TSTP Solution File: RNG005-2 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : RNG005-2 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n022.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 03:17:36 EDT 2022
% Result : Unsatisfiable 11.81s 7.82s
% Output : Proof 11.90s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.13 % Problem : RNG005-2 : TPTP v8.1.0. Released v1.0.0.
% 0.04/0.14 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.35 % Computer : n022.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Sep 2 21:17:10 EDT 2022
% 0.14/0.35 % CPUTime :
% 0.14/0.36 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.14/0.36 Usage: tptp [options] [-file:]file
% 0.14/0.36 -h, -? prints this message.
% 0.14/0.36 -smt2 print SMT-LIB2 benchmark.
% 0.14/0.36 -m, -model generate model.
% 0.14/0.36 -p, -proof generate proof.
% 0.14/0.36 -c, -core generate unsat core of named formulas.
% 0.14/0.36 -st, -statistics display statistics.
% 0.14/0.36 -t:timeout set timeout (in second).
% 0.14/0.36 -smt2status display status in smt2 format instead of SZS.
% 0.14/0.36 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.14/0.36 -<param>:<value> configuration parameter and value.
% 0.14/0.36 -o:<output-file> file to place output in.
% 11.81/7.82 % SZS status Unsatisfiable
% 11.81/7.82 % SZS output start Proof
% 11.81/7.82 tff(product_type, type, (
% 11.81/7.82 product: ( $i * $i * $i ) > $o)).
% 11.81/7.82 tff(add_type, type, (
% 11.81/7.82 add: ( $i * $i ) > $i)).
% 11.81/7.82 tff(d_type, type, (
% 11.81/7.82 d: $i)).
% 11.81/7.82 tff(c_type, type, (
% 11.81/7.82 c: $i)).
% 11.81/7.82 tff(b_type, type, (
% 11.81/7.82 b: $i)).
% 11.81/7.82 tff(additive_identity_type, type, (
% 11.81/7.82 additive_identity: $i)).
% 11.81/7.82 tff(sum_type, type, (
% 11.81/7.82 sum: ( $i * $i * $i ) > $o)).
% 11.81/7.82 tff(a_type, type, (
% 11.81/7.82 a: $i)).
% 11.81/7.82 tff(additive_inverse_type, type, (
% 11.81/7.82 additive_inverse: $i > $i)).
% 11.81/7.82 tff(1,plain,
% 11.81/7.82 (^[Y: $i, X: $i] : refl(sum(X, Y, add(X, Y)) <=> sum(X, Y, add(X, Y)))),
% 11.81/7.82 inference(bind,[status(th)],[])).
% 11.81/7.82 tff(2,plain,
% 11.81/7.82 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 11.81/7.82 inference(quant_intro,[status(thm)],[1])).
% 11.81/7.82 tff(3,plain,
% 11.81/7.82 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 11.81/7.82 inference(rewrite,[status(thm)],[])).
% 11.81/7.82 tff(4,axiom,(![Y: $i, X: $i] : sum(X, Y, add(X, Y))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','closure_of_addition')).
% 11.81/7.82 tff(5,plain,
% 11.81/7.82 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 11.81/7.82 inference(modus_ponens,[status(thm)],[4, 3])).
% 11.81/7.82 tff(6,plain,(
% 11.81/7.82 ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 11.81/7.82 inference(skolemize,[status(sab)],[5])).
% 11.81/7.82 tff(7,plain,
% 11.81/7.82 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 11.81/7.82 inference(modus_ponens,[status(thm)],[6, 2])).
% 11.81/7.82 tff(8,plain,
% 11.81/7.82 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(c, d, add(c, d))),
% 11.81/7.82 inference(quant_inst,[status(thm)],[])).
% 11.81/7.82 tff(9,plain,
% 11.81/7.82 (sum(c, d, add(c, d))),
% 11.81/7.82 inference(unit_resolution,[status(thm)],[8, 7])).
% 11.81/7.82 tff(10,plain,
% 11.81/7.82 (^[X: $i] : refl(sum(additive_inverse(X), X, additive_identity) <=> sum(additive_inverse(X), X, additive_identity))),
% 11.81/7.82 inference(bind,[status(th)],[])).
