TSTP Solution File: BOO005-1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : BOO005-1 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n021.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 6 17:18:36 EDT 2022
% Result : Unsatisfiable 2.15s 1.65s
% Output : Proof 2.15s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : BOO005-1 : TPTP v8.1.0. Released v1.0.0.
% 0.07/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n021.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Tue Aug 30 02:39:55 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 2.15/1.65 % SZS status Unsatisfiable
% 2.15/1.65 % SZS output start Proof
% 2.15/1.65 tff(product_type, type, (
% 2.15/1.65 product: ( $i * $i * $i ) > $o)).
% 2.15/1.65 tff(add_type, type, (
% 2.15/1.65 add: ( $i * $i ) > $i)).
% 2.15/1.65 tff(x_type, type, (
% 2.15/1.65 x: $i)).
% 2.15/1.65 tff(multiplicative_identity_type, type, (
% 2.15/1.65 multiplicative_identity: $i)).
% 2.15/1.65 tff(inverse_type, type, (
% 2.15/1.65 inverse: $i > $i)).
% 2.15/1.65 tff(sum_type, type, (
% 2.15/1.65 sum: ( $i * $i * $i ) > $o)).
% 2.15/1.65 tff(1,assumption,(~product(multiplicative_identity, inverse(x), inverse(x))), introduced(assumption)).
% 2.15/1.65 tff(2,plain,
% 2.15/1.65 (^[X: $i] : refl(product(multiplicative_identity, X, X) <=> product(multiplicative_identity, X, X))),
% 2.15/1.65 inference(bind,[status(th)],[])).
% 2.15/1.65 tff(3,plain,
% 2.15/1.65 (![X: $i] : product(multiplicative_identity, X, X) <=> ![X: $i] : product(multiplicative_identity, X, X)),
% 2.15/1.65 inference(quant_intro,[status(thm)],[2])).
% 2.15/1.65 tff(4,plain,
% 2.15/1.65 (![X: $i] : product(multiplicative_identity, X, X) <=> ![X: $i] : product(multiplicative_identity, X, X)),
% 2.15/1.65 inference(rewrite,[status(thm)],[])).
% 2.15/1.65 tff(5,axiom,(![X: $i] : product(multiplicative_identity, X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_identity1')).
% 2.15/1.65 tff(6,plain,
% 2.15/1.65 (![X: $i] : product(multiplicative_identity, X, X)),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[5, 4])).
% 2.15/1.65 tff(7,plain,(
% 2.15/1.65 ![X: $i] : product(multiplicative_identity, X, X)),
% 2.15/1.65 inference(skolemize,[status(sab)],[6])).
% 2.15/1.65 tff(8,plain,
% 2.15/1.65 (![X: $i] : product(multiplicative_identity, X, X)),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[7, 3])).
% 2.15/1.65 tff(9,plain,
% 2.15/1.65 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, inverse(x), inverse(x))),
% 2.15/1.65 inference(quant_inst,[status(thm)],[])).
% 2.15/1.65 tff(10,plain,
% 2.15/1.65 ($false),
% 2.15/1.65 inference(unit_resolution,[status(thm)],[9, 8, 1])).
% 2.15/1.65 tff(11,plain,(product(multiplicative_identity, inverse(x), inverse(x))), inference(lemma,lemma(discharge,[]))).
% 2.15/1.65 tff(12,plain,
% 2.15/1.65 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, multiplicative_identity, multiplicative_identity)),
% 2.15/1.65 inference(quant_inst,[status(thm)],[])).
% 2.15/1.65 tff(13,plain,
% 2.15/1.65 (product(multiplicative_identity, multiplicative_identity, multiplicative_identity)),
% 2.15/1.65 inference(unit_resolution,[status(thm)],[12, 8])).
% 2.15/1.65 tff(14,plain,
% 2.15/1.65 (^[Y: $i, X: $i] : refl(sum(X, Y, add(X, Y)) <=> sum(X, Y, add(X, Y)))),
% 2.15/1.65 inference(bind,[status(th)],[])).
