])).
% 5.87/4.03 tff(220,plain,
% 5.87/4.03 (![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 5.87/4.03 inference(modus_ponens,[status(thm)],[219, 215])).
% 5.87/4.03 tff(221,plain,
% 5.87/4.03 ((~![X: $i] : sum(additive_inverse(X), X, additive_identity)) | sum(additive_inverse(a), a, additive_identity)),
% 5.87/4.03 inference(quant_inst,[status(thm)],[])).
% 5.87/4.03 tff(222,plain,
% 5.87/4.03 (sum(additive_inverse(a), a, additive_identity)),
% 5.87/4.03 inference(unit_resolution,[status(thm)],[221, 220])).
% 5.87/4.03 tff(223,plain,
% 5.87/4.03 (sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.03 inference(modus_ponens,[status(thm)],[222, 213])).
% 5.87/4.03 tff(224,plain,
% 5.87/4.03 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl(((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1))) <=> ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1))))),
% 5.87/4.03 inference(bind,[status(th)],[])).
% 5.87/4.03 tff(225,plain,
% 5.87/4.03 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | 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(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 5.87/4.03 inference(quant_intro,[status(thm)],[224])).
% 5.87/4.03 tff(226,plain,
% 5.87/4.03 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | 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(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 5.87/4.03 inference(rewrite,[status(thm)],[])).
% 5.87/4.03 tff(227,plain,
% 5.87/4.03 (^[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))) | (~sum(V1, V2, V4))) <=> (((~sum(Y, Z, V3)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | (~sum(V1, V2, V4))))), rewrite((((~sum(Y, Z, V3)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | (~sum(V1, V2, V4))) <=> ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))), (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) <=> ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))))), ((((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(V3, X, V4)) <=> (((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | product(V3, X, V4)))), rewrite((((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | product(V3, X, V4)) <=> ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))), ((((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(V3, X, V4)) <=> ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))))),
% 5.87/4.03 inference(bind,[status(th)],[])).
% 5.87/4.03 tff(228,plain,
% 5.87/4.03 (![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))) | (~sum(V1, V2, V4))) | product(V3, X, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 5.87/4.03 inference(quant_intro,[status(thm)],[227])).
% 5.87/4.03 tff(229,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))) | (~sum(V1, V2, V4))) | product(V3, X, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','distributivity4')).
% 5.87/4.03 tff(230,plain,
% 5.87/4.03 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 5.87/4.03 inference(modus_ponens,[status(thm)],[229, 228])).
% 5.87/4.03 tff(231,plain,
% 5.87/4.03 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 5.87/4.03 inference(modus_ponens,[status(thm)],[230, 226])).
% 5.87/4.03 tff(232,plain,(
% 5.87/4.03 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 5.87/4.03 inference(skolemize,[status(sab)],[231])).
% 5.87/4.03 tff(233,plain,
% 5.87/4.03 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 5.87/4.03 inference(modus_ponens,[status(thm)],[232, 225])).
% 5.87/4.03 tff(234,plain,
% 5.87/4.03 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(additive_inverse(a), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(additive_inverse(a), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))))),
% 5.87/4.03 inference(rewrite,[status(thm)],[])).
% 5.87/4.03 tff(235,plain,
% 5.87/4.03 (((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(additive_inverse(a), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))))) <=> ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(additive_inverse(a), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))))),
% 5.87/4.03 inference(rewrite,[status(thm)],[])).
% 5.87/4.03 tff(236,plain,
% 5.87/4.03 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(additive_inverse(a), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(additive_inverse(a), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))))))),
% 5.87/4.03 inference(monotonicity,[status(thm)],[235])).
% 5.87/4.03 tff(237,plain,
% 5.87/4.03 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(additive_inverse(a), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(additive_inverse(a), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))))),
% 5.87/4.03 inference(transitivity,[status(thm)],[236, 234])).
% 5.87/4.03 tff(238,plain,
% 5.87/4.03 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(additive_inverse(a), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))))),
% 5.87/4.03 inference(quant_inst,[status(thm)],[])).
% 5.87/4.03 tff(239,plain,
% 5.87/4.03 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(additive_inverse(a), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity))))),
% 5.87/4.03 inference(modus_ponens,[status(thm)],[238, 237])).
% 5.87/4.03 tff(240,plain,
% 5.87/4.03 ($false),
% 5.87/4.03 inference(unit_resolution,[status(thm)],[239, 233, 223, 211, 208, 203, 198])).
% 5.87/4.03 tff(241,plain,(product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))), inference(lemma,lemma(discharge,[]))).
% 5.87/4.03 tff(242,plain,
% 5.87/4.03 ((~![X: $i] : product(X, X, X)) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.03 inference(quant_inst,[status(thm)],[])).
% 5.87/4.03 tff(243,plain,
% 5.87/4.03 (product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.03 inference(unit_resolution,[status(thm)],[242, 25])).
% 5.87/4.03 tff(244,plain,
% 5.87/4.03 (^[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)))),
% 5.87/4.03 inference(bind,[status(th)],[])).
% 5.87/4.03 tff(245,plain,
% 5.87/4.03 (![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))),
% 5.87/4.03 inference(quant_intro,[status(thm)],[244])).
% 5.87/4.03 tff(246,plain,
% 5.87/4.03 (![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))),
% 5.87/4.03 inference(rewrite,[status(thm)],[])).
% 5.87/4.03 tff(247,plain,
% 5.87/4.03 (^[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)))),
% 5.87/4.03 inference(bind,[status(th)],[])).
% 5.87/4.03 tff(248,plain,
% 5.87/4.03 (![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))),
% 5.87/4.03 inference(quant_intro,[status(thm)],[247])).
% 5.87/4.03 tff(249,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')).
% 5.87/4.03 tff(250,plain,
% 5.87/4.03 (![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))),
% 5.87/4.03 inference(modus_ponens,[status(thm)],[249, 248])).
% 5.87/4.03 tff(251,plain,
% 5.87/4.03 (![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))),
% 5.87/4.03 inference(modus_ponens,[status(thm)],[250, 246])).
% 5.87/4.03 tff(252,plain,(
% 5.87/4.03 ![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))),
% 5.87/4.03 inference(skolemize,[status(sab)],[251])).
% 5.87/4.03 tff(253,plain,
% 5.87/4.03 (![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))),
% 5.87/4.03 inference(modus_ponens,[status(thm)],[252, 245])).
% 5.87/4.03 tff(254,plain,
% 5.87/4.03 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))) | ((~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))) | (multiply(additive_inverse(a), multiply(a, additive_identity)) = add(multiply(a, additive_identity), multiply(a, additive_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))) | (multiply(additive_inverse(a), multiply(a, additive_identity)) = add(multiply(a, additive_identity), multiply(a, additive_identity))))),
% 5.87/4.03 inference(rewrite,[status(thm)],[])).