% 11.81/7.82 tff(11,plain,
% 11.81/7.82 (![X: $i] : sum(additive_inverse(X), X, additive_identity) <=> ![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 11.81/7.82 inference(quant_intro,[status(thm)],[10])).
% 11.81/7.82 tff(12,plain,
% 11.81/7.82 (![X: $i] : sum(additive_inverse(X), X, additive_identity) <=> ![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 11.81/7.82 inference(rewrite,[status(thm)],[])).
% 11.81/7.82 tff(13,axiom,(![X: $i] : sum(additive_inverse(X), X, additive_identity)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','left_inverse')).
% 11.81/7.82 tff(14,plain,
% 11.81/7.82 (![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 11.81/7.82 inference(modus_ponens,[status(thm)],[13, 12])).
% 11.81/7.82 tff(15,plain,(
% 11.81/7.82 ![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 11.81/7.82 inference(skolemize,[status(sab)],[14])).
% 11.81/7.82 tff(16,plain,
% 11.81/7.82 (![X: $i] : sum(additive_inverse(X), X, additive_identity)),
% 11.81/7.82 inference(modus_ponens,[status(thm)],[15, 11])).
% 11.81/7.82 tff(17,plain,
% 11.81/7.82 ((~![X: $i] : sum(additive_inverse(X), X, additive_identity)) | sum(additive_inverse(a), a, additive_identity)),
% 11.81/7.82 inference(quant_inst,[status(thm)],[])).
% 11.81/7.82 tff(18,plain,
% 11.81/7.82 (sum(additive_inverse(a), a, additive_identity)),
% 11.81/7.82 inference(unit_resolution,[status(thm)],[17, 16])).
% 11.81/7.82 tff(19,plain,
% 11.81/7.82 (product(additive_inverse(a), b, c) <=> product(additive_inverse(a), b, c)),
% 11.81/7.82 inference(rewrite,[status(thm)],[])).
% 11.81/7.82 tff(20,axiom,(product(additive_inverse(a), b, c)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','a_inverse_times_b')).
% 11.81/7.82 tff(21,plain,
% 11.81/7.82 (product(additive_inverse(a), b, c)),
% 11.81/7.82 inference(modus_ponens,[status(thm)],[20, 19])).
% 11.81/7.82 tff(22,plain,
% 11.81/7.82 (product(a, b, d) <=> product(a, b, d)),
% 11.81/7.82 inference(rewrite,[status(thm)],[])).
% 11.81/7.82 tff(23,axiom,(product(a, b, d)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','a_times_b')).
% 11.81/7.82 tff(24,plain,
% 11.81/7.82 (product(a, b, d)),
% 11.81/7.82 inference(modus_ponens,[status(thm)],[23, 22])).
% 11.81/7.82 tff(25,plain,
% 11.81/7.82 (^[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))))),
% 11.81/7.82 inference(bind,[status(th)],[])).
% 11.81/7.82 tff(26,plain,
% 11.81/7.82 (![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)))),
% 11.81/7.82 inference(quant_intro,[status(thm)],[25])).
% 11.81/7.82 tff(27,plain,
% 11.81/7.82 (![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)))),
% 11.81/7.82 inference(rewrite,[status(thm)],[])).
% 11.81/7.82 tff(28,plain,
% 11.81/7.82 (^[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)))))),
% 11.81/7.82 inference(bind,[status(th)],[])).
% 11.81/7.82 tff(29,plain,
% 11.81/7.82 (![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)))),
% 11.81/7.82 inference(quant_intro,[status(thm)],[28])).
% 11.81/7.82 tff(30,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/sandbox/benchmark/theBenchmark.p','distributivity4')).
% 11.81/7.82 tff(31,plain,
% 11.81/7.82 (![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)))),
% 11.81/7.82 inference(modus_ponens,[status(thm)],[30, 29])).
% 11.81/7.82 tff(32,plain,
% 11.81/7.82 (![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)))),
% 11.81/7.82 inference(modus_ponens,[status(thm)],[31, 27])).
% 11.81/7.82 tff(33,plain,(
% 11.81/7.82 ![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)))),
% 11.81/7.82 inference(skolemize,[status(sab)],[32])).
% 11.81/7.82 tff(34,plain,
% 11.81/7.82 (![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)))),
% 11.81/7.82 inference(modus_ponens,[status(thm)],[33, 26])).