% 2.15/1.65 tff(15,plain,
% 2.15/1.65 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 2.15/1.65 inference(quant_intro,[status(thm)],[14])).
% 2.15/1.65 tff(16,plain,
% 2.15/1.65 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 2.15/1.65 inference(rewrite,[status(thm)],[])).
% 2.15/1.65 tff(17,axiom,(![Y: $i, X: $i] : sum(X, Y, add(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','closure_of_addition')).
% 2.15/1.65 tff(18,plain,
% 2.15/1.65 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[17, 16])).
% 2.15/1.65 tff(19,plain,(
% 2.15/1.65 ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 2.15/1.65 inference(skolemize,[status(sab)],[18])).
% 2.15/1.65 tff(20,plain,
% 2.15/1.65 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[19, 15])).
% 2.15/1.65 tff(21,plain,
% 2.15/1.65 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(multiplicative_identity, x, add(multiplicative_identity, x))),
% 2.15/1.65 inference(quant_inst,[status(thm)],[])).
% 2.15/1.65 tff(22,plain,
% 2.15/1.65 (sum(multiplicative_identity, x, add(multiplicative_identity, x))),
% 2.15/1.65 inference(unit_resolution,[status(thm)],[21, 20])).
% 2.15/1.65 tff(23,plain,
% 2.15/1.65 (^[Z: $i, Y: $i, X: $i] : refl(((~sum(X, Y, Z)) | sum(Y, X, Z)) <=> ((~sum(X, Y, Z)) | sum(Y, X, Z)))),
% 2.15/1.65 inference(bind,[status(th)],[])).
% 2.15/1.65 tff(24,plain,
% 2.15/1.65 (![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))),
% 2.15/1.65 inference(quant_intro,[status(thm)],[23])).
% 2.15/1.65 tff(25,plain,
% 2.15/1.65 (![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))),
% 2.15/1.65 inference(rewrite,[status(thm)],[])).
% 2.15/1.65 tff(26,axiom,(![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','commutativity_of_addition')).
% 2.15/1.65 tff(27,plain,
% 2.15/1.65 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[26, 25])).
% 2.15/1.65 tff(28,plain,(
% 2.15/1.65 ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 2.15/1.65 inference(skolemize,[status(sab)],[27])).
% 2.15/1.65 tff(29,plain,
% 2.15/1.65 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[28, 24])).
% 2.15/1.65 tff(30,plain,
% 2.15/1.65 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(multiplicative_identity, x, add(multiplicative_identity, x))) | sum(x, multiplicative_identity, add(multiplicative_identity, x)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(multiplicative_identity, x, add(multiplicative_identity, x))) | sum(x, multiplicative_identity, add(multiplicative_identity, x)))),
% 2.15/1.65 inference(rewrite,[status(thm)],[])).
% 2.15/1.65 tff(31,plain,
% 2.15/1.65 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(multiplicative_identity, x, add(multiplicative_identity, x))) | sum(x, multiplicative_identity, add(multiplicative_identity, x)))),
% 2.15/1.65 inference(quant_inst,[status(thm)],[])).
% 2.15/1.65 tff(32,plain,
% 2.15/1.65 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(multiplicative_identity, x, add(multiplicative_identity, x))) | sum(x, multiplicative_identity, add(multiplicative_identity, x))),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[31, 30])).
% 2.15/1.65 tff(33,plain,
% 2.15/1.65 (sum(x, multiplicative_identity, add(multiplicative_identity, x))),
% 2.15/1.65 inference(unit_resolution,[status(thm)],[32, 29, 22])).
% 2.15/1.65 tff(34,plain,
% 2.15/1.65 ((~sum(x, multiplicative_identity, multiplicative_identity)) <=> (~sum(x, multiplicative_identity, multiplicative_identity))),
% 2.15/1.65 inference(rewrite,[status(thm)],[])).
% 2.15/1.65 tff(35,axiom,(~sum(x, multiplicative_identity, multiplicative_identity)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_equations')).