% 5.87/4.03 tff(255,plain,
% 5.87/4.03 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))) | ((~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))) | (multiply(additive_inverse(a), multiply(a, additive_identity)) = add(multiply(a, additive_identity), multiply(a, additive_identity))))),
% 5.87/4.03 inference(quant_inst,[status(thm)],[])).
% 5.87/4.03 tff(256,plain,
% 5.87/4.03 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~product(X, Y, V)) | (~product(X, Y, U)) | (U = V))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), multiply(additive_inverse(a), multiply(a, additive_identity)))) | (multiply(additive_inverse(a), multiply(a, additive_identity)) = add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.03 inference(modus_ponens,[status(thm)],[255, 254])).
% 5.87/4.03 tff(257,plain,
% 5.87/4.03 (multiply(additive_inverse(a), multiply(a, additive_identity)) = add(multiply(a, additive_identity), multiply(a, additive_identity))),
% 5.87/4.03 inference(unit_resolution,[status(thm)],[256, 253, 243, 241])).
% 5.87/4.03 tff(258,plain,
% 5.87/4.03 (multiply(additive_inverse(a), multiply(a, additive_identity)) = additive_identity),
% 5.87/4.03 inference(transitivity,[status(thm)],[257, 178, 132])).
% 5.87/4.03 tff(259,plain,
% 5.87/4.03 (product(additive_inverse(a), multiply(a, additive_identity), multiply(additive_inverse(a), multiply(a, additive_identity))) <=> product(additive_inverse(a), additive_identity, additive_identity)),
% 5.87/4.03 inference(monotonicity,[status(thm)],[132, 258])).
% 5.87/4.03 tff(260,plain,
% 5.87/4.03 (product(additive_inverse(a), multiply(a, additive_identity), multiply(additive_inverse(a), multiply(a, additive_identity))) <=> product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.03 inference(transitivity,[status(thm)],[259, 197])).
% 5.87/4.03 tff(261,plain,
% 5.87/4.03 (product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.03 inference(modus_ponens,[status(thm)],[207, 260])).
% 5.87/4.03 tff(262,plain,
% 5.87/4.03 (sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))) <=> sum(additive_identity, additive_inverse(a), additive_inverse(a))),
% 5.87/4.03 inference(monotonicity,[status(thm)],[179, 195, 195])).
% 5.87/4.03 tff(263,plain,
% 5.87/4.03 (sum(additive_identity, additive_inverse(a), additive_inverse(a)) <=> sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))),
% 5.87/4.03 inference(symmetry,[status(thm)],[262])).
% 5.87/4.03 tff(264,plain,
% 5.87/4.03 (sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))),
% 5.87/4.03 inference(modus_ponens,[status(thm)],[190, 263])).
% 5.87/4.03 tff(265,plain,
% 5.87/4.03 (additive_identity = multiply(a, additive_identity)),
% 5.87/4.03 inference(symmetry,[status(thm)],[132])).
% 5.87/4.03 tff(266,plain,
% 5.87/4.03 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, additive_inverse(additive_identity), additive_inverse(additive_identity))),
% 5.87/4.03 inference(quant_inst,[status(thm)],[])).
% 5.87/4.03 tff(267,plain,
% 5.87/4.03 (sum(additive_identity, additive_inverse(additive_identity), additive_inverse(additive_identity))),
% 5.87/4.03 inference(unit_resolution,[status(thm)],[266, 188])).
% 5.87/4.03 tff(268,plain,
% 5.87/4.03 ((~![X: $i] : sum(X, additive_inverse(X), additive_identity)) | sum(additive_identity, additive_inverse(additive_identity), additive_identity)),
% 5.87/4.03 inference(quant_inst,[status(thm)],[])).
% 5.87/4.03 tff(269,plain,
% 5.87/4.03 (sum(additive_identity, additive_inverse(additive_identity), additive_identity)),
% 5.87/4.03 inference(unit_resolution,[status(thm)],[268, 102])).
% 5.87/4.03 tff(270,plain,
% 5.87/4.03 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | (~sum(additive_identity, additive_inverse(additive_identity), additive_inverse(additive_identity))) | (additive_identity = additive_inverse(additive_identity)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | (~sum(additive_identity, additive_inverse(additive_identity), additive_inverse(additive_identity))) | (additive_identity = additive_inverse(additive_identity)))),
% 5.87/4.03 inference(rewrite,[status(thm)],[])).
% 5.87/4.03 tff(271,plain,
% 5.87/4.03 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | ((~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | (~sum(additive_identity, additive_inverse(additive_identity), additive_inverse(additive_identity))) | (additive_identity = additive_inverse(additive_identity)))),
% 5.87/4.03 inference(quant_inst,[status(thm)],[])).
% 5.87/4.03 tff(272,plain,
% 5.87/4.03 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~sum(X, Y, U)) | (~sum(X, Y, V)) | (U = V))) | (~sum(additive_identity, additive_inverse(additive_identity), additive_identity)) | (~sum(additive_identity, additive_inverse(additive_identity), additive_inverse(additive_identity))) | (additive_identity = additive_inverse(additive_identity))),
% 5.87/4.03 inference(modus_ponens,[status(thm)],[271, 270])).
% 5.87/4.03 tff(273,plain,
% 5.87/4.03 (additive_identity = additive_inverse(additive_identity)),
% 5.87/4.03 inference(unit_resolution,[status(thm)],[272, 67, 269, 267])).
% 5.87/4.03 tff(274,plain,
% 5.87/4.03 (additive_inverse(additive_identity) = additive_identity),
% 5.87/4.03 inference(symmetry,[status(thm)],[273])).
% 5.87/4.03 tff(275,plain,
% 5.87/4.03 (additive_inverse(add(multiply(a, additive_identity), multiply(a, additive_identity))) = additive_inverse(additive_identity)),
% 5.87/4.03 inference(monotonicity,[status(thm)],[179])).
% 5.87/4.03 tff(276,plain,
% 5.87/4.03 (multiply(a, additive_identity) = add(multiply(a, additive_identity), multiply(a, additive_identity))),
% 5.87/4.03 inference(symmetry,[status(thm)],[178])).
% 5.87/4.03 tff(277,plain,
% 5.87/4.03 (additive_inverse(multiply(a, additive_identity)) = additive_inverse(add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.03 inference(monotonicity,[status(thm)],[276])).
% 5.87/4.03 tff(278,plain,
% 5.87/4.03 (additive_inverse(multiply(a, additive_identity)) = multiply(a, additive_identity)),
% 5.87/4.03 inference(transitivity,[status(thm)],[277, 275, 274, 265])).
% 5.87/4.03 tff(279,plain,
% 5.87/4.03 (add(additive_identity, additive_inverse(multiply(a, additive_identity))) = add(multiply(a, additive_identity), multiply(a, additive_identity))),
% 5.87/4.03 inference(monotonicity,[status(thm)],[265, 278])).