% 11.81/7.82 tff(35,plain,
% 11.81/7.82 (((~![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(a, b, d)) | (~product(additive_inverse(a), b, c)) | (~sum(additive_inverse(a), a, additive_identity)) | (~sum(c, d, add(c, d))) | product(additive_identity, b, add(c, d)))) <=> ((~![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(a, b, d)) | (~product(additive_inverse(a), b, c)) | (~sum(additive_inverse(a), a, additive_identity)) | (~sum(c, d, add(c, d))) | product(additive_identity, b, add(c, d)))),
% 11.81/7.82 inference(rewrite,[status(thm)],[])).
% 11.81/7.82 tff(36,plain,
% 11.81/7.82 (((~sum(additive_inverse(a), a, additive_identity)) | (~sum(c, d, add(c, d))) | product(additive_identity, b, add(c, d)) | (~product(a, b, d)) | (~product(additive_inverse(a), b, c))) <=> ((~product(a, b, d)) | (~product(additive_inverse(a), b, c)) | (~sum(additive_inverse(a), a, additive_identity)) | (~sum(c, d, add(c, d))) | product(additive_identity, b, add(c, d)))),
% 11.81/7.82 inference(rewrite,[status(thm)],[])).
% 11.81/7.82 tff(37,plain,
% 11.81/7.82 (((~![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(additive_inverse(a), a, additive_identity)) | (~sum(c, d, add(c, d))) | product(additive_identity, b, add(c, d)) | (~product(a, b, d)) | (~product(additive_inverse(a), b, c)))) <=> ((~![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(a, b, d)) | (~product(additive_inverse(a), b, c)) | (~sum(additive_inverse(a), a, additive_identity)) | (~sum(c, d, add(c, d))) | product(additive_identity, b, add(c, d))))),
% 11.81/7.82 inference(monotonicity,[status(thm)],[36])).
% 11.81/7.82 tff(38,plain,
% 11.81/7.82 (((~![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(additive_inverse(a), a, additive_identity)) | (~sum(c, d, add(c, d))) | product(additive_identity, b, add(c, d)) | (~product(a, b, d)) | (~product(additive_inverse(a), b, c)))) <=> ((~![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(a, b, d)) | (~product(additive_inverse(a), b, c)) | (~sum(additive_inverse(a), a, additive_identity)) | (~sum(c, d, add(c, d))) | product(additive_identity, b, add(c, d)))),
% 11.81/7.82 inference(transitivity,[status(thm)],[37, 35])).
% 11.81/7.82 tff(39,plain,
% 11.81/7.82 ((~![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(additive_inverse(a), a, additive_identity)) | (~sum(c, d, add(c, d))) | product(additive_identity, b, add(c, d)) | (~product(a, b, d)) | (~product(additive_inverse(a), b, c)))),
% 11.81/7.82 inference(quant_inst,[status(thm)],[])).
% 11.81/7.82 tff(40,plain,
% 11.81/7.82 ((~![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(a, b, d)) | (~product(additive_inverse(a), b, c)) | (~sum(additive_inverse(a), a, additive_identity)) | (~sum(c, d, add(c, d))) | product(additive_identity, b, add(c, d))),
% 11.81/7.82 inference(modus_ponens,[status(thm)],[39, 38])).
% 11.81/7.82 tff(41,plain,
% 11.81/7.82 (product(additive_identity, b, add(c, d))),
% 11.81/7.82 inference(unit_resolution,[status(thm)],[40, 34, 24, 21, 18, 9])).
% 11.81/7.82 tff(42,assumption,(~product(additive_identity, c, additive_identity)), introduced(assumption)).
% 11.81/7.82 tff(43,plain,
% 11.81/7.82 (^[X: $i] : refl(product(additive_identity, X, additive_identity) <=> product(additive_identity, X, additive_identity))),
% 11.81/7.82 inference(bind,[status(th)],[])).
% 11.81/7.82 tff(44,plain,
% 11.81/7.82 (![X: $i] : product(additive_identity, X, additive_identity) <=> ![X: $i] : product(additive_identity, X, additive_identity)),
% 11.81/7.82 inference(quant_intro,[status(thm)],[43])).