% 2.15/1.65 tff(36,plain,
% 2.15/1.65 (~sum(x, multiplicative_identity, multiplicative_identity)),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[35, 34])).
% 2.15/1.65 tff(37,plain,
% 2.15/1.65 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4))) <=> (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4))))),
% 2.15/1.65 inference(bind,[status(th)],[])).
% 2.15/1.65 tff(38,plain,
% 2.15/1.65 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))),
% 2.15/1.65 inference(quant_intro,[status(thm)],[37])).
% 2.15/1.65 tff(39,plain,
% 2.15/1.65 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))),
% 2.15/1.65 inference(rewrite,[status(thm)],[])).
% 2.15/1.65 tff(40,plain,
% 2.15/1.65 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) <=> ((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))), (((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) <=> (((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) | (~product(V1, V2, V4))))), rewrite((((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) | (~product(V1, V2, V4))) <=> ((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))), (((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) <=> ((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4))))), ((((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(X, V3, V4)) <=> (((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4))) | sum(X, V3, V4)))), rewrite((((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4))) | sum(X, V3, V4)) <=> (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))), ((((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(X, V3, V4)) <=> (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))))),
% 2.15/1.65 inference(bind,[status(th)],[])).
% 2.15/1.65 tff(41,plain,
% 2.15/1.65 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(X, V3, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))),
% 2.15/1.65 inference(quant_intro,[status(thm)],[40])).
% 2.15/1.65 tff(42,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(X, V3, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity6')).
% 2.15/1.65 tff(43,plain,
% 2.15/1.65 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[42, 41])).
% 2.15/1.65 tff(44,plain,
% 2.15/1.65 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[43, 39])).
% 2.15/1.65 tff(45,plain,(
% 2.15/1.65 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))),
% 2.15/1.65 inference(skolemize,[status(sab)],[44])).
% 2.15/1.65 tff(46,plain,
% 2.15/1.65 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[45, 38])).
% 2.15/1.65 tff(47,plain,
% 2.15/1.65 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (sum(x, multiplicative_identity, multiplicative_identity) | (~product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | sum(x, multiplicative_identity, multiplicative_identity) | (~product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)))),
% 2.15/1.65 inference(rewrite,[status(thm)],[])).
% 2.15/1.65 tff(48,plain,
% 2.15/1.65 ((sum(x, multiplicative_identity, multiplicative_identity) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity))) <=> (sum(x, multiplicative_identity, multiplicative_identity) | (~product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)))),
% 2.15/1.65 inference(rewrite,[status(thm)],[])).
% 2.15/1.65 tff(49,plain,
% 2.15/1.65 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (sum(x, multiplicative_identity, multiplicative_identity) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (sum(x, multiplicative_identity, multiplicative_identity) | (~product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity))))),
% 2.15/1.65 inference(monotonicity,[status(thm)],[48])).
% 2.15/1.65 tff(50,plain,
% 2.15/1.65 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (sum(x, multiplicative_identity, multiplicative_identity) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | sum(x, multiplicative_identity, multiplicative_identity) | (~product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)))),
% 2.15/1.65 inference(transitivity,[status(thm)],[49, 47])).
% 2.15/1.65 tff(51,plain,
% 2.15/1.65 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (sum(x, multiplicative_identity, multiplicative_identity) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity)))),
% 2.15/1.65 inference(quant_inst,[status(thm)],[])).
% 2.15/1.65 tff(52,plain,
% 2.15/1.65 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | sum(x, multiplicative_identity, multiplicative_identity) | (~product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity))),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[51, 50])).
% 2.15/1.65 tff(53,plain,
% 2.15/1.65 (~product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity)),
% 2.15/1.65 inference(unit_resolution,[status(thm)],[52, 46, 36, 33, 13])).
% 2.15/1.65 tff(54,plain,
% 2.15/1.65 (^[X: $i] : refl(sum(X, inverse(X), multiplicative_identity) <=> sum(X, inverse(X), multiplicative_identity))),
% 2.15/1.65 inference(bind,[status(th)],[])).