% 5.87/4.03 tff(280,plain,
% 5.87/4.03 (add(additive_identity, additive_inverse(multiply(a, additive_identity))) = additive_identity),
% 5.87/4.03 inference(transitivity,[status(thm)],[279, 178, 132])).
% 5.87/4.03 tff(281,plain,
% 5.87/4.03 (product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity)))) <=> product(additive_inverse(a), additive_identity, additive_identity)),
% 5.87/4.04 inference(monotonicity,[status(thm)],[195, 280])).
% 5.87/4.04 tff(282,plain,
% 5.87/4.04 (product(additive_inverse(a), additive_identity, additive_identity) <=> product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity))))),
% 5.87/4.04 inference(symmetry,[status(thm)],[281])).
% 5.87/4.04 tff(283,plain,
% 5.87/4.04 (product(additive_inverse(a), multiply(a, additive_identity), multiply(additive_inverse(a), multiply(a, additive_identity))) <=> product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity))))),
% 5.87/4.04 inference(transitivity,[status(thm)],[259, 282])).
% 5.87/4.04 tff(284,plain,
% 5.87/4.04 (product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity))))),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[207, 283])).
% 5.87/4.04 tff(285,plain,
% 5.87/4.04 (product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))) <=> product(additive_inverse(a), additive_inverse(a), additive_inverse(a))),
% 5.87/4.04 inference(monotonicity,[status(thm)],[195, 195, 195])).
% 5.87/4.04 tff(286,plain,
% 5.87/4.04 (product(additive_inverse(a), additive_inverse(a), additive_inverse(a)) <=> product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))),
% 5.87/4.04 inference(symmetry,[status(thm)],[285])).
% 5.87/4.04 tff(287,plain,
% 5.87/4.04 ((~![X: $i] : product(X, X, X)) | product(additive_inverse(a), additive_inverse(a), additive_inverse(a))),
% 5.87/4.04 inference(quant_inst,[status(thm)],[])).
% 5.87/4.04 tff(288,plain,
% 5.87/4.04 (product(additive_inverse(a), additive_inverse(a), additive_inverse(a))),
% 5.87/4.04 inference(unit_resolution,[status(thm)],[287, 25])).
% 5.87/4.04 tff(289,plain,
% 5.87/4.04 (product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[288, 286])).
% 5.87/4.04 tff(290,plain,
% 5.87/4.04 (sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a))) <=> sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), additive_inverse(a))),
% 5.87/4.04 inference(monotonicity,[status(thm)],[195])).
% 5.87/4.04 tff(291,plain,
% 5.87/4.04 (sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), additive_inverse(a)) <=> sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a)))),
% 5.87/4.04 inference(symmetry,[status(thm)],[290])).
% 5.87/4.04 tff(292,plain,
% 5.87/4.04 ((~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), additive_inverse(a))) <=> (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a))))),
% 5.87/4.04 inference(monotonicity,[status(thm)],[291])).
% 5.87/4.04 tff(293,plain,
% 5.87/4.04 (sum(add(a, multiply(a, additive_identity)), additive_identity, add(a, multiply(a, additive_identity))) <=> sum(a, additive_identity, a)),
% 5.87/4.04 inference(monotonicity,[status(thm)],[72, 72])).
% 5.87/4.04 tff(294,plain,
% 5.87/4.04 (sum(a, additive_identity, a) <=> sum(add(a, multiply(a, additive_identity)), additive_identity, add(a, multiply(a, additive_identity)))),
% 5.87/4.04 inference(symmetry,[status(thm)],[293])).
% 5.87/4.04 tff(295,plain,
% 5.87/4.04 (sum(add(a, multiply(a, additive_identity)), additive_identity, add(a, multiply(a, additive_identity)))),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[38, 294])).
% 5.87/4.04 tff(296,plain,
% 5.87/4.04 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(additive_identity, additive_inverse(multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))))),
% 5.87/4.04 inference(quant_inst,[status(thm)],[])).
% 5.87/4.04 tff(297,plain,
% 5.87/4.04 (sum(additive_identity, additive_inverse(multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))))),
% 5.87/4.04 inference(unit_resolution,[status(thm)],[296, 16])).
% 5.87/4.04 tff(298,plain,
% 5.87/4.04 (sum(additive_inverse(multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), additive_inverse(a)) <=> sum(additive_inverse(multiply(a, additive_identity)), a, additive_inverse(a))),
% 5.87/4.04 inference(monotonicity,[status(thm)],[72])).
% 5.87/4.04 tff(299,plain,
% 5.87/4.04 (sum(additive_inverse(multiply(a, additive_identity)), a, additive_inverse(a)) <=> sum(additive_inverse(multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), additive_inverse(a))),
% 5.87/4.04 inference(symmetry,[status(thm)],[298])).
% 5.87/4.04 tff(300,plain,
% 5.87/4.04 ((~sum(additive_inverse(multiply(a, additive_identity)), a, additive_inverse(a))) <=> (~sum(additive_inverse(multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), additive_inverse(a)))),
% 5.87/4.04 inference(monotonicity,[status(thm)],[299])).
% 5.87/4.04 tff(301,plain,
% 5.87/4.04 ((~![X: $i] : sum(additive_inverse(X), X, additive_identity)) | sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)),
% 5.87/4.04 inference(quant_inst,[status(thm)],[])).
% 5.87/4.04 tff(302,plain,
% 5.87/4.04 (sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)),
% 5.87/4.04 inference(unit_resolution,[status(thm)],[301, 220])).
% 5.87/4.04 tff(303,plain,
% 5.87/4.04 ((~sum(a, a, additive_identity)) <=> (~sum(a, a, additive_identity))),
% 5.87/4.04 inference(rewrite,[status(thm)],[])).
% 5.87/4.04 tff(304,axiom,(~sum(a, a, additive_identity)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_a_plus_a_is_id')).
% 5.87/4.04 tff(305,plain,
% 5.87/4.04 (~sum(a, a, additive_identity)),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[304, 303])).
% 5.87/4.04 tff(306,plain,
% 5.87/4.04 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(a, a, additive_identity) | (~sum(additive_inverse(a), a, additive_identity)) | (~sum(additive_identity, a, a)) | (~sum(a, additive_identity, additive_inverse(a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | sum(a, a, additive_identity) | (~sum(additive_inverse(a), a, additive_identity)) | (~sum(additive_identity, a, a)) | (~sum(a, additive_identity, additive_inverse(a))))),
% 5.87/4.04 inference(rewrite,[status(thm)],[])).
% 5.87/4.04 tff(307,plain,
% 5.87/4.04 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(a, a, additive_identity) | (~sum(additive_inverse(a), a, additive_identity)) | (~sum(additive_identity, a, a)) | (~sum(a, additive_identity, additive_inverse(a))))),
% 5.87/4.04 inference(quant_inst,[status(thm)],[])).