% 11.81/7.82 tff(45,plain,
% 11.81/7.82 (![X: $i] : product(additive_identity, X, additive_identity) <=> ![X: $i] : product(additive_identity, X, additive_identity)),
% 11.81/7.83 inference(rewrite,[status(thm)],[])).
% 11.81/7.83 tff(46,axiom,(![X: $i] : product(additive_identity, X, additive_identity)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','multiplicative_identity1')).
% 11.81/7.83 tff(47,plain,
% 11.81/7.83 (![X: $i] : product(additive_identity, X, additive_identity)),
% 11.81/7.83 inference(modus_ponens,[status(thm)],[46, 45])).
% 11.81/7.83 tff(48,plain,(
% 11.81/7.83 ![X: $i] : product(additive_identity, X, additive_identity)),
% 11.81/7.83 inference(skolemize,[status(sab)],[47])).
% 11.81/7.83 tff(49,plain,
% 11.81/7.83 (![X: $i] : product(additive_identity, X, additive_identity)),
% 11.81/7.83 inference(modus_ponens,[status(thm)],[48, 44])).
% 11.81/7.83 tff(50,plain,
% 11.81/7.83 ((~![X: $i] : product(additive_identity, X, additive_identity)) | product(additive_identity, c, additive_identity)),
% 11.81/7.83 inference(quant_inst,[status(thm)],[])).
% 11.81/7.83 tff(51,plain,
% 11.81/7.83 ($false),
% 11.81/7.83 inference(unit_resolution,[status(thm)],[50, 49, 42])).
% 11.81/7.83 tff(52,plain,(product(additive_identity, c, additive_identity)), inference(lemma,lemma(discharge,[]))).
% 11.81/7.83 tff(53,assumption,(~product(additive_identity, additive_inverse(a), additive_identity)), introduced(assumption)).
% 11.81/7.83 tff(54,plain,
% 11.81/7.83 ((~![X: $i] : product(additive_identity, X, additive_identity)) | product(additive_identity, additive_inverse(a), additive_identity)),
% 11.81/7.83 inference(quant_inst,[status(thm)],[])).
% 11.81/7.83 tff(55,plain,
% 11.81/7.83 ($false),
% 11.81/7.83 inference(unit_resolution,[status(thm)],[54, 49, 53])).
% 11.81/7.83 tff(56,plain,(product(additive_identity, additive_inverse(a), additive_identity)), inference(lemma,lemma(discharge,[]))).
% 11.81/7.83 tff(57,plain,
% 11.81/7.83 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : refl((product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) <=> (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))))),
% 11.81/7.83 inference(bind,[status(th)],[])).
% 11.81/7.83 tff(58,plain,
% 11.81/7.83 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 11.81/7.83 inference(quant_intro,[status(thm)],[57])).
% 11.81/7.83 tff(59,plain,
% 11.81/7.83 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 11.81/7.83 inference(rewrite,[status(thm)],[])).
% 11.81/7.83 tff(60,plain,
% 11.81/7.83 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : trans(monotonicity(rewrite((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) <=> ((~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))), (((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W)) <=> (((~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) | product(U, Z, W)))), rewrite((((~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) | product(U, Z, W)) <=> (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))), (((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W)) <=> (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))))),
% 11.81/7.83 inference(bind,[status(th)],[])).
% 11.81/7.83 tff(61,plain,
% 11.81/7.83 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W)) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 11.81/7.83 inference(quant_intro,[status(thm)],[60])).
% 11.81/7.83 tff(62,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','associativity_of_multiplication2')).
% 11.81/7.83 tff(63,plain,
% 11.81/7.83 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 11.81/7.83 inference(modus_ponens,[status(thm)],[62, 61])).
% 11.81/7.83 tff(64,plain,
% 11.81/7.83 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 11.81/7.83 inference(modus_ponens,[status(thm)],[63, 59])).
% 11.81/7.83 tff(65,plain,(
% 11.81/7.83 ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 11.81/7.83 inference(skolemize,[status(sab)],[64])).
% 11.81/7.83 tff(66,plain,
% 11.81/7.83 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 11.81/7.83 inference(modus_ponens,[status(thm)],[65, 58])).