% 2.15/1.65 tff(55,plain,
% 2.15/1.65 (![X: $i] : sum(X, inverse(X), multiplicative_identity) <=> ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 2.15/1.65 inference(quant_intro,[status(thm)],[54])).
% 2.15/1.65 tff(56,plain,
% 2.15/1.65 (![X: $i] : sum(X, inverse(X), multiplicative_identity) <=> ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 2.15/1.65 inference(rewrite,[status(thm)],[])).
% 2.15/1.65 tff(57,axiom,(![X: $i] : sum(X, inverse(X), multiplicative_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_inverse2')).
% 2.15/1.65 tff(58,plain,
% 2.15/1.65 (![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[57, 56])).
% 2.15/1.65 tff(59,plain,(
% 2.15/1.65 ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 2.15/1.65 inference(skolemize,[status(sab)],[58])).
% 2.15/1.65 tff(60,plain,
% 2.15/1.65 (![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[59, 55])).
% 2.15/1.65 tff(61,plain,
% 2.15/1.65 ((~![X: $i] : sum(X, inverse(X), multiplicative_identity)) | sum(x, inverse(x), multiplicative_identity)),
% 2.15/1.65 inference(quant_inst,[status(thm)],[])).
% 2.15/1.65 tff(62,plain,
% 2.15/1.65 (sum(x, inverse(x), multiplicative_identity)),
% 2.15/1.65 inference(unit_resolution,[status(thm)],[61, 60])).
% 2.15/1.65 tff(63,plain,
% 2.15/1.65 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) <=> (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))))),
% 2.15/1.65 inference(bind,[status(th)],[])).
% 2.15/1.65 tff(64,plain,
% 2.15/1.65 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))),
% 2.15/1.65 inference(quant_intro,[status(thm)],[63])).
% 2.15/1.65 tff(65,plain,
% 2.15/1.65 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))),
% 2.15/1.65 inference(rewrite,[status(thm)],[])).
% 2.15/1.65 tff(66,plain,
% 2.15/1.65 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) <=> ((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))), (((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~sum(X, V3, V4))) <=> (((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) | (~sum(X, V3, V4))))), rewrite((((~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) | (~sum(X, V3, V4))) <=> ((~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))), (((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~sum(X, V3, V4))) <=> ((~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))))), ((((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~sum(X, V3, V4))) | product(V1, V2, V4)) <=> (((~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) | product(V1, V2, V4)))), rewrite((((~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1))) | product(V1, V2, V4)) <=> (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))), ((((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~sum(X, V3, V4))) | product(V1, V2, V4)) <=> (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))))),
% 2.15/1.65 inference(bind,[status(th)],[])).
% 2.15/1.65 tff(67,plain,
% 2.15/1.65 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~sum(X, V3, V4))) | product(V1, V2, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))),
% 2.15/1.65 inference(quant_intro,[status(thm)],[66])).
% 2.15/1.65 tff(68,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(X, Y, V1)) | (~sum(X, Z, V2))) | (~product(Y, Z, V3))) | (~sum(X, V3, V4))) | product(V1, V2, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity5')).
% 2.15/1.65 tff(69,plain,
% 2.15/1.65 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[68, 67])).
% 2.15/1.65 tff(70,plain,
% 2.15/1.65 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[69, 65])).
% 2.15/1.65 tff(71,plain,(
% 2.15/1.65 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))),
% 2.15/1.65 inference(skolemize,[status(sab)],[70])).
% 2.15/1.65 tff(72,plain,
% 2.15/1.65 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))),
% 2.15/1.65 inference(modus_ponens,[status(thm)],[71, 64])).