% 5.87/4.04 tff(308,plain,
% 5.87/4.04 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | sum(a, a, additive_identity) | (~sum(additive_inverse(a), a, additive_identity)) | (~sum(additive_identity, a, a)) | (~sum(a, additive_identity, additive_inverse(a)))),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[307, 306])).
% 5.87/4.04 tff(309,plain,
% 5.87/4.04 (~sum(a, additive_identity, additive_inverse(a))),
% 5.87/4.04 inference(unit_resolution,[status(thm)],[308, 115, 305, 222, 85])).
% 5.87/4.04 tff(310,plain,
% 5.87/4.04 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(additive_identity, a, additive_inverse(a))) | sum(a, additive_identity, additive_inverse(a)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(additive_identity, a, additive_inverse(a))) | sum(a, additive_identity, additive_inverse(a)))),
% 5.87/4.04 inference(rewrite,[status(thm)],[])).
% 5.87/4.04 tff(311,plain,
% 5.87/4.04 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(additive_identity, a, additive_inverse(a))) | sum(a, additive_identity, additive_inverse(a)))),
% 5.87/4.04 inference(quant_inst,[status(thm)],[])).
% 5.87/4.04 tff(312,plain,
% 5.87/4.04 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(additive_identity, a, additive_inverse(a))) | sum(a, additive_identity, additive_inverse(a))),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[311, 310])).
% 5.87/4.04 tff(313,plain,
% 5.87/4.04 (~sum(additive_identity, a, additive_inverse(a))),
% 5.87/4.04 inference(unit_resolution,[status(thm)],[312, 81, 309])).
% 5.87/4.04 tff(314,plain,
% 5.87/4.04 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : refl((sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) <=> (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))))),
% 5.87/4.04 inference(bind,[status(th)],[])).
% 5.87/4.04 tff(315,plain,
% 5.87/4.04 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 5.87/4.04 inference(quant_intro,[status(thm)],[314])).
% 5.87/4.04 tff(316,plain,
% 5.87/4.04 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 5.87/4.04 inference(rewrite,[status(thm)],[])).
% 5.87/4.04 tff(317,plain,
% 5.87/4.04 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : trans(monotonicity(rewrite((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) <=> ((~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))), (((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) | sum(U, Z, W)) <=> (((~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) | sum(U, Z, W)))), rewrite((((~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W))) | sum(U, Z, W)) <=> (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))), (((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) | sum(U, Z, W)) <=> (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))))),
% 5.87/4.04 inference(bind,[status(th)],[])).
% 5.87/4.04 tff(318,plain,
% 5.87/4.04 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) | sum(U, Z, W)) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 5.87/4.04 inference(quant_intro,[status(thm)],[317])).
% 5.87/4.04 tff(319,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) | sum(U, Z, W))), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','associativity_of_addition2')).
% 5.87/4.04 tff(320,plain,
% 5.87/4.04 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[319, 318])).
% 5.87/4.04 tff(321,plain,
% 5.87/4.04 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[320, 316])).
% 5.87/4.04 tff(322,plain,(
% 5.87/4.04 ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 5.87/4.04 inference(skolemize,[status(sab)],[321])).
% 5.87/4.04 tff(323,plain,
% 5.87/4.04 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[322, 315])).
% 5.87/4.04 tff(324,plain,
% 5.87/4.04 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(additive_identity, a, additive_inverse(a)) | (~sum(additive_inverse(multiply(a, additive_identity)), a, additive_inverse(a))) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | (~sum(multiply(a, additive_identity), a, a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | sum(additive_identity, a, additive_inverse(a)) | (~sum(additive_inverse(multiply(a, additive_identity)), a, additive_inverse(a))) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | (~sum(multiply(a, additive_identity), a, a)))),
% 5.87/4.04 inference(rewrite,[status(thm)],[])).
% 5.87/4.04 tff(325,plain,
% 5.87/4.04 ((sum(additive_identity, a, additive_inverse(a)) | (~sum(multiply(a, additive_identity), a, a)) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | (~sum(additive_inverse(multiply(a, additive_identity)), a, additive_inverse(a)))) <=> (sum(additive_identity, a, additive_inverse(a)) | (~sum(additive_inverse(multiply(a, additive_identity)), a, additive_inverse(a))) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | (~sum(multiply(a, additive_identity), a, a)))),
% 5.87/4.04 inference(rewrite,[status(thm)],[])).
% 5.87/4.04 tff(326,plain,
% 5.87/4.04 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(additive_identity, a, additive_inverse(a)) | (~sum(multiply(a, additive_identity), a, a)) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | (~sum(additive_inverse(multiply(a, additive_identity)), a, additive_inverse(a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(additive_identity, a, additive_inverse(a)) | (~sum(additive_inverse(multiply(a, additive_identity)), a, additive_inverse(a))) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | (~sum(multiply(a, additive_identity), a, a))))),
% 5.87/4.04 inference(monotonicity,[status(thm)],[325])).
% 5.87/4.04 tff(327,plain,
% 5.87/4.04 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(additive_identity, a, additive_inverse(a)) | (~sum(multiply(a, additive_identity), a, a)) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | (~sum(additive_inverse(multiply(a, additive_identity)), a, additive_inverse(a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | sum(additive_identity, a, additive_inverse(a)) | (~sum(additive_inverse(multiply(a, additive_identity)), a, additive_inverse(a))) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | (~sum(multiply(a, additive_identity), a, a)))),
% 5.87/4.04 inference(transitivity,[status(thm)],[326, 324])).
% 5.87/4.04 tff(328,plain,
% 5.87/4.04 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(additive_identity, a, additive_inverse(a)) | (~sum(multiply(a, additive_identity), a, a)) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | (~sum(additive_inverse(multiply(a, additive_identity)), a, additive_inverse(a))))),
% 5.87/4.04 inference(quant_inst,[status(thm)],[])).
% 5.87/4.04 tff(329,plain,
% 5.87/4.04 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | sum(additive_identity, a, additive_inverse(a)) | (~sum(additive_inverse(multiply(a, additive_identity)), a, additive_inverse(a))) | (~sum(additive_inverse(multiply(a, additive_identity)), multiply(a, additive_identity), additive_identity)) | (~sum(multiply(a, additive_identity), a, a))),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[328, 327])).
% 5.87/4.04 tff(330,plain,
% 5.87/4.04 (~sum(additive_inverse(multiply(a, additive_identity)), a, additive_inverse(a))),
% 5.87/4.04 inference(unit_resolution,[status(thm)],[329, 323, 313, 302, 92])).
% 5.87/4.04 tff(331,plain,
% 5.87/4.04 (~sum(additive_inverse(multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), additive_inverse(a))),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[330, 300])).
% 5.87/4.04 tff(332,plain,
% 5.87/4.04 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a))) | sum(additive_inverse(multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), additive_inverse(a)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a))) | sum(additive_inverse(multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), additive_inverse(a)))),
% 5.87/4.04 inference(rewrite,[status(thm)],[])).