% 11.81/7.83 tff(67,plain,
% 11.81/7.83 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(additive_inverse(a), b, c)) | (~product(additive_identity, additive_inverse(a), additive_identity)) | (~product(additive_identity, c, additive_identity)) | product(additive_identity, b, additive_identity))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(additive_inverse(a), b, c)) | (~product(additive_identity, additive_inverse(a), additive_identity)) | (~product(additive_identity, c, additive_identity)) | product(additive_identity, b, additive_identity))),
% 11.81/7.83 inference(rewrite,[status(thm)],[])).
% 11.81/7.83 tff(68,plain,
% 11.81/7.83 ((product(additive_identity, b, additive_identity) | (~product(additive_inverse(a), b, c)) | (~product(additive_identity, additive_inverse(a), additive_identity)) | (~product(additive_identity, c, additive_identity))) <=> ((~product(additive_inverse(a), b, c)) | (~product(additive_identity, additive_inverse(a), additive_identity)) | (~product(additive_identity, c, additive_identity)) | product(additive_identity, b, additive_identity))),
% 11.81/7.83 inference(rewrite,[status(thm)],[])).
% 11.81/7.83 tff(69,plain,
% 11.81/7.83 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(additive_identity, b, additive_identity) | (~product(additive_inverse(a), b, c)) | (~product(additive_identity, additive_inverse(a), additive_identity)) | (~product(additive_identity, c, additive_identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(additive_inverse(a), b, c)) | (~product(additive_identity, additive_inverse(a), additive_identity)) | (~product(additive_identity, c, additive_identity)) | product(additive_identity, b, additive_identity)))),
% 11.81/7.83 inference(monotonicity,[status(thm)],[68])).
% 11.81/7.83 tff(70,plain,
% 11.81/7.83 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(additive_identity, b, additive_identity) | (~product(additive_inverse(a), b, c)) | (~product(additive_identity, additive_inverse(a), additive_identity)) | (~product(additive_identity, c, additive_identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(additive_inverse(a), b, c)) | (~product(additive_identity, additive_inverse(a), additive_identity)) | (~product(additive_identity, c, additive_identity)) | product(additive_identity, b, additive_identity))),
% 11.81/7.83 inference(transitivity,[status(thm)],[69, 67])).
% 11.81/7.83 tff(71,plain,
% 11.81/7.83 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(additive_identity, b, additive_identity) | (~product(additive_inverse(a), b, c)) | (~product(additive_identity, additive_inverse(a), additive_identity)) | (~product(additive_identity, c, additive_identity)))),
% 11.81/7.83 inference(quant_inst,[status(thm)],[])).
% 11.81/7.83 tff(72,plain,
% 11.81/7.83 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(additive_inverse(a), b, c)) | (~product(additive_identity, additive_inverse(a), additive_identity)) | (~product(additive_identity, c, additive_identity)) | product(additive_identity, b, additive_identity)),
% 11.81/7.83 inference(modus_ponens,[status(thm)],[71, 70])).
% 11.81/7.83 tff(73,plain,
% 11.81/7.83 (product(additive_identity, b, additive_identity)),
% 11.81/7.83 inference(unit_resolution,[status(thm)],[72, 66, 21, 56, 52])).
% 11.81/7.83 tff(74,plain,
% 11.81/7.83 (^[X: $i] : refl(sum(X, additive_identity, X) <=> sum(X, additive_identity, X))),
% 11.81/7.83 inference(bind,[status(th)],[])).
% 11.81/7.83 tff(75,plain,
% 11.81/7.83 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 11.81/7.83 inference(quant_intro,[status(thm)],[74])).
% 11.81/7.83 tff(76,plain,
% 11.81/7.83 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 11.81/7.83 inference(rewrite,[status(thm)],[])).
% 11.81/7.83 tff(77,axiom,(![X: $i] : sum(X, additive_identity, X)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','additive_identity2')).
% 11.81/7.83 tff(78,plain,
% 11.81/7.83 (![X: $i] : sum(X, additive_identity, X)),
% 11.81/7.83 inference(modus_ponens,[status(thm)],[77, 76])).
% 11.81/7.83 tff(79,plain,(
% 11.81/7.83 ![X: $i] : sum(X, additive_identity, X)),
% 11.81/7.83 inference(skolemize,[status(sab)],[78])).
% 11.81/7.83 tff(80,plain,
% 11.81/7.83 (![X: $i] : sum(X, additive_identity, X)),
% 11.81/7.83 inference(modus_ponens,[status(thm)],[79, 75])).