% 2.15/1.65 tff(73,plain,
% 2.15/1.65 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | ((~sum(x, inverse(x), multiplicative_identity)) | product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~sum(x, inverse(x), add(multiplicative_identity, x))) | (~product(multiplicative_identity, inverse(x), inverse(x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (~sum(x, inverse(x), multiplicative_identity)) | product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~sum(x, inverse(x), add(multiplicative_identity, x))) | (~product(multiplicative_identity, inverse(x), inverse(x))))),
% 2.15/1.65 inference(rewrite,[status(thm)],[])).
% 2.15/1.65 tff(74,plain,
% 2.15/1.65 ((product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity) | (~sum(x, inverse(x), multiplicative_identity)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~sum(x, inverse(x), add(multiplicative_identity, x))) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x)))) <=> ((~sum(x, inverse(x), multiplicative_identity)) | product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~sum(x, inverse(x), add(multiplicative_identity, x))) | (~product(multiplicative_identity, inverse(x), inverse(x))))),
% 2.15/1.65 inference(rewrite,[status(thm)],[])).
% 2.15/1.65 tff(75,plain,
% 2.15/1.65 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity) | (~sum(x, inverse(x), multiplicative_identity)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~sum(x, inverse(x), add(multiplicative_identity, x))) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | ((~sum(x, inverse(x), multiplicative_identity)) | product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~sum(x, inverse(x), add(multiplicative_identity, x))) | (~product(multiplicative_identity, inverse(x), inverse(x)))))),
% 2.15/1.65 inference(monotonicity,[status(thm)],[74])).
% 2.15/1.65 tff(76,plain,
% 2.15/1.65 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity) | (~sum(x, inverse(x), multiplicative_identity)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~sum(x, inverse(x), add(multiplicative_identity, x))) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (~sum(x, inverse(x), multiplicative_identity)) | product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~sum(x, inverse(x), add(multiplicative_identity, x))) | (~product(multiplicative_identity, inverse(x), inverse(x))))),
% 2.15/1.66 inference(transitivity,[status(thm)],[75, 73])).
% 2.15/1.66 tff(77,plain,
% 2.15/1.66 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity) | (~sum(x, inverse(x), multiplicative_identity)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~sum(x, inverse(x), add(multiplicative_identity, x))) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))))),
% 2.15/1.66 inference(quant_inst,[status(thm)],[])).
% 2.15/1.66 tff(78,plain,
% 2.15/1.66 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (~sum(x, inverse(x), multiplicative_identity)) | product(add(multiplicative_identity, x), add(multiplicative_identity, x), multiplicative_identity) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~sum(x, inverse(x), add(multiplicative_identity, x))) | (~product(multiplicative_identity, inverse(x), inverse(x)))),
% 2.15/1.66 inference(modus_ponens,[status(thm)],[77, 76])).
% 2.15/1.66 tff(79,plain,
% 2.15/1.66 (~sum(x, inverse(x), add(multiplicative_identity, x))),
% 2.15/1.66 inference(unit_resolution,[status(thm)],[78, 72, 62, 33, 53, 11])).
% 2.15/1.66 tff(80,plain,
% 2.15/1.66 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | ((~sum(x, inverse(x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | sum(x, inverse(x), add(multiplicative_identity, x)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (~sum(x, inverse(x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | sum(x, inverse(x), add(multiplicative_identity, x)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x))))),
% 2.15/1.66 inference(rewrite,[status(thm)],[])).
% 2.15/1.66 tff(81,plain,
% 2.15/1.66 ((sum(x, inverse(x), add(multiplicative_identity, x)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~sum(x, inverse(x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x)))) <=> ((~sum(x, inverse(x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | sum(x, inverse(x), add(multiplicative_identity, x)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x))))),
% 2.15/1.66 inference(rewrite,[status(thm)],[])).
% 2.15/1.66 tff(82,plain,
% 2.15/1.66 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (sum(x, inverse(x), add(multiplicative_identity, x)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~sum(x, inverse(x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | ((~sum(x, inverse(x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | sum(x, inverse(x), add(multiplicative_identity, x)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x)))))),
% 2.15/1.66 inference(monotonicity,[status(thm)],[81])).