% 5.87/4.04 tff(333,plain,
% 5.87/4.04 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a))) | sum(additive_inverse(multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), additive_inverse(a)))),
% 5.87/4.04 inference(quant_inst,[status(thm)],[])).
% 5.87/4.04 tff(334,plain,
% 5.87/4.04 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a))) | sum(additive_inverse(multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), additive_inverse(a))),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[333, 332])).
% 5.87/4.04 tff(335,plain,
% 5.87/4.04 ((~sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a))) | sum(additive_inverse(multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), additive_inverse(a))),
% 5.87/4.04 inference(unit_resolution,[status(thm)],[334, 81])).
% 5.87/4.04 tff(336,plain,
% 5.87/4.04 (~sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a))),
% 5.87/4.04 inference(unit_resolution,[status(thm)],[335, 331])).
% 5.87/4.04 tff(337,plain,
% 5.87/4.04 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a)) | (~sum(add(a, multiply(a, additive_identity)), additive_identity, add(a, multiply(a, additive_identity)))) | (~sum(additive_identity, additive_inverse(multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), additive_inverse(a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a)) | (~sum(add(a, multiply(a, additive_identity)), additive_identity, add(a, multiply(a, additive_identity)))) | (~sum(additive_identity, additive_inverse(multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), additive_inverse(a))))),
% 5.87/4.04 inference(rewrite,[status(thm)],[])).
% 5.87/4.04 tff(338,plain,
% 5.87/4.04 ((sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a)) | (~sum(additive_identity, additive_inverse(multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))))) | (~sum(add(a, multiply(a, additive_identity)), additive_identity, add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), additive_inverse(a)))) <=> (sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a)) | (~sum(add(a, multiply(a, additive_identity)), additive_identity, add(a, multiply(a, additive_identity)))) | (~sum(additive_identity, additive_inverse(multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), additive_inverse(a))))),
% 5.87/4.04 inference(rewrite,[status(thm)],[])).
% 5.87/4.04 tff(339,plain,
% 5.87/4.04 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a)) | (~sum(additive_identity, additive_inverse(multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))))) | (~sum(add(a, multiply(a, additive_identity)), additive_identity, add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), additive_inverse(a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a)) | (~sum(add(a, multiply(a, additive_identity)), additive_identity, add(a, multiply(a, additive_identity)))) | (~sum(additive_identity, additive_inverse(multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), additive_inverse(a)))))),
% 5.87/4.04 inference(monotonicity,[status(thm)],[338])).
% 5.87/4.04 tff(340,plain,
% 5.87/4.04 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a)) | (~sum(additive_identity, additive_inverse(multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))))) | (~sum(add(a, multiply(a, additive_identity)), additive_identity, add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), additive_inverse(a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a)) | (~sum(add(a, multiply(a, additive_identity)), additive_identity, add(a, multiply(a, additive_identity)))) | (~sum(additive_identity, additive_inverse(multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), additive_inverse(a))))),
% 5.87/4.04 inference(transitivity,[status(thm)],[339, 337])).
% 5.87/4.04 tff(341,plain,
% 5.87/4.04 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a)) | (~sum(additive_identity, additive_inverse(multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))))) | (~sum(add(a, multiply(a, additive_identity)), additive_identity, add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), additive_inverse(a))))),
% 5.87/4.04 inference(quant_inst,[status(thm)],[])).
% 5.87/4.04 tff(342,plain,
% 5.87/4.04 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | sum(add(a, multiply(a, additive_identity)), additive_inverse(multiply(a, additive_identity)), additive_inverse(a)) | (~sum(add(a, multiply(a, additive_identity)), additive_identity, add(a, multiply(a, additive_identity)))) | (~sum(additive_identity, additive_inverse(multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), additive_inverse(a)))),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[341, 340])).
% 5.87/4.04 tff(343,plain,
% 5.87/4.04 (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), additive_inverse(a))),
% 5.87/4.04 inference(unit_resolution,[status(thm)],[342, 323, 336, 297, 295])).
% 5.87/4.04 tff(344,plain,
% 5.87/4.04 (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a)))),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[343, 292])).
% 5.87/4.04 tff(345,plain,
% 5.87/4.04 (sum(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(a))) <=> sum(additive_inverse(a), additive_identity, additive_inverse(a))),
% 5.87/4.04 inference(monotonicity,[status(thm)],[195, 195])).
% 5.87/4.04 tff(346,plain,
% 5.87/4.04 (sum(additive_inverse(a), additive_identity, additive_inverse(a)) <=> sum(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(a)))),
% 5.87/4.04 inference(symmetry,[status(thm)],[345])).
% 5.87/4.04 tff(347,plain,
% 5.87/4.04 ((~![X: $i] : sum(X, additive_identity, X)) | sum(additive_inverse(a), additive_identity, additive_inverse(a))),
% 5.87/4.04 inference(quant_inst,[status(thm)],[])).
% 5.87/4.04 tff(348,plain,
% 5.87/4.04 (sum(additive_inverse(a), additive_identity, additive_inverse(a))),
% 5.87/4.04 inference(unit_resolution,[status(thm)],[347, 36])).
% 5.87/4.04 tff(349,plain,
% 5.87/4.04 (sum(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(a)))),
% 5.87/4.04 inference(modus_ponens,[status(thm)],[348, 346])).
% 5.87/4.04 tff(350,plain,
% 5.87/4.04 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))) | (~product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity))))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))) | (~product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity))))))),
% 5.87/4.04 inference(rewrite,[status(thm)],[])).
% 5.87/4.04 tff(351,plain,
% 5.87/4.04 ((sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity))))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))))) <=> (sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))) | (~product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity))))))),
% 5.87/4.05 inference(rewrite,[status(thm)],[])).
% 5.87/4.05 tff(352,plain,
% 5.87/4.05 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity))))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))) | (~product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity)))))))),
% 5.87/4.05 inference(monotonicity,[status(thm)],[351])).
% 5.87/4.05 tff(353,plain,
% 5.87/4.05 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity))))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))) | (~product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity))))))),
% 5.87/4.05 inference(transitivity,[status(thm)],[352, 350])).
% 5.87/4.05 tff(354,plain,
% 5.87/4.05 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity))))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))))),
% 5.87/4.05 inference(quant_inst,[status(thm)],[])).
% 5.87/4.05 tff(355,plain,
% 5.87/4.05 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))) | (~product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity)))))),
% 5.87/4.05 inference(modus_ponens,[status(thm)],[354, 353])).
% 5.87/4.05 tff(356,plain,
% 5.87/4.05 (sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(multiply(a, additive_identity))), add(additive_identity, additive_inverse(a))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))) | (~product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity)))))),
% 5.87/4.05 inference(unit_resolution,[status(thm)],[355, 48])).