% 11.81/7.83 tff(81,plain,
% 11.81/7.83 ((~![X: $i] : sum(X, additive_identity, X)) | sum(additive_identity, additive_identity, additive_identity)),
% 11.81/7.83 inference(quant_inst,[status(thm)],[])).
% 11.81/7.83 tff(82,plain,
% 11.81/7.83 (sum(additive_identity, additive_identity, additive_identity)),
% 11.81/7.83 inference(unit_resolution,[status(thm)],[81, 80])).
% 11.81/7.83 tff(83,plain,
% 11.81/7.83 ((~![X: $i] : sum(X, additive_identity, X)) | sum(c, additive_identity, c)),
% 11.81/7.83 inference(quant_inst,[status(thm)],[])).
% 11.81/7.83 tff(84,plain,
% 11.81/7.83 (sum(c, additive_identity, c)),
% 11.81/7.83 inference(unit_resolution,[status(thm)],[83, 80])).
% 11.81/7.83 tff(85,plain,
% 11.81/7.83 (^[Z: $i, Y: $i, X: $i] : refl(((~sum(X, Y, Z)) | sum(Y, X, Z)) <=> ((~sum(X, Y, Z)) | sum(Y, X, Z)))),
% 11.81/7.83 inference(bind,[status(th)],[])).
% 11.81/7.83 tff(86,plain,
% 11.81/7.83 (![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))),
% 11.81/7.83 inference(quant_intro,[status(thm)],[85])).
% 11.81/7.83 tff(87,plain,
% 11.81/7.83 (![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))),
% 11.81/7.83 inference(rewrite,[status(thm)],[])).
% 11.81/7.83 tff(88,axiom,(![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','commutativity_of_addition')).
% 11.81/7.83 tff(89,plain,
% 11.81/7.83 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 11.81/7.83 inference(modus_ponens,[status(thm)],[88, 87])).
% 11.81/7.83 tff(90,plain,(
% 11.81/7.83 ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 11.81/7.83 inference(skolemize,[status(sab)],[89])).
% 11.81/7.83 tff(91,plain,
% 11.81/7.83 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 11.81/7.83 inference(modus_ponens,[status(thm)],[90, 86])).
% 11.81/7.83 tff(92,plain,
% 11.81/7.83 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(c, additive_identity, c)) | sum(additive_identity, c, c))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(c, additive_identity, c)) | sum(additive_identity, c, c))),
% 11.81/7.83 inference(rewrite,[status(thm)],[])).
% 11.81/7.83 tff(93,plain,
% 11.81/7.83 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(c, additive_identity, c)) | sum(additive_identity, c, c))),
% 11.81/7.83 inference(quant_inst,[status(thm)],[])).
% 11.81/7.83 tff(94,plain,
% 11.81/7.83 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(c, additive_identity, c)) | sum(additive_identity, c, c)),
% 11.81/7.83 inference(modus_ponens,[status(thm)],[93, 92])).
% 11.81/7.83 tff(95,plain,
% 11.81/7.83 (sum(additive_identity, c, c)),
% 11.81/7.83 inference(unit_resolution,[status(thm)],[94, 91, 84])).
% 11.81/7.83 tff(96,plain,
% 11.81/7.83 ((~sum(c, d, additive_identity)) <=> (~sum(c, d, additive_identity))),
% 11.81/7.83 inference(rewrite,[status(thm)],[])).
% 11.81/7.83 tff(97,axiom,(~sum(c, d, additive_identity)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_sum_is_additive_identity')).
% 11.81/7.83 tff(98,plain,
% 11.81/7.83 (~sum(c, d, additive_identity)),
% 11.81/7.83 inference(modus_ponens,[status(thm)],[97, 96])).
% 11.81/7.83 tff(99,plain,
% 11.81/7.83 (^[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))))),
% 11.90/7.83 inference(bind,[status(th)],[])).
% 11.90/7.83 tff(100,plain,
% 11.90/7.83 (![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)))),
% 11.90/7.83 inference(quant_intro,[status(thm)],[99])).
% 11.90/7.83 tff(101,plain,
% 11.90/7.83 (![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)))),
% 11.90/7.83 inference(rewrite,[status(thm)],[])).