% 2.15/1.66 tff(83,plain,
% 2.15/1.66 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (sum(x, inverse(x), add(multiplicative_identity, x)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~sum(x, inverse(x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (~sum(x, inverse(x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | sum(x, inverse(x), add(multiplicative_identity, x)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x))))),
% 2.15/1.66 inference(transitivity,[status(thm)],[82, 80])).
% 2.15/1.66 tff(84,plain,
% 2.15/1.66 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (sum(x, inverse(x), add(multiplicative_identity, x)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~sum(x, inverse(x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x))))),
% 2.15/1.66 inference(quant_inst,[status(thm)],[])).
% 2.15/1.66 tff(85,plain,
% 2.15/1.66 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (~sum(x, inverse(x), multiplicative_identity)) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | sum(x, inverse(x), add(multiplicative_identity, x)) | (~product(multiplicative_identity, inverse(x), inverse(x))) | (~product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x)))),
% 2.15/1.66 inference(modus_ponens,[status(thm)],[84, 83])).
% 2.15/1.66 tff(86,plain,
% 2.15/1.66 (~product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x))),
% 2.15/1.66 inference(unit_resolution,[status(thm)],[85, 46, 62, 33, 79, 11])).
% 2.15/1.66 tff(87,plain,
% 2.15/1.66 (^[X: $i] : refl(product(X, multiplicative_identity, X) <=> product(X, multiplicative_identity, X))),
% 2.15/1.66 inference(bind,[status(th)],[])).
% 2.15/1.66 tff(88,plain,
% 2.15/1.66 (![X: $i] : product(X, multiplicative_identity, X) <=> ![X: $i] : product(X, multiplicative_identity, X)),
% 2.15/1.66 inference(quant_intro,[status(thm)],[87])).
% 2.15/1.66 tff(89,plain,
% 2.15/1.66 (![X: $i] : product(X, multiplicative_identity, X) <=> ![X: $i] : product(X, multiplicative_identity, X)),
% 2.15/1.66 inference(rewrite,[status(thm)],[])).
% 2.15/1.66 tff(90,axiom,(![X: $i] : product(X, multiplicative_identity, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_identity2')).
% 2.15/1.66 tff(91,plain,
% 2.15/1.66 (![X: $i] : product(X, multiplicative_identity, X)),
% 2.15/1.66 inference(modus_ponens,[status(thm)],[90, 89])).
% 2.15/1.66 tff(92,plain,(
% 2.15/1.66 ![X: $i] : product(X, multiplicative_identity, X)),
% 2.15/1.66 inference(skolemize,[status(sab)],[91])).
% 2.15/1.66 tff(93,plain,
% 2.15/1.66 (![X: $i] : product(X, multiplicative_identity, X)),
% 2.15/1.66 inference(modus_ponens,[status(thm)],[92, 88])).
% 2.15/1.66 tff(94,plain,
% 2.15/1.66 ((~![X: $i] : product(X, multiplicative_identity, X)) | product(x, multiplicative_identity, x)),
% 2.15/1.66 inference(quant_inst,[status(thm)],[])).
% 2.15/1.66 tff(95,plain,
% 2.15/1.66 (product(x, multiplicative_identity, x)),
% 2.15/1.66 inference(unit_resolution,[status(thm)],[94, 93])).
% 2.15/1.66 tff(96,plain,
% 2.15/1.66 (^[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))))),
% 2.15/1.66 inference(bind,[status(th)],[])).
% 2.15/1.66 tff(97,plain,
% 2.15/1.66 (![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)))),
% 2.15/1.66 inference(quant_intro,[status(thm)],[96])).
% 2.15/1.66 tff(98,plain,
% 2.15/1.66 (![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)))),
% 2.15/1.66 inference(rewrite,[status(thm)],[])).
% 2.15/1.66 tff(99,plain,
% 2.15/1.66 (^[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)))))),
% 2.15/1.66 inference(bind,[status(th)],[])).