% 5.87/4.05 tff(357,plain,
% 5.87/4.05 ((~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))) | (~product(add(additive_identity, additive_inverse(a)), additive_identity, add(additive_identity, additive_inverse(multiply(a, additive_identity)))))),
% 5.87/4.05 inference(unit_resolution,[status(thm)],[356, 349, 344, 289])).
% 5.87/4.05 tff(358,plain,
% 5.87/4.05 (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))),
% 5.87/4.05 inference(unit_resolution,[status(thm)],[357, 284])).
% 5.87/4.05 tff(359,plain,
% 5.87/4.05 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) <=> (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))))),
% 5.87/4.05 inference(bind,[status(th)],[])).
% 5.87/4.05 tff(360,plain,
% 5.87/4.05 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 5.87/4.05 inference(quant_intro,[status(thm)],[359])).
% 5.87/4.05 tff(361,plain,
% 5.87/4.05 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 5.87/4.05 inference(rewrite,[status(thm)],[])).
% 5.87/4.05 tff(362,plain,
% 5.87/4.05 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) <=> (((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~sum(V1, V2, V4))))), rewrite((((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~sum(V1, V2, V4))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> (((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) | product(X, V3, V4)))), rewrite((((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))))),
% 5.87/4.05 inference(bind,[status(th)],[])).
% 5.87/4.05 tff(363,plain,
% 5.87/4.05 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 5.87/4.05 inference(quant_intro,[status(thm)],[362])).
% 5.87/4.05 tff(364,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax','distributivity2')).
% 5.87/4.05 tff(365,plain,
% 5.87/4.05 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 5.87/4.05 inference(modus_ponens,[status(thm)],[364, 363])).
% 5.87/4.05 tff(366,plain,
% 5.87/4.05 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 5.87/4.05 inference(modus_ponens,[status(thm)],[365, 361])).
% 5.87/4.05 tff(367,plain,(
% 5.87/4.05 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 5.87/4.05 inference(skolemize,[status(sab)],[366])).
% 5.87/4.05 tff(368,plain,
% 5.87/4.05 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 5.87/4.05 inference(modus_ponens,[status(thm)],[367, 360])).
% 5.87/4.05 tff(369,plain,
% 5.87/4.05 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | ((~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))))),
% 5.87/4.05 inference(rewrite,[status(thm)],[])).
% 5.87/4.05 tff(370,plain,
% 5.87/4.05 ((product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))))) <=> ((~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))))),
% 5.87/4.05 inference(rewrite,[status(thm)],[])).
% 5.87/4.05 tff(371,plain,
% 5.87/4.05 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | ((~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))))))),
% 5.87/4.05 inference(monotonicity,[status(thm)],[370])).
% 5.87/4.05 tff(372,plain,
% 5.87/4.05 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))))),
% 5.87/4.05 inference(transitivity,[status(thm)],[371, 369])).
% 5.87/4.05 tff(373,plain,
% 5.87/4.05 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))))),
% 5.87/4.05 inference(quant_inst,[status(thm)],[])).
% 5.87/4.05 tff(374,plain,
% 5.87/4.05 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~product(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))))),
% 5.87/4.05 inference(modus_ponens,[status(thm)],[373, 372])).
% 5.87/4.05 tff(375,plain,
% 5.87/4.05 (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)))),
% 5.87/4.05 inference(unit_resolution,[status(thm)],[374, 368, 289, 358, 264, 261])).
% 5.87/4.05 tff(376,plain,
% 5.87/4.05 (sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity))) <=> sum(a, additive_inverse(a), additive_identity)),
% 5.87/4.05 inference(monotonicity,[status(thm)],[72, 195, 179])).
% 5.87/4.05 tff(377,plain,
% 5.87/4.05 (sum(a, additive_inverse(a), additive_identity) <=> sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.05 inference(symmetry,[status(thm)],[376])).
% 5.87/4.05 tff(378,plain,
% 5.87/4.05 (sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.05 inference(modus_ponens,[status(thm)],[104, 377])).
% 5.87/4.05 tff(379,plain,
% 5.87/4.05 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | ((~sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))))),
% 5.87/4.06 inference(rewrite,[status(thm)],[])).
% 5.87/4.06 tff(380,plain,
% 5.87/4.06 ((sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity))))) <=> ((~sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))))),
% 5.87/4.06 inference(rewrite,[status(thm)],[])).
% 5.87/4.06 tff(381,plain,
% 5.87/4.06 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | ((~sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))))))),
% 5.87/4.06 inference(monotonicity,[status(thm)],[380])).
% 5.87/4.06 tff(382,plain,
% 5.87/4.06 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))))),
% 5.87/4.06 inference(transitivity,[status(thm)],[381, 379])).
% 5.87/4.06 tff(383,plain,
% 5.87/4.06 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)))))),
% 5.87/4.06 inference(quant_inst,[status(thm)],[])).
% 5.87/4.06 tff(384,plain,
% 5.87/4.06 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(U, Z, W) | (~sum(Y, Z, V)) | (~sum(X, Y, U)) | (~sum(X, V, W)))) | (~sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))))),
% 5.87/4.06 inference(modus_ponens,[status(thm)],[383, 382])).
% 5.87/4.06 tff(385,plain,
% 5.87/4.06 (~sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)))),
% 5.87/4.06 inference(unit_resolution,[status(thm)],[384, 323, 378, 375, 18])).
% 5.87/4.06 tff(386,plain,
% 5.87/4.06 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity))))),
% 5.87/4.06 inference(rewrite,[status(thm)],[])).
% 5.87/4.06 tff(387,plain,
% 5.87/4.06 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity))))),
% 5.87/4.06 inference(quant_inst,[status(thm)],[])).
% 5.87/4.06 tff(388,plain,
% 5.87/4.06 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(add(a, multiply(a, additive_identity)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)))),
% 5.87/4.06 inference(modus_ponens,[status(thm)],[387, 386])).
% 5.87/4.06 tff(389,plain,
% 5.87/4.06 (~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))),
% 5.87/4.06 inference(unit_resolution,[status(thm)],[388, 81, 385])).
% 5.87/4.06 tff(390,plain,
% 5.87/4.06 (product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) <=> product(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))),
% 5.87/4.06 inference(monotonicity,[status(thm)],[178])).
% 5.87/4.06 tff(391,plain,
% 5.87/4.06 (product(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) <=> product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))),
% 5.87/4.06 inference(symmetry,[status(thm)],[390])).
% 5.87/4.06 tff(392,plain,
% 5.87/4.06 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))),
% 5.87/4.06 inference(quant_inst,[status(thm)],[])).
% 5.87/4.06 tff(393,plain,
% 5.87/4.06 (product(multiply(a, additive_identity), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))),
% 5.87/4.06 inference(unit_resolution,[status(thm)],[392, 7])).
% 5.87/4.06 tff(394,plain,
% 5.87/4.06 (product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))),
% 5.87/4.06 inference(modus_ponens,[status(thm)],[393, 391])).