% 11.90/7.83 tff(102,plain,
% 11.90/7.83 (^[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)))))),
% 11.90/7.83 inference(bind,[status(th)],[])).
% 11.90/7.83 tff(103,plain,
% 11.90/7.83 (![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)))),
% 11.90/7.83 inference(quant_intro,[status(thm)],[102])).
% 11.90/7.83 tff(104,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~sum(X, Y, U)) | (~sum(Y, Z, V))) | (~sum(X, V, W))) | sum(U, Z, W))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','associativity_of_addition2')).
% 11.90/7.83 tff(105,plain,
% 11.90/7.83 (![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)))),
% 11.90/7.83 inference(modus_ponens,[status(thm)],[104, 103])).
% 11.90/7.83 tff(106,plain,
% 11.90/7.83 (![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)))),
% 11.90/7.83 inference(modus_ponens,[status(thm)],[105, 101])).
% 11.90/7.83 tff(107,plain,(
% 11.90/7.83 ![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)))),
% 11.90/7.83 inference(skolemize,[status(sab)],[106])).
% 11.90/7.83 tff(108,plain,
% 11.90/7.83 (![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)))),
% 11.90/7.83 inference(modus_ponens,[status(thm)],[107, 100])).
% 11.90/7.83 tff(109,plain,
% 11.90/7.83 (((~![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(c, d, additive_identity) | (~sum(additive_identity, c, c)) | (~sum(c, d, add(c, d))) | (~sum(additive_identity, add(c, d), 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(c, d, additive_identity) | (~sum(additive_identity, c, c)) | (~sum(c, d, add(c, d))) | (~sum(additive_identity, add(c, d), additive_identity)))),
% 11.90/7.83 inference(rewrite,[status(thm)],[])).
% 11.90/7.83 tff(110,plain,
% 11.90/7.83 ((sum(c, d, additive_identity) | (~sum(c, d, add(c, d))) | (~sum(additive_identity, c, c)) | (~sum(additive_identity, add(c, d), additive_identity))) <=> (sum(c, d, additive_identity) | (~sum(additive_identity, c, c)) | (~sum(c, d, add(c, d))) | (~sum(additive_identity, add(c, d), additive_identity)))),
% 11.90/7.83 inference(rewrite,[status(thm)],[])).
% 11.90/7.83 tff(111,plain,
% 11.90/7.83 (((~![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(c, d, additive_identity) | (~sum(c, d, add(c, d))) | (~sum(additive_identity, c, c)) | (~sum(additive_identity, add(c, d), 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(c, d, additive_identity) | (~sum(additive_identity, c, c)) | (~sum(c, d, add(c, d))) | (~sum(additive_identity, add(c, d), additive_identity))))),
% 11.90/7.83 inference(monotonicity,[status(thm)],[110])).
% 11.90/7.83 tff(112,plain,
% 11.90/7.83 (((~![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(c, d, additive_identity) | (~sum(c, d, add(c, d))) | (~sum(additive_identity, c, c)) | (~sum(additive_identity, add(c, d), 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(c, d, additive_identity) | (~sum(additive_identity, c, c)) | (~sum(c, d, add(c, d))) | (~sum(additive_identity, add(c, d), additive_identity)))),
% 11.90/7.83 inference(transitivity,[status(thm)],[111, 109])).
% 11.90/7.83 tff(113,plain,
% 11.90/7.83 ((~![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(c, d, additive_identity) | (~sum(c, d, add(c, d))) | (~sum(additive_identity, c, c)) | (~sum(additive_identity, add(c, d), additive_identity)))),
% 11.90/7.83 inference(quant_inst,[status(thm)],[])).
% 11.90/7.83 tff(114,plain,
% 11.90/7.83 ((~![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(c, d, additive_identity) | (~sum(additive_identity, c, c)) | (~sum(c, d, add(c, d))) | (~sum(additive_identity, add(c, d), additive_identity))),
% 11.90/7.83 inference(modus_ponens,[status(thm)],[113, 112])).
% 11.90/7.83 tff(115,plain,
% 11.90/7.83 (~sum(additive_identity, add(c, d), additive_identity)),
% 11.90/7.83 inference(unit_resolution,[status(thm)],[114, 108, 98, 9, 95])).