% 2.15/1.66 tff(100,plain,
% 2.15/1.66 (![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)))),
% 2.15/1.66 inference(quant_intro,[status(thm)],[99])).
% 2.15/1.66 tff(101,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/BOO002-0.ax','distributivity4')).
% 2.15/1.66 tff(102,plain,
% 2.15/1.66 (![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)))),
% 2.15/1.66 inference(modus_ponens,[status(thm)],[101, 100])).
% 2.15/1.66 tff(103,plain,
% 2.15/1.66 (![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)))),
% 2.15/1.66 inference(modus_ponens,[status(thm)],[102, 98])).
% 2.15/1.66 tff(104,plain,(
% 2.15/1.66 ![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)))),
% 2.15/1.66 inference(skolemize,[status(sab)],[103])).
% 2.15/1.66 tff(105,plain,
% 2.15/1.66 (![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)))),
% 2.15/1.66 inference(modus_ponens,[status(thm)],[104, 97])).
% 2.15/1.66 tff(106,plain,
% 2.15/1.66 (((~![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(multiplicative_identity, x, add(multiplicative_identity, x))) | (~product(x, multiplicative_identity, x)) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x)))) <=> ((~![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(multiplicative_identity, x, add(multiplicative_identity, x))) | (~product(x, multiplicative_identity, x)) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x)))),
% 2.15/1.66 inference(rewrite,[status(thm)],[])).
% 2.15/1.66 tff(107,plain,
% 2.15/1.66 (((~sum(multiplicative_identity, x, add(multiplicative_identity, x))) | (~sum(multiplicative_identity, x, add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x)) | (~product(x, multiplicative_identity, x)) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity))) <=> ((~sum(multiplicative_identity, x, add(multiplicative_identity, x))) | (~product(x, multiplicative_identity, x)) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x)))),
% 2.15/1.66 inference(rewrite,[status(thm)],[])).
% 2.15/1.66 tff(108,plain,
% 2.15/1.66 (((~![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(multiplicative_identity, x, add(multiplicative_identity, x))) | (~sum(multiplicative_identity, x, add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x)) | (~product(x, multiplicative_identity, x)) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_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(multiplicative_identity, x, add(multiplicative_identity, x))) | (~product(x, multiplicative_identity, x)) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x))))),
% 2.15/1.66 inference(monotonicity,[status(thm)],[107])).
% 2.15/1.66 tff(109,plain,
% 2.15/1.66 (((~![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(multiplicative_identity, x, add(multiplicative_identity, x))) | (~sum(multiplicative_identity, x, add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x)) | (~product(x, multiplicative_identity, x)) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_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(multiplicative_identity, x, add(multiplicative_identity, x))) | (~product(x, multiplicative_identity, x)) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x)))),
% 2.15/1.66 inference(transitivity,[status(thm)],[108, 106])).
% 2.15/1.66 tff(110,plain,
% 2.15/1.66 ((~![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(multiplicative_identity, x, add(multiplicative_identity, x))) | (~sum(multiplicative_identity, x, add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x)) | (~product(x, multiplicative_identity, x)) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)))),
% 2.15/1.66 inference(quant_inst,[status(thm)],[])).
% 2.15/1.66 tff(111,plain,
% 2.15/1.66 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~sum(Y, Z, V3)) | (~sum(V1, V2, V4)) | product(V3, X, V4) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~sum(multiplicative_identity, x, add(multiplicative_identity, x))) | (~product(x, multiplicative_identity, x)) | (~product(multiplicative_identity, multiplicative_identity, multiplicative_identity)) | product(add(multiplicative_identity, x), multiplicative_identity, add(multiplicative_identity, x))),
% 2.15/1.66 inference(modus_ponens,[status(thm)],[110, 109])).
% 2.15/1.66 tff(112,plain,
% 2.15/1.66 ($false),
% 2.15/1.66 inference(unit_resolution,[status(thm)],[111, 105, 22, 95, 13, 86])).
% 2.15/1.66 % SZS output end Proof
%------------------------------------------------------------------------------