% 5.87/4.06 tff(395,plain,
% 5.87/4.06 (product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) <=> product(a, a, a)),
% 5.87/4.06 inference(monotonicity,[status(thm)],[72, 72, 72])).
% 5.87/4.06 tff(396,plain,
% 5.87/4.06 (product(a, a, a) <=> product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))),
% 5.87/4.06 inference(symmetry,[status(thm)],[395])).
% 5.87/4.06 tff(397,plain,
% 5.87/4.06 (product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))),
% 5.87/4.06 inference(modus_ponens,[status(thm)],[27, 396])).
% 5.87/4.06 tff(398,plain,
% 5.87/4.06 (sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity))) <=> sum(a, multiply(a, additive_identity), add(a, multiply(a, additive_identity)))),
% 5.87/4.06 inference(monotonicity,[status(thm)],[72])).
% 5.87/4.06 tff(399,plain,
% 5.87/4.06 (sum(a, multiply(a, additive_identity), add(a, multiply(a, additive_identity))) <=> sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))),
% 5.87/4.06 inference(symmetry,[status(thm)],[398])).
% 5.87/4.06 tff(400,plain,
% 5.87/4.06 (sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))),
% 5.87/4.06 inference(modus_ponens,[status(thm)],[57, 399])).
% 5.87/4.06 tff(401,plain,
% 5.87/4.06 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | ((~sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) | sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) | sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity))))),
% 5.87/4.06 inference(rewrite,[status(thm)],[])).
% 5.87/4.06 tff(402,plain,
% 5.87/4.06 ((sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity))))) <=> ((~sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) | sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity))))),
% 5.87/4.06 inference(rewrite,[status(thm)],[])).
% 5.87/4.06 tff(403,plain,
% 5.87/4.06 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | ((~sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) | sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))))),
% 5.87/4.06 inference(monotonicity,[status(thm)],[402])).
% 5.87/4.06 tff(404,plain,
% 5.87/4.06 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) | sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity))))),
% 5.87/4.06 inference(transitivity,[status(thm)],[403, 401])).
% 5.87/4.06 tff(405,plain,
% 5.87/4.06 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))))),
% 5.87/4.06 inference(quant_inst,[status(thm)],[])).
% 5.87/4.06 tff(406,plain,
% 5.87/4.06 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(multiply(a, additive_identity), multiply(a, additive_identity), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(a, multiply(a, additive_identity)), multiply(a, additive_identity), add(a, multiply(a, additive_identity)))) | sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))),
% 5.87/4.06 inference(modus_ponens,[status(thm)],[405, 404])).
% 5.87/4.06 tff(407,plain,
% 5.87/4.06 (sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))),
% 5.87/4.06 inference(unit_resolution,[status(thm)],[406, 115, 174, 400])).
% 5.87/4.06 tff(408,plain,
% 5.87/4.06 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))))),
% 5.87/4.06 inference(rewrite,[status(thm)],[])).
% 5.87/4.06 tff(409,plain,
% 5.87/4.06 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))))),
% 5.87/4.06 inference(quant_inst,[status(thm)],[])).
% 5.87/4.06 tff(410,plain,
% 5.87/4.06 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))),
% 5.87/4.06 inference(modus_ponens,[status(thm)],[409, 408])).
% 5.87/4.06 tff(411,plain,
% 5.87/4.06 (sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))),
% 5.87/4.06 inference(unit_resolution,[status(thm)],[410, 81, 407])).
% 5.87/4.06 tff(412,plain,
% 5.87/4.06 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, 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(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))))),
% 5.87/4.06 inference(rewrite,[status(thm)],[])).
% 5.87/4.06 tff(413,plain,
% 5.87/4.06 ((sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity)))))) <=> (sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))))),
% 5.87/4.06 inference(rewrite,[status(thm)],[])).
% 5.87/4.06 tff(414,plain,
% 5.87/4.06 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, 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(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity)))))))),
% 5.87/4.07 inference(monotonicity,[status(thm)],[413])).
% 5.87/4.07 tff(415,plain,
% 5.87/4.07 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, 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(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))))),
% 5.87/4.07 inference(transitivity,[status(thm)],[414, 412])).
% 5.87/4.07 tff(416,plain,
% 5.87/4.07 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))))),
% 5.87/4.07 inference(quant_inst,[status(thm)],[])).
% 5.87/4.07 tff(417,plain,
% 5.87/4.07 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity)))))),
% 5.87/4.07 inference(modus_ponens,[status(thm)],[416, 415])).
% 5.87/4.07 tff(418,plain,
% 5.87/4.07 (sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))),
% 5.87/4.07 inference(unit_resolution,[status(thm)],[417, 164, 411, 397, 394])).
% 5.87/4.07 tff(419,plain,
% 5.87/4.07 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | ((~sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity)))))),
% 5.87/4.07 inference(rewrite,[status(thm)],[])).
% 5.87/4.07 tff(420,plain,
% 5.87/4.07 ((sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity))))) <=> ((~sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity)))))),
% 5.87/4.07 inference(rewrite,[status(thm)],[])).
% 5.87/4.07 tff(421,plain,
% 5.87/4.07 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | ((~sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity))))))),
% 5.87/4.07 inference(monotonicity,[status(thm)],[420])).
% 5.87/4.07 tff(422,plain,
% 5.87/4.07 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity)))))),
% 5.87/4.07 inference(transitivity,[status(thm)],[421, 419])).
% 5.87/4.07 tff(423,plain,
% 5.87/4.07 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity)))))),
% 5.87/4.07 inference(quant_inst,[status(thm)],[])).
% 5.87/4.07 tff(424,plain,
% 5.87/4.07 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity))) | (~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity))))),
% 5.87/4.07 inference(modus_ponens,[status(thm)],[423, 422])).
% 5.87/4.07 tff(425,plain,
% 5.87/4.07 (~sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.07 inference(unit_resolution,[status(thm)],[424, 115, 411, 418, 389])).
% 5.87/4.07 tff(426,plain,
% 5.87/4.07 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity))))),
% 5.87/4.07 inference(rewrite,[status(thm)],[])).
% 5.87/4.07 tff(427,plain,
% 5.87/4.07 ((sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))))) <=> ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity))))),
% 5.87/4.07 inference(rewrite,[status(thm)],[])).
% 5.87/4.07 tff(428,plain,
% 5.87/4.07 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity)))))),
% 5.87/4.08 inference(monotonicity,[status(thm)],[427])).
% 5.87/4.08 tff(429,plain,
% 5.87/4.08 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity))))),
% 5.87/4.08 inference(transitivity,[status(thm)],[428, 426])).
% 5.87/4.08 tff(430,plain,
% 5.87/4.08 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))))),
% 5.87/4.08 inference(quant_inst,[status(thm)],[])).