% 11.90/7.83 tff(116,plain,
% 11.90/7.83 (^[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))))),
% 11.90/7.83 inference(bind,[status(th)],[])).
% 11.90/7.83 tff(117,plain,
% 11.90/7.83 (![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)))),
% 11.90/7.83 inference(quant_intro,[status(thm)],[116])).
% 11.90/7.83 tff(118,plain,
% 11.90/7.83 (![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)))),
% 11.90/7.83 inference(rewrite,[status(thm)],[])).
% 11.90/7.83 tff(119,plain,
% 11.90/7.83 (^[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)))))),
% 11.90/7.83 inference(bind,[status(th)],[])).
% 11.90/7.83 tff(120,plain,
% 11.90/7.83 (![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)))),
% 11.90/7.83 inference(quant_intro,[status(thm)],[119])).
% 11.90/7.83 tff(121,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) | sum(V1, V2, V4))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','distributivity3')).
% 11.90/7.83 tff(122,plain,
% 11.90/7.83 (![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)))),
% 11.90/7.83 inference(modus_ponens,[status(thm)],[121, 120])).
% 11.90/7.83 tff(123,plain,
% 11.90/7.83 (![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)))),
% 11.90/7.83 inference(modus_ponens,[status(thm)],[122, 118])).
% 11.90/7.83 tff(124,plain,(
% 11.90/7.83 ![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)))),
% 11.90/7.83 inference(skolemize,[status(sab)],[123])).
% 11.90/7.83 tff(125,plain,
% 11.90/7.83 (![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)))),
% 11.90/7.83 inference(modus_ponens,[status(thm)],[124, 117])).
% 11.90/7.83 tff(126,plain,
% 11.90/7.83 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(additive_identity, add(c, d), additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, b, additive_identity)) | (~product(additive_identity, b, add(c, d))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | sum(additive_identity, add(c, d), additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, b, additive_identity)) | (~product(additive_identity, b, add(c, d))))),
% 11.90/7.83 inference(rewrite,[status(thm)],[])).
% 11.90/7.83 tff(127,plain,
% 11.90/7.83 ((sum(additive_identity, add(c, d), additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, b, additive_identity)) | (~product(additive_identity, b, add(c, d))) | (~product(additive_identity, b, additive_identity))) <=> (sum(additive_identity, add(c, d), additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, b, additive_identity)) | (~product(additive_identity, b, add(c, d))))),
% 11.90/7.83 inference(rewrite,[status(thm)],[])).
% 11.90/7.83 tff(128,plain,
% 11.90/7.83 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(additive_identity, add(c, d), additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, b, additive_identity)) | (~product(additive_identity, b, add(c, d))) | (~product(additive_identity, b, additive_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(additive_identity, add(c, d), additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, b, additive_identity)) | (~product(additive_identity, b, add(c, d)))))),
% 11.90/7.84 inference(monotonicity,[status(thm)],[127])).
% 11.90/7.84 tff(129,plain,
% 11.90/7.84 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(additive_identity, add(c, d), additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, b, additive_identity)) | (~product(additive_identity, b, add(c, d))) | (~product(additive_identity, b, additive_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | sum(additive_identity, add(c, d), additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, b, additive_identity)) | (~product(additive_identity, b, add(c, d))))),
% 11.90/7.84 inference(transitivity,[status(thm)],[128, 126])).
% 11.90/7.84 tff(130,plain,
% 11.90/7.84 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(additive_identity, add(c, d), additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, b, additive_identity)) | (~product(additive_identity, b, add(c, d))) | (~product(additive_identity, b, additive_identity)))),
% 11.90/7.84 inference(quant_inst,[status(thm)],[])).
% 11.90/7.84 tff(131,plain,
% 11.90/7.84 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | sum(additive_identity, add(c, d), additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, b, additive_identity)) | (~product(additive_identity, b, add(c, d)))),
% 11.90/7.84 inference(modus_ponens,[status(thm)],[130, 129])).
% 11.90/7.84 tff(132,plain,
% 11.90/7.84 ($false),
% 11.90/7.84 inference(unit_resolution,[status(thm)],[131, 125, 115, 82, 73, 41])).
% 11.90/7.84 % SZS output end Proof
%------------------------------------------------------------------------------