% 5.87/4.08 tff(431,plain,
% 5.87/4.08 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))))) | (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | sum(add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))), multiply(multiply(a, additive_identity), add(a, multiply(a, additive_identity))), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.87/4.08 inference(modus_ponens,[status(thm)],[430, 429])).
% 5.87/4.08 tff(432,plain,
% 5.87/4.08 (~product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))),
% 5.87/4.08 inference(unit_resolution,[status(thm)],[431, 48, 243, 223, 394, 425])).
% 5.87/4.08 tff(433,plain,
% 5.87/4.08 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))))),
% 5.95/4.08 inference(rewrite,[status(thm)],[])).
% 5.95/4.08 tff(434,plain,
% 5.95/4.08 (((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) <=> ((~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))))),
% 5.95/4.08 inference(rewrite,[status(thm)],[])).
% 5.95/4.08 tff(435,plain,
% 5.95/4.08 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))))),
% 5.95/4.08 inference(monotonicity,[status(thm)],[434])).
% 5.95/4.08 tff(436,plain,
% 5.95/4.08 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))))),
% 5.95/4.08 inference(transitivity,[status(thm)],[435, 433])).
% 5.95/4.08 tff(437,plain,
% 5.95/4.08 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))))),
% 5.95/4.08 inference(quant_inst,[status(thm)],[])).
% 5.95/4.08 tff(438,plain,
% 5.95/4.08 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))) | product(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))))),
% 5.95/4.08 inference(modus_ponens,[status(thm)],[437, 436])).
% 5.95/4.08 tff(439,plain,
% 5.95/4.08 (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))),
% 5.95/4.08 inference(unit_resolution,[status(thm)],[438, 233, 289, 223, 18, 432])).
% 5.95/4.08 tff(440,plain,
% 5.95/4.08 (sum(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))) <=> sum(additive_inverse(a), additive_identity, additive_inverse(a))),
% 5.95/4.08 inference(monotonicity,[status(thm)],[195, 179, 195])).
% 5.95/4.08 tff(441,plain,
% 5.95/4.08 (sum(additive_inverse(a), additive_identity, additive_inverse(a)) <=> sum(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))),
% 5.95/4.08 inference(symmetry,[status(thm)],[440])).
% 5.95/4.08 tff(442,plain,
% 5.95/4.08 (sum(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))),
% 5.95/4.08 inference(modus_ponens,[status(thm)],[348, 441])).
% 5.95/4.08 tff(443,plain,
% 5.95/4.08 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | ((~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))) | (~sum(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))) | (~sum(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))))),
% 5.95/4.08 inference(rewrite,[status(thm)],[])).
% 5.95/4.08 tff(444,plain,
% 5.95/4.08 ((product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))) | (~sum(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) <=> ((~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))) | (~sum(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))))),
% 5.95/4.08 inference(rewrite,[status(thm)],[])).
% 5.95/4.08 tff(445,plain,
% 5.95/4.08 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))) | (~sum(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | ((~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))) | (~sum(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))))),
% 5.95/4.09 inference(monotonicity,[status(thm)],[444])).
% 5.95/4.09 tff(446,plain,
% 5.95/4.09 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))) | (~sum(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))) | (~sum(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))))),
% 5.95/4.09 inference(transitivity,[status(thm)],[445, 443])).
% 5.95/4.09 tff(447,plain,
% 5.95/4.09 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))) | (~sum(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))))),
% 5.95/4.09 inference(quant_inst,[status(thm)],[])).
% 5.95/4.09 tff(448,plain,
% 5.95/4.09 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a))) | (~sum(add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))),
% 5.95/4.09 inference(modus_ponens,[status(thm)],[447, 446])).
% 5.95/4.09 tff(449,plain,
% 5.95/4.09 (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))),
% 5.95/4.09 inference(unit_resolution,[status(thm)],[448, 368, 211, 442, 439, 9])).
% 5.95/4.09 tff(450,plain,
% 5.95/4.09 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | ((~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))))),
% 5.95/4.09 inference(rewrite,[status(thm)],[])).
% 5.95/4.09 tff(451,plain,
% 5.95/4.09 ((sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity))))) <=> ((~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))))),
% 5.95/4.09 inference(rewrite,[status(thm)],[])).
% 5.95/4.09 tff(452,plain,
% 5.95/4.09 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | ((~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity))))))),
% 5.95/4.09 inference(monotonicity,[status(thm)],[451])).
% 5.95/4.09 tff(453,plain,
% 5.95/4.09 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))))),
% 5.95/4.09 inference(transitivity,[status(thm)],[452, 450])).
% 5.95/4.09 tff(454,plain,
% 5.95/4.09 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))))),
% 5.95/4.09 inference(quant_inst,[status(thm)],[])).
% 5.95/4.09 tff(455,plain,
% 5.95/4.09 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (sum(X, V, W) | (~sum(U, Z, W)) | (~sum(Y, Z, V)) | (~sum(X, Y, U)))) | (~sum(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), add(additive_identity, additive_inverse(a)))) | sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))) | (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity))))),
% 5.95/4.09 inference(modus_ponens,[status(thm)],[454, 453])).
% 5.95/4.09 tff(456,plain,
% 5.95/4.09 (~sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.95/4.09 inference(unit_resolution,[status(thm)],[455, 115, 378, 264, 449])).
% 5.95/4.09 tff(457,plain,
% 5.95/4.09 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity))))),
% 5.95/4.09 inference(rewrite,[status(thm)],[])).
% 5.95/4.09 tff(458,plain,
% 5.95/4.09 ((sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)))))) <=> ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity))))),
% 5.95/4.09 inference(rewrite,[status(thm)],[])).
% 5.95/4.09 tff(459,plain,
% 5.95/4.09 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))))),
% 5.95/4.09 inference(monotonicity,[status(thm)],[458])).
% 5.95/4.09 tff(460,plain,
% 5.95/4.09 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity))))),
% 5.95/4.09 inference(transitivity,[status(thm)],[459, 457])).
% 5.95/4.09 tff(461,plain,
% 5.95/4.09 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))))),
% 5.95/4.09 inference(quant_inst,[status(thm)],[])).
% 5.95/4.09 tff(462,plain,
% 5.95/4.09 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(add(additive_identity, additive_inverse(a)), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)), add(a, multiply(a, additive_identity)))) | (~product(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a)), multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))))) | sum(multiply(add(a, multiply(a, additive_identity)), add(additive_identity, additive_inverse(a))), add(a, multiply(a, additive_identity)), add(multiply(a, additive_identity), multiply(a, additive_identity)))),
% 5.95/4.10 inference(modus_ponens,[status(thm)],[461, 460])).
% 5.95/4.10 tff(463,plain,
% 5.95/4.10 ($false),
% 5.95/4.10 inference(unit_resolution,[status(thm)],[462, 48, 223, 211, 397, 9, 456])).
% 5.95/4.10 % SZS output end Proof
%------------------------------------------------------------------